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1. Introduction to Automata Theory,
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2. Introduction to the Theory of
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3. How to Prove It : A Structured
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4. The Art and Craft of Problem Solving
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5. Symbolic Logic (5th Edition)
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6. Gödel, Escher, Bach: An Eternal
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7. An Introduction to Formal Languages
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8. The Logic Book with Student Solutions
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9. Fuzzy Sets and Fuzzy Logic: Theory
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10. An Introduction to Probability
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11. An Introduction to Kolmogorov
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12. How to Solve It : A New Aspect
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13. An Introduction to Mathematical
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14. Greek, Indian and Arabic Logic
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15. Elements of the Theory of Computation
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16. Foundations and Fundamental Concepts
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17. Attacking Faulty Reasoning: A
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18. Languages and Machines: An Introduction
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19. Friendly Introduction to Mathematical
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20. Basic Category Theory for Computer

1. Introduction to Automata Theory, Languages, and Computation (2nd Edition)
by John E. Hopcroft, Rajeev Motwani, Jeffrey D. Ullman
list price: $108.20
our price: $108.20
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Asin: 0201441241
Catlog: Book (2000-11-14)
Publisher: Addison Wesley
Sales Rank: 22942
Average Customer Review: 3.62 out of 5 stars
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Reviews (29)

5-0 out of 5 stars Very useful book for GRE CS Subject preparation (part III)
I needed a book which would speed me up with my GRE computer science subject test (part III: Theory). Having non US and non english language based Bachelor Degrees in CS and Math, I needed something to both learn the more precise terminology and at the same time to gather my prevous knoweldge of the subject. After little bit of browsing and examining of reviews, book contents and browsing pages in bookstores, I decided to buy this one.

I admit that I had a solid knowledge of almost all chapters of the book and that the book might be hard to swallow for someone who is not a little bit familiar in mathematical logic and elementary math but otherwise, the book is excellent. Even authors admit that previous editions were more demanding and in this one they introduced many easier examples and appropriate pictures and diagrams so I really did not have any problems understanding every concept.

After each chapter exercises are given and while they are useful, I would prefer solutions embedded into the book (as in Knuth's Art of Programming). Rather that doing that, authors put solutions (to selected exercises) on their web page which is not bad but the book would be more complete (and probably more expensive) with solutions inside. I would pay $20 more for that version though...

I also must say that I really appreciate hard cover and excellent quality paper (these unfortunately raised the price)

Overall, this is an excellent book and if you are in a similar situation as me, I would recommend this one.

4-0 out of 5 stars A very good book for theory of computing
Hopcroft's book is a very good introduction to the theory of computing, from finite automata to undecidability. He introduces the text with a crash course in proofs, which is useful for a text of this nature. They have several examples with illustrations to facilitate quicker learning of deterministic finite automata, pushdown automata, and Turing machines. These illustrations proved very helpful for me, a visual learner. The book itself is chock full of examples and theorems with proofs. Problems with the book: more explanation on Homomorphisms would be nice. The exercises can get very much harder than the simple material the book teaches, so running through them takes considerable amount of time often. Overall it's a good book, and a lot easier to understand than their first edition in 1979. The material can at times seem a bit outdated since the computing world has changed by several orders of magnitude since their original work, but it still provides a solid foundation in the philosophy and mathematics of computing. Perhaps if you're a Cornell student you'll get the privilege of taking this theory class with Hopcroft as your instructor; he's very nice and willing to help students understand the material.

1-0 out of 5 stars first edition is a classic, the second one unremarkable
The first edition is one of the best book in its field. A classic. A reference for many advanced courses in computer theory.

Sadly, the second edition misses a great deal of the first edition. Many chapters were removed. Important lemmas and theorems are missing.

I would gladly exchange my second edition for the first one, if it wasn't out of print.

J.

4-0 out of 5 stars Excellent introductory text, but has several weaknesses
This was my textbook for an introductory course on Finite Automata and Languages - I enjoyed it a lot and I think that the chapters until the Turing Machines are covered very well, along with good examples. As one previous reviewer has already mentioned, the exercises can get very hard as compared to what's actually presented - this I found not too good.

The topics of complexity classes and NP-Completeness, as well as the chapter on Turing Machines are rather succint and do not cover the full depth. Papadimitriou's "Computational Complexity" does a better job in this respect, even though it is not at all flawless. Some might say that there is a reason why this book is introductory, but I argue that instead of doing a poor job, the authors should have maybe just made another book dealing with the above-mentioned topics.

PS: My professor told me that the first edition was much better - maybe you could find it somewhere in the library, if interested.

2-0 out of 5 stars Could be better
As a student using this book, I simply found it a little too difficult at times to grasp what the concepts were. The examples, at times were just too complicated, and could have been done better with easy to understand examples. Not so sure about this one. However, if you are already tamed in automata theory concepts, I'm sure you'll love it. ... Read more


2. Introduction to the Theory of Computation
by Michael Sipser
list price: $103.95
our price: $103.95
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Asin: 053494728X
Catlog: Book (1996-12-13)
Publisher: Course Technology
Sales Rank: 36260
Average Customer Review: 4.69 out of 5 stars
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Amazon.com

"Intended as an upper-level undergraduate or introductory graduate text in computer science theory," this book lucidly covers the key concepts and theorems of the theory of computation. The presentation is remarkably clear; for example, the "proof idea," which offers the reader an intuitive feel for how the proof was constructed, accompanies many of the theorems and a proof. Introduction to the Theory of Computation covers the usual topics for this type of text plus it features a solid section on complexity theory--including an entire chapter on space complexity. The final chapter introduces more advanced topics, such as the discussion of complexity classes associated with probabilistic algorithms. ... Read more

Reviews (35)

5-0 out of 5 stars Life saver
I used this book as a supplement in a college class. It was VERY helpful in understanding computability theory, Turing Machines, and finite languages. Everything is put forth in a straightforward easy to understand manner.

The main thing that made this book stand out above the rest is that it's written in language that is easily understood, while other text books burden you down with a multitude of symbols and equations. The proof ideas are very helpfull in understanding concepts

Thank you Mr. Sipser!

5-0 out of 5 stars BEST Computer Theory book
This book is by far the best book that I read!!! It presents topics in a very interesting and readable way.

My advice is read this book if you an undergrad student, even though instructor might be using a different book. If you are a grad student this books makes an excellent reference for refreshing your knowledge of Computer Theory. Computer Theory is not my area of interest, but this book makes it very interesting and fun area; which is quiet unusual for Computer Theory books.

I am a grad student taking advanced "Computer Theory" class. I have bought couple books including this one, and checked out from library another 6. This book in an introductory book and it has excellent coverage of the basics, and it has some brief but very good coverage of advanced topics as well. I read this book every time to refresh my knowledge before I go on to more in depth topics. The only thing that I wish, is that the undergrad course that I have taken a number years ago was using this book; and/or I read this book when I was an undergrad.

5-0 out of 5 stars Probably the best computation theory text for students
In my opinion this is one of the best written books in the CS discipline, a must have for every computer scientist. The topics are presented clearly, with emphasis in understanding the concept, which most of the times is missed in other books amongst the equation line up of theorems that nobody will further investigate. Probably not comprehensive enough for a researcher of the field, but definately the right text to start on the subject and comprehend the basics, which is more than most students in the CS field will need.

5-0 out of 5 stars A Near Perfect Computer Theory Textbook
This book is suitable for beginners and graduate students who want to explor the theory of computation . It explains the hard theory and logic by easy sentences and words. Even if you use English as foreign language , you can read this book by yourself and understand its contents easily. This book is near perfect.

5-0 out of 5 stars An EXCELLENT Automata/Theory of Computation book
This book is one of the best written books on Automata/Theory of Computation that I have ever seen. It is a great introduction to the subject. It's also a great way to review the key topics.

One of the greatest things about this book is its focus on developing an intuitive understanding of the concepts and proofs. Other books do a better job of formal proofs but this book is light years ahead of any other in terms of helping you develop an intuitive understanding of why a given proof or construction is correct. It's a lot better than the memorize/regurgitate model necessitated by the emphasis on minutiae of other books.

Lastly, this book provides great tips on how to approach problem solving (especially proofs). ... Read more


3. How to Prove It : A Structured Approach
by Daniel J. Velleman
list price: $28.99
our price: $28.99
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Asin: 0521446635
Catlog: Book (1994-11-25)
Publisher: Cambridge University Press
Sales Rank: 193781
Average Customer Review: 4.75 out of 5 stars
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Book Description

Many mathematics students have trouble the first time they take acourse, such as linear algebra, abstract algebra, introductory analysis, or discrete mathematics, in which they are asked to prove various theorems.This textbook will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs.The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted.These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs.The author shows how complex proofs are built up from these smaller steps, using detailed "scratchwork" sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets.Numerous exercises give students the opportunity to construct their own proofs.No background beyond standard high school mathematics is assumed.This book will be useful to anyone interested in logic and proofs:computer scientists, philosophers, linguists, and of course mathematicians. ... Read more

Reviews (16)

3-0 out of 5 stars A good start on writing proofs, but falls short!
I found that this book utilized a little too much set theory for beginning students. If the author could have given more concrete examples, perhaps from group theory or simpler ones from analysis or number theory, it would have been much better. For students wanting a more lucid exposition of proof techniques, I highly recommend, "100% Mathematical Proof" by Rowan Garnier and someone else,whos name escapes me at the moment. "100% Mathematical Proof" is far superior to this book, and it has answers to the exercises which is crucial to the beginning student learning on his/her own. Velleman needs to bring the abstract nearer to the concrete for the beginning student.

5-0 out of 5 stars An excellent book
This is an excellent book for the early undergraduate student. It is actually two books in one. The first half is a careful review of Logic and the essentials of Set Theory with an emphasis on precise language. Thereafter a structured development of proof techniques is clearly presented using these tools. The second half of the book is a detailed presentation of introductory material about functions, relations, and a few aspects of more advanced set theory. These chapters serve as a wonderful introduction and show applications of the proof techniques developed earlier.
I have referred back to this book often in my own study of analysis and number theory. I recommend it highly. It will be very useful to any undergraduate proceeding through a mathematics curriculum. I recommend studying it early in the first semester, and re-reading it as time goes on.

4-0 out of 5 stars Starts off good, and then goes off on a tangent.
I bought this book in the hopes that it would help me improve my proof writing skills. Being only a high school graduate (a month ago), I had practically no knowledge of set theory. The initial proof structures were great, and I enjoyed following the proofs from the premises and, through logical steps, to the desired conclusion. However, then the Set Theory came in. I can understand why a certain amount of set theory was necessary in order to be able to talk about certain types of proofs, but he goes so far into set theory in the book, that by a certain point, instead of following the logical flow of the proofs, I was trying to remember abstruse terminology he had mentioned briefly and trying, successfully for the most part, to understand what the actual proof meant, and why it would make sense that it was correct. Its possible that the reason I feel this way is because when I do proofs, I usually need to understand it intuitively first and then go from there, and it could be the case that this isn't possible with more abstract proofs. Overall, it was a good read, but unfortunately, he went a little too far into the set theory than was necessary. Reading it twice would fix that problem though. Another criticism is that there are no solutions to the exercises.

5-0 out of 5 stars Breakthrough and Original ......
I recall it was a few years back when I encountered this little gem at my first analysis class. In fact this book wasn't assigned and instead we used Analysis by Lay. I didn't get essential proof tactics/strategies out of Lay's so I plunged myself into Library and after looking up one after another, I finally found this book. It is about as title says and not about Analysis. The book does not cover as much as one expects from Analysis books. But many of them I've seen seem to fail on teaching "how to prove" to study Analysis.

Velleman uses structured style as a technique. Two columns are prepared. The left column is Givens and right Goal. By restructuring Givens and Goal using relationships and definitions, some parts of Goal statement is moved to Givens, like peeling skins of onion. This process iterates until one finds the proving obvious. The whole process is a "scratch work" and a reader is able to see how the author structures the proof step by step, both from Goal and Givens viewpoints.

In past, there was only a Macintosh proofing program, but now Java version called Proof Designer is out. So Windows and Linux users alike can now enjoy this little program in conjunction with the book. Two disappointments with Proof Designer are that the output is only in the form of a traditional proof style which does not expose "the scratch work" and that the program does not use the two column style used in the book.

There are additional materials such as supplementary exercises, documentation, and a list of proof strategies (which is also available at the end of the book as a good reminder and reference), all available from author's site for free. [search in google like this: velleman "how to prove it" inurl:amherst]

After completion of this book, don't throw it away! Advance to Rudin's Principles of Mathematical Analysis and keep Velleman aside. Now one can work on complete proof of materials in Rudin with rigor and study how he constructs logical structures step by step in your own "structured" words!

4-0 out of 5 stars Probably the best book out there but not perfect
A good basic introduction to understanding math proofs by understanding logic first. Only lacking in its connection to math proofs that one might actually see, in other words too basic (which is as much complement as a critcism.) ... Read more


4. The Art and Craft of Problem Solving
by PaulZeitz
list price: $65.95
our price: $65.95
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Asin: 0471135712
Catlog: Book (1999-02-09)
Publisher: Wiley
Sales Rank: 38955
Average Customer Review: 4.5 out of 5 stars
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Book Description

This text on mathematical problem solving provides a comprehensive outline of "problemsolving-ology," concentrating on strategy and tactics.It discusses a number of standard mathematical subjects such as combinatorics and calculus from a problem solver's perspective. ... Read more

Reviews (14)

5-0 out of 5 stars General Problem Solving Strategies.
Perfect match for all math problem solvers.
Wonderful Book with around 660 problems.
Level National Math competition, IMO, Putnam.
If I have to pick the best two problem solving books so far publish in the English Language
Problem-Solving Strategies (Problem Books in Mathematics) by Arthur Engel and this Book by Paul Zeitz are the clear winners.

This particular book has very clear explanations of the main problem solving strategies illustrated with carefully sample problems. Reading this book brings to my memory the works of Polya. One of the only things I think the book is lacking is on strategies to solve Geometry problems in particular or to use the same strategies in the book to solve more Geometrically flavor problems. Nevertheless is a Joy to read.
Please Paul keep writing this beautiful problem solving books.

5-0 out of 5 stars One of the best
This book is indeed one of the best problem-solving textbook so far. As a frequent lecturer of Taiwan IMO team, I have many many MO books. Most of the books available are well-written by professionals and excellent mathematicians. However, since IMO does really prevail in recent years, these authors could not be the participants themselves (^^). Furthermore, usually these books (except those are merely problems collections) contains a good proportion of "harder" and beautiful problems, and the easier and basic training problems are relatively few. It often get the beginners frustrate.

Now this maybe is the first book written by a member of former MO team, and now a training lecturer. (The author himself won the USAMO and IMO in 1974, and helped train several USA IMO teams, including the 1994 "perfect score team"). So here is the precious experience! Besides, the ratio between the harder problems and the easier problems is really good. In my opinion this is an excellent textbook for ambitious beginners (both teachers and students), for self-studys and problem-solving fans. Highly recommended.

5-0 out of 5 stars Essential for budding (and experienced) problem-solvers
I join the ranks of previous reviewers here who honestly feel that having read this book in high school would have almost certainly changed my life. I, too, did very well in high school math competitions, but the maturity I am gleaning from this gem may have vaulted me into a different league.

It contains hundreds of problems from various levels of competition, from AIME problems all the way through some of the toughest Putnam problems (which, if you know anything about the Putnam, are about as hard as competition problems come). But the biggest help are the vital insights and exciting ways of looking at these problems. Don't take my word for it-- many past IMO contestants have suggested this book too.

You don't have to be a math competition buff to gain from this book, however. If you're simply interested in mathematical puzzles and problems, and looking to expand your repertoire, this book will help you. Anyone with a good dose of intelligence and motivation will benefit.

For an additional problem book, check out Mathematical Olympiad Challenges by Andreescu and Gelca. For purely Putnam treatment, there are several volumes written by Kedlaya. And if you're a CS student, looking for honing those CS math skills to be razor sharp, you should definitely look into Concrete Mathematics by Graham, Knuth, and Patashnik.

Happy solving.

5-0 out of 5 stars The Book I wish I had in High School
When I was in high school, I placed second in the Alabama State Mathematics Contest and won many others. However, I might could have been competitive with the IMO style problems had I had this book and would be much better off today had I seen this book earlier.

This book is for the exceptionally brilliant and the mentally tough. It is absolutely necessary to approach this book in a different way from a standard math textbook. You MUST attempt the examples BEFORE looking at the example solutions, NO MATTER HOW DIFFICULT OR FRUSTRATING. You may be bamboozled by the problems, but even trying to understand the problems before looking at the solutions and thinking about how a solution might proceed will pay huge dividends in the long run.

For example, in the first chapter Zeitz presents an example asking the reader to prove that the product of four consecutive integers cannot be a perfect square. The solution involves some clever algebraic trickery not visible to the inexperienced, but persistence and getting your hands dirty is key.

If you persist in spite of the considerable difficulty, you will find that you get better very, very quickly. You will also notice that it isn't just contest problems it helps you solve. I have found that I have solved my homework sets in the Berkeley graduate engineering program much more easily since working these problems. You will start to see creative and clever solutions where they exist in everything problem oriented.

PATIENCE PATIENCE PATIENCE!

5-0 out of 5 stars Brilliant
As a high school student that is essentially bored with the regular, ho-hum classes that my school offers, this book is perfect. It gives a problem-solving foundation for math enthusiasts desiring to compete nationally in contests like the AMC, AIME, and USAMO. The problems are excellent and cover a wide range of difficulty (past ASHMEs, USAMOs, and, finally, IMOs); and the solutions are well-written, logical, and intelligible. In short, if you are looking to "get better" at problem solving, this is the book for you.

Note: I also bought Problem-Solving Strategies by Arthur Engle. Those, perhaps more advanced, problem-solvers that want even more of a challenge should purchase this book as well (as both books give very challenging problems, but Engel's is undoubtedly more advanced). ... Read more


5. Symbolic Logic (5th Edition)
by Irving M. Copi
list price: $95.33
our price: $95.33
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Asin: 0023249803
Catlog: Book (1979-04-01)
Publisher: Prentice Hall
Sales Rank: 435910
Average Customer Review: 4.75 out of 5 stars
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Reviews (4)

5-0 out of 5 stars Great, But Outdated
I thourougly enjoyed learning from this book, and it became the foundation of my analytic philosophy knowledge.

That said, I do not recommend this book as a text for those attempting to learn logic today. The symbolic language that is used and the mode of problem-solving demonstrated by Copi in this work is long since outdated and using this text will only confuse a logic amateur when they move on to more current and complicated logic.

4-0 out of 5 stars a classic textbook on logic
This was my first textbook on logic and it has a warm place in my heart. It is not very current or modern, and it's probably not appropriate for teaching logic in the math or computer science departments, but otherwise, it's a lovely book. The two nicest features of this book are the wealth of interesting exercises and the emphasis on language: The correspondence between sentences in English and propositions in logic.

Re the logical structure of English sentences, I would like to note that I used many of the exercises from this book in a logic class I taught a few years ago, and was stunned to see the difficulties students were having: Difficulties in comprehending the logical structure of a sentence in English and then expressing this structure using Boolean connectives and quantifiers. I found this discovery both alarming and curious.

5-0 out of 5 stars This is "the" book to use.
This was the book I used when I took symbolic logic in college. Very self explanitory - the book can be used to learn symbolic logic on your own. Why? Simply because it's fun!

5-0 out of 5 stars Excellent text dealing with 2nd order predicate calculus.
My Background: Graduate Computer Science student, emphasis in complex programming.

Most programmers never get beyond the first-order (unquantified) predicate calculus introduced in the standard finite math course. This text goes to the next level in formal logic, teaching how to prove or disprove that a quantified expression follows logically from a group of premises.

Copi's notation is concise, leads to elegant proofs, and to proofs which are much shorter than many of the tree methods.

Even if you don't feel that you have the stamina to take on quantified logic, the book is an excellent text to unquantified rules of inference. But the real wealth here is the treatment of UI, UG, EI, and EG. To become fluent with this notation requires diligently working the host of example problems in each chapter, but the result will be problem-solving abilities that are much more flexible than the abilities of mathematics alone. You may find yourself becoming addicted to formal logic! Steve ... Read more


6. Gödel, Escher, Bach: An Eternal Golden Braid
by Douglas R. Hofstadter
list price: $22.00
our price: $14.96
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Asin: 0465026567
Catlog: Book (1999-01-01)
Publisher: Basic Books
Sales Rank: 2752
Average Customer Review: 4.5 out of 5 stars
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Amazon.com

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 (197)

4-0 out of 5 stars Multi-faceted Thesis
Ancient runic languages scrawled onto South Pacific stones. Gödel's Incompleteness Theorum. Shifted perspectives in artistic pencil pictures. Modern artificial intelligence research. Masterpieces of Baroque harmony.
It's not often that bestselling books manage to link all of the above items in a highly satisfying blend of fact and philosophy, but Gödel, Escher, Bach: An Eternal Golden Braid defies both convention and classification.
The book is such a sprawling, wide-ranging argument that it's difficult to know where to start. Personally, I most enjoyed the chapters on the location of meaning within symbols; Hofstadter's description of the essential elements of a message's structure caught my interest because it seemed applicable in many fields: literature, cryptography, and psychology, to start. I was also quite intrigued by his exploration of the brain's mode of operation: sense impressions stored as complex 'symbols.' Fascinating. The long sections on mathematics and the often goofy dialogue chapters were trying, yes, but persevere; better parts lie in store.
Hofstadter's case is best made when he follow a topic through many disciplines. Though I ultimately disagree with his position on the feasibility of artificial intelligence, he has produced a stimulating read, and I am thankful for it. It is far superior to my other late-night literary conquest of the summer (Harry Potter and the Order of the Phoenix) and I recommend it to anyone with pondering time to spare.
Oh, and as a side note: don't buy Yudkowsky's review. Nothing personal, but this isn't the only thinking man's book out there. It just investigates so many nooks and crannies that almost anyone can find something to further pursue.

5-0 out of 5 stars Essential reading
GEB is a great jumping point into issues of the philosophy of the mind, the underpinnings of mathemetical logic and the possibility of artificial intelligence and consciousness.

The book explores a number of themes - one of the most important is joining together disparate forms of 'strange loops' - paradoxical self referential constructs that pop up in in art (Escher and Bach fugues), mathematics (Godel's theorem), religion (Zen buddhism), AI and various other places.

I agree with another reviewer - everything in GEB leads towards an understanding the mind (Hofstadter's field is of course AI/cog science) - it's not just a random romp - but it's a misleading exagerration to say GEB is trying to provide a bottom-up theory!

It is true, some of the foundations of AI such as propositional logic are explored and various metaphors for the mind are developed as well as the importance of circular self-referentiality, and emergence of complex behavior from simple primitives - but the implications for AI and cognitive science are always rather vague and the HOW is mostly left as an open-ended question.

This open-endedness perhaps contributes to the rambling feeling of the book. Of course these questions are great mysteries and it's not surprising that GEB doesn't provide a neat theory to tie it all together.

At it's size it is a rather daunting book to read in one go, but since a lot of chapters are rather independent it is possible to dip into it from time to time, i find myself picking it up occasionally and re-reading random chapters, usually i notice something new to ponder on.

For me the most unique contribution of this book is the pointing out the importance of 'strange loops' in so many areas of thought (although they're never formally defined). I found myself constantly linking this idea to other things - for example Jacques Derrida's notion of deconstruction seems to me most easily understood as about creating a linguistic strange-loop to point out the limitations of language and philosophy itself.

I don't think the book has really dated much at all the central ideas are timeless and AI and cognitive science haven't advanced to a point that invalidates anything, although Fermat's theorem has now been solved.

5-0 out of 5 stars A wonderful read for all aspiring thinkers
The Atlanta Journal Constitution describes Gödel, Escher, Bach (GEB) as "A huge, sprawling literary marvel, a philosophy book, disguised as a book of entertainment, disguised as a book of instruction." That is the best one line description of this book that anybody could give. GEB is without a doubt the most interesting mathematical book that I have ever read, quickly making its place into the Top 5 books I have ever read.
The introduction of the book, "Introduction: A Musico-Logical Offering" begins by quickly discussing the three main participants in the book, Gödel, Escher, and Bach. Gödel was a mathematician who founded Gödel's Incompleteness Theorem, which states, as Hofstadter paraphrases, "All consistent axiomatic formulations of number theory include undecidable propositions." This is what Hofstadter calls the pearl. This is one example of one of the recurring themes in GEB, strange loops.
Strange loops occur when you move up or down in a hierarchical manner and eventually end up exactly where you started. The first example of a strange loop comes from Bach's Endlessly rising canon. This is a musical piece that continues to rise in key, modulating through the entire chromatic scale, ending at the same key with which he began. To emphasize the loop Bach wrote in the margin, "As the modulation rises, so may the King's Glory."
The third loop in the introduction comes from an artist, Escher. Escher is famous for his paintings of paradoxes. A good example is his Waterfall; Hofstadter gives many examples of Escher's work, which truly exemplify the strange loop phenomenon.
One feature of GEB, which I was particularly fond of, is the 'little stories' in between each chapter of the book. These stories which star Achilles and the Tortoise of Lewis Carroll fame, are illustrations of the points which Hofstadter brings out in the chapters. They also serve as a guidepost to the careful reader who finds clues buried inside of these sections. Hofstadter introduces these stories by reproducing "What the Tortoise Said to Achilles" by Lewis Carroll. This illustrates Zeno's paradox, another example of a strange loop.
In GEB Hofstadter comments on the trouble author's have with people skipping to the end of the book and reading the ending. He suggests that a solution to this would be to print a series of blank pages at the end, but then the reader would turn through the blank pages and find the last one with text on it. So he says to print gibberish throughout those blank pages, again a human would be smart enough to find the end of the gibberish and read there. He finally suggests that authors need to write many pages more of text than the book requires just fooling the reader into having to read the entire book. Perhaps Hofstadter employs this technique.
GEB is in itself a strange loop. It talks about the interconnectedness of things always getting more and more in depth about the topic at hand. However you are frequently brought back to the same point, similarly to Escher's paintings, Bach's rising canon, and Gödel's Incompleteness theorem. A book, which is filled with puzzles and riddles for the reader to find and answer, GEB, is a magnificently captivating book.

5-0 out of 5 stars A readable Mobius strip
If you have never read this book, then I'd like to say that it has a lot of the most greatest knowledge out there. It doesn't just deal with math, art, and music, but also with zen, philosophy, self-ref, self-rep, holism, reductionism, and everything else that is considered pure knowledge of cognitive science and general intelligence. I don't know why some of the people rating it have no idea of what's it about; it's not about Godel's theorem like many think it is, it's about consciousness and how the power of the mind and the "I" comes out of the inanimate matter that creates us. That's not it, the second part of the book talks about computer programming and AI. Can a computer program ever have a sense of self or compose meaningful music? Hofstadter's response to the second one was: "Only if that AI could go through the maze of life on it's own, fighting it's way through it and feeling the cold of a chilly night, the longing for a cherished hand, the inaccessibility of a distant town, the regenaration after a human death, the...and only then can it be considered to do so."
This book really has more than that. I can't say all of the things mentioned in it, not in this tiny little review, but I can say that you should probably read it and hopefully understand it because it truly is a masterpiece.

1-0 out of 5 stars Pseudo-science at best
I quite agree with the reviewer from East Hartford. Maybe I am not extremely eligible to comment on the portions dealing with Escher and Bach, respectively (I have no appetite for Escher. I like chamber music of Bach and somtimes play his keyboard music but my performance level is, of course, that of amateur.)
But I must say the part dealing with Gödel's Theorem of Incompleteness is *complete garbage*. I am convinced anyone with a degree of mathematics will agree with me: for those who have no background in mathematics, I assure you that Gödel's theorem concerns a problem in "formal logic" and has nothing to do with human-cogno-something.
If this book were meant to be a cult literature, that would be okay: I don't care anyway.
But if this is meant to be an entertainment for people with no scientific background, I rate this alchemy or pseudo-science at best. ... Read more


7. An Introduction to Formal Languages and Automata
by Peter Linz
list price: $93.95
our price: $93.95
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Asin: 0763714224
Catlog: Book (2000-10-01)
Publisher: Jones & Bartlett Publishers
Sales Rank: 128009
Average Customer Review: 2.48 out of 5 stars
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Book Description

This text covers all the material essential to an introductorytheory of computation course for undergraduate students. The text has a solidmathematical base, and provides precise mathematical statements of theorems anddefinitions, giving an intuitive motivation for constructions and proofs. Proofsand arguments are clearly stated, without excessive mathematical detail, to helpstudents understand the basic principles. The text is illustrated withintegrated examples of new concepts as well as an abundance of exercises to aidin the development of problem solving skills. ... Read more

Reviews (25)

2-0 out of 5 stars Not an "Introduction"
This book contains all of the essential information and the author does have a fairly good writing style. However, where it fails is in the title: "An Introduction to ..." If it was intended to be an introduction to automata and formal language it should have a lot more clear explanation and none of the proofs. This book is filled with proofs that no one but a mathematics major could possibly understand. And the proofs aren't even relevant to the fundamental understanding of the concepts.

It is clear that the author isn't writing this book as a tutorial for students, but as a reference for professors. It seems like he is trying to show off his proof-writing ability throughout the book and has no concern for the poor student who might be struggling to understand new concepts. That's why I give it a low rating.

I was lucky to have an excellent instructor for this class so that after her lectures I could read this book and understand most of it. But, it would be almost impossible to learn anything from this book by self study only. So my advice, like others who have posted is: Unless this book is required for a class, don't buy it!

1-0 out of 5 stars Simply godawful
I had to purchase this for my school's Intro to CS Theory course.

Linz' utter ineptitude towards writing is what gives this book 1 star. Examples throughout chapters are sparse and relatively worthless. Sample problems at the end of the chapter, in contrast, are ridiculously difficult, and the solutions in the back don't offer any explanation whatsoever towards the answers.

This is the only book I have ever read that actually made me feel dumber for reading it. It's simply demeaning. Rather than explaining or justifying his logic, as he should to the target audience of this book, he simply uses "it's obvious that..." repeatedly for sample problems and solutions. A ridiculously complex problem's solution in the back of the book will be whittled down to two lines at best, half of which says something along the line of "It's blatantly obvious that the answer is ___, and you're stupid for not realizing it."

If you're actually assigned graded work from this book, may god have mercy on your soul.

3-0 out of 5 stars Boring subject
This subject is confusing in general, I have this professor and he's really confusing, but when I read his own book it's actually better that him.

2-0 out of 5 stars Not a good book
First off, let me say this book did not confuse me. It's just very poorly written. If this was the only Automata book I ever read, my review would not mean as much. On the contrary, I have read 4-5 Automata books and have taught the topic numerous times. I urge teachers and students to avoid this book. If you would like a great book covering this material, get Dexter Kozen's Automata and Computability. That book is so well written and elegant that it puts most of the other books to shame. It is one of the top CS books on my list along with SICP, CLRS, and a few others. In addition, the "OLD" version of Hopcroft and Ullman is pretty good but the newer version with Motwani is bad. Anyways, I hope this helps some of you who are looking for a good book to read.

5-0 out of 5 stars for the brainy one
Lot of review didn't like this book simply because it was "confusing", well this subject is inherently confused. I had to use this book for one of my course, and to my opinion is it a good book, it does a good job of explaining the concept, providing enough proof but not to the point that it bogged down the reader. If you put some effort in and actually think about all the concept/algorithm, you will like it much better, to those that couldn't understand this book, get a new major, i don't think your brain is fit for CS or any engineering major, may i suggest liberal art? or some type of social sci? ... Read more


8. The Logic Book with Student Solutions Manual
by MerrieBergmann, JamesMoor, JackNelson
list price: $80.00
our price: $80.00
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Asin: 0079130836
Catlog: Book (1997-03-01)
Publisher: McGraw-Hill Humanities/Social Sciences/Languages
Sales Rank: 211344
Average Customer Review: 3.38 out of 5 stars
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Book Description

This outstanding book is a leading text for symbolic or formal logic courses All techniques and concepts are presented with clear, comprehensive explanations and numerous, carefully constructed examples.Its flexible organization (all chapters are complete and self-contained) allows instructors the freedom to cover the topics they want in the order they choose.

The third edition incorporates many new and updated exercises and expanded discussions on evaluating arguments and symbolization in predicate logic. A free Student Solutions Manual is packaged with every copy of the textbook. Two logic programs, Bertie III and Twootie, are available as a free download from the University of Connecticut Philosophy Department’s Web site. The Web address for downloading the software is http://www.ucc.uconn.edu/~wwwphil/software.html. Bertie 3 is a proof checker for the natural deduction method and Twootie is a proof checker for the truth tree method. ... Read more

Reviews (8)

4-0 out of 5 stars not concise, but still a good learning aide
This book might be a good addition to the library of a self-studier who has an ample amount of time or perhaps has little experience in logic or mathematics. Its explicitness really helps on the more difficult chapters, although this can frustrate, too, because it's hard to study when the rules aren't laid out in one particular spot. But overall, it's not so bad.

4-0 out of 5 stars Was Moriss Thinking?
This book has remained in my library (thumbed to pieces, which is why I now seek a new copy) since I took symbolic logic in the summer of 1995 on the way to my undergraduate degree at Arizona State University. I have to say that out of the many logic texts I have seen, this is by far one of the finest around. I won't belabor any of the excellent points made by the gentleman from Lubbock, Texas, but I will add that this is an immaculate reference for any student of logic, though he or she should have the benefit of being able to seek the guidance of a seasoned professional should difficulties arise.

Contrary to my compatriot from Lubbock,though, I find the time spent on truth trees to be entirely beneficial. While the derivations are definitely the grandest exercise in formal logic, their focus on the validity or invalidity of an argument leaves them to fail to show the one thing that truth trees reveal, that is, the truth conditions of any given proposition.

Please do not let the naysayers detract you from using this book as a text if you are undertaking the task of teaching logic. It is rather common that symbolic logic is the ruination of many a philosophy major's GPA. For me, however, I found this book extremely easy to follow and comprehend. The ample exercises, most with answers and explanations provided, only add to the worth of this classic text. Few logic books can compare with this fine instrument.

1-0 out of 5 stars One of the worst written books I came across
For Goodness sakes!!! What were Bergmann, Moor, and Nelson thinking of when they wrote this book? Morrisss you hit the nail on the head. As a philosophy student, I also have spent long hours study this book, and I don't know anyone in class who understands this unpenetrable, reader-enemy, harder to decipher than Egyptian scrolls, book. Good grief!!! I prefer to read Chinese arithmetic upside down than to decipher this book. Save your money.

4-0 out of 5 stars Not a Classic for No Reason
It is unfortunate that Mr. Morriss had such difficulty with his logic course, especially since he is both a philosophy student and has been able to maintain a respectable GPA. As any student of philosophy knows, however, any one of a number of things could have brought about the 'D' on the midterm for the unfortunate fifty percent of his fellow students.

My experience with this text and (perhaps, therefore) with my undergraduate logic class in general was considerably more enjoyable than Mr. Morriss's experience. Certainly my professor was very good at presenting the material, but presumably he was not making up for a poorly written book. For, when I needed to consult the text, I found it actually to be quite clear and helpful.

Turning to the book itself, since I am not a fan of truth trees, when I have the opportunity to teach my own logic course, I will utilize proofs. I have viewed many undergraduate logic texts, and the proof method presented in this text seems the clearest.

Furthermore, as my logic professor told me and his professor told him: Logic is not in the head; it's in the fingers. Therefore, I think that the considerable amount of exercises contained in this text is greatly beneficial.

Third, definitions of terms and important points are presented clearly in text boxes throughout the book. Students will benefit greatly by committing these terms and points to memory.

Finally, although the price is prohibitive (hence, four stars), Bergmann, Moor, and Nelson's text has withstood the test of time. (In fact, the 4th edition is supposed to be out this summer.) Such evidence is not conclusive, but it suggests that perhaps lying behind Mr. Morriss's frustration is a cause more complex than merely this classic text.

4-0 out of 5 stars Good text, but of course with some flaws
Although I concede the point of another reviewer, that perhaps the book is a bit dense and inaccessible depending on the background of the student, I have found this text to be an excellent choice. The coverage is thorough and examples and exercises are hardly lacking. In fact, some easier points are explained in what is in my opinion too much detail. My only serious complaint is that there is no glossary, and with so many definitions this is a definite disadvantage. Despite this, The Logic Book remains an adequate an INTERESTING introduction to formal logic. ... Read more


9. Fuzzy Sets and Fuzzy Logic: Theory and Applications
by George J. Klir, Bo Yuan
list price: $93.33
our price: $93.33
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Asin: 0131011715
Catlog: Book (1995-05-11)
Publisher: Prentice Hall PTR
Sales Rank: 563139
Average Customer Review: 4.67 out of 5 stars
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Book Description

Reflecting the tremendous advances that havetaken place in the study of fuzzy set theory and fuzzy logic from1988 to the present, this book not only details the theoretical advancesin these areas, but considers a broad variety of applications of fuzzysets and fuzzy logic as well.Theoretical aspectsof fuzzy set theory and fuzzy logic are covered in Part I of the text,including: basic types of fuzzy sets; connections between fuzzy setsand crisp sets; the various aggregation operations of fuzzy sets;fuzzy numbers and arithmetic operations on fuzzy numbers; fuzzy relationsand the study of fuzzy relation equations.Part II is devoted toapplications of fuzzy set theory and fuzzy logic, including: variousmethods for constructing membership functions of fuzzy sets; the useof fuzzy logic for approximate reasoning in expert systems; fuzzysystems and controllers; fuzzy databases; fuzzy decision making; andengineering applications. For everyone interested inan introduction to fuzzy set theory and fuzzy logic. ... Read more

Reviews (3)

4-0 out of 5 stars First bible of fuzzy systems theory since Dubois and Prade.
A comprehensive and authoritative presentation of developments in the mathematics of fuzzy systems theory over the past thiry years. While the basic mathematics are presented, this book is not for the casual reader, but for those seriously interested in fuzzy systems theory. If the reader does not have a good mathematical background, he or she will find this book tough going. Coverage of theoretical fuzzy concepts is quite complete, including theory of fuzzy sets, fuzzy arithmetic, fuzzy relations, possiblity theory, fuzzy logic and uncertainty-based information.

The applications section presents theory which could be useful in applications rather than the applications themselves. References are given, but no distinction is made between theoretical work and real-world applications, and many of the references are old and out-of-date.

For a reference book on fuzzy mathematics, this book is superb; as a pointer to real-world applications, it leaves something to be desired.

5-0 out of 5 stars Robust treatment of fuzzy logic has interdisciplinary appeal
George and Bo have been as thorough and lucid in preparing this book as well as George explicated systems thinking in the very first book of his I read, "An Approach to General Systems Theory." Here, as there, without compromising mathematical rigor, the goal of this book is to elaborate its subject matter in such a robust manner that it has multidisciplinary appeal. As always, the reader is given a flexible, almost interactive, access to the what, why and how of fuzzy thinking. Despite the exception taken by Professor Lotfi A. Zadeh, the "founder of fuzzy logic," the percipient reader will appreciate the authors' unusual association of "fuzzy measure," that is, the degree of belief that a particular element belongs to a crisp set, (not the degree of membership in the set), with Possibility Theory so as to clarify the differences between fuzzy set theory and probability theory. The illustrative applications are not only case studies that one may pick and choose from for examination and emulation but also constitute incontrovertible evidence of the successful and promising realization of the fuzzy paradigm. As a former professor of engineering at Rutgers University, I found the 79-page Instructor's manual helpful for self- or extended study and I assume it would be valuable for teaching. I have read many books on fuzzy logic and I judge this to be the most balanced to date, (early 1998), - not filled with C++ code or trying to sell a software package nor is it theoretically daunting - it is simply an inviting demonstration of how fuzzy logic clears up foggy modeling and analysis.

5-0 out of 5 stars One of the most important book to learn about fuzzy logic
The book presents the mathematical theory of fuzzy logic including theorems and demonstrations. There are one part of applications of this logic in many distint areas like engineering, medicine, economics and others. ... Read more


10. An Introduction to Probability and Inductive Logic
by Ian Hacking
list price: $24.99
our price: $24.99
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Asin: 0521775019
Catlog: Book (2001-07-02)
Publisher: Cambridge University Press
Sales Rank: 54352
Average Customer Review: 4.67 out of 5 stars
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Book Description

This is an introductory textbook on probability and induction written by one of the world's foremost philosophers of science.The book has been designed to offer maximal accessibility to the widest range of students (not only those majoring in philosophy) and assumes no formal training in elementary symbolic logic.It offers a comprehensive course covering all basic definitions of induction and probability, and considers such topics as decision theory, Bayesianism, frequency ideas, and the philosophical problem of induction.The key features of the book are:* A lively and vigorous prose style* Lucid and systematic organization and presentation of the ideas* Many practical applications* A rich supply of exercises drawing on examples from such fields as psychology, ecology, economics, bioethics, engineering, and political science* Numerous brief historical accounts of how fundamental ideas of probability and induction developed.* A full bibliography of further readingAlthough designed primarily for courses in philosophy, the book could certainly be read and enjoyed by those in the social sciences (particularly psychology, economics, political science and sociology) or medical sciences such as epidemiology seeking a reader-friendly account of the basic ideas of probability and induction.Ian Hacking is University Professor, University of Toronto. He is Fellow of the Royal Society of Canada, Fellow of the British Academy, and Fellow of the American Academy of Arts and Sciences. he is author of many books including five previous books with Cambridge (The Logic of Statistical Inference, Why Does Language Matter to Philosophy?, The Emergence of Probability, Representing and Intervening, and The Taming of Chance). ... Read more

Reviews (3)

5-0 out of 5 stars Hacking gets everything right except for Keynes
Hacking's book is a job well done.He blends history,philosophy,logic,mathematics,statistics and science with wit and judicious scrutiny in general.Unfortunately,the book is slightly marred by inaccurate and/or incorrect statements about J. M. Keynes and/or his logical theory of probability.Describing Keynes as a"belief dogmatist"is way off the mark given Keynes's penchant for changing his mind as new and/or relevant information and analysis became available over his lifetime.Secondly,it is bizarre for Hacking to claim that Keynes had no use for frequency-type probability theories and jeered at the idea of relative frequency holding in the long run because in the long run we are all dead.(Hacking,pp.146-151).The only frequency theory Keynes ever rejected was that of John Venn.Keynes always considered frequency theories to be accurate and correct for some cases.However,they were not general in scope but limited in their applicability.The interested reader should consult chapter 8 of Keynes's A Treatise on Probability(1921).Finally, Keynes rejected the fallacy of long runism or conditional apriorism because of its unsound argument.The fact that in the long run some process may converge to a particular outcome in the limit offers no support to a do-nothing policy in the present.If the only available relevant evidence bearing on the probability of a proposition is frequency data then the logical probability is the same as the relative frequency estimate.The only caveat Keynes would add would be that the frequency data should have passed the Lexis Q Test for stability.

5-0 out of 5 stars For anyone, any thinker
I would HIGHLY recommend this book for anyone (including business men) who must make decisions with incomplete information and under uncertainty. Instead of focusing on the mechanics of statistics, it focuses on how to think about risky propositions.

I bought this book while working on a particular problem in machine learning, at a point where I had started realizing that I was losing clarity on my definition of probability. I was using the mechanics, but didn't clearly understand why the use was valid. This seemed an odd and embarrassing circumstance at the time, how could I not understand what "probability" means? As it turns out this confusion is one shared broadly in history of science, and in current applications of statistical mechanics.

Prof Hacking's writing is clear and entertaining, clearly aimed at engaging the reading audience.

4-0 out of 5 stars What do you mean, "probably"?
The best thing about this book is that it teachs basic probability theory while keeping the reader constantly aware of the on-going debate regarding what it means to talk in terms of probabilities, and of how that debate has shaped the development of probability theory. If you are a student taking a course in probability and statistics who would like to genuinely understand the conceptual basis of all those formulas they are teaching you, I suggest you read this book.

Some readers will be disappointed by this book. Since the book concentrates on the conceptual basis of probability and inductive logic, it does not give the reader enough technical tools to really do much applied mathematics. On the other hand, by the time Hacking gets around to discussing what students of philosophy will likely view as the big philosophical pay-off of probability theory (i.e. Bayesian and frequentist contributions to the problem of justifying induction) he devotes to them a mere 20 pages of not terribly deep discussion. ... Read more


11. An Introduction to Kolmogorov Complexity and Its Applications (Graduate Texts in Computer Science)
by Ming Li, P. M. B. Vitanyi
list price: $79.95
our price: $67.96
(price subject to change: see help)
Asin: 0387948686
Catlog: Book (1997-01-01)
Publisher: Springer-Verlag
Sales Rank: 441701
Average Customer Review: 5 out of 5 stars
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Book Description

Written by two experts in the field, this is the only comprehensive and unified treatment of the central ideas and their applications of Kolmogorov complexity. The book presents a thorough treatment of the subject with a wide range of illustrative applications. Such applications include the randomness of finite objects or infinite sequences, Martin-Loef tests for randomness, information theory, computational learning theory, the complexity of algorithms, and the thermodynamics of computing. In this new edition the authors have added new material on circuit theory, distributed algorithms, data compression, and other topics. ... Read more

Reviews (6)

5-0 out of 5 stars A must
The book provides all the tools needed for a productive use of the theory. Written by leading experts in the field, the book is both a fascinating introduction as well as a comprehensive reference for experts.

The authors are careful to place the development of the theory in its historical context, give a face to the main players in the field and explore frictions with other lines of thought. But the main storyline is the mathematical world of Kolmogorov complexity. Neccessary background knowledge is provided, most proofs are given and the open problems are presented. Most chapters are more or less self sufficient, making it possible to skip those that are of less relevance to you. In the later chapters much thought is given to the different fields of application.

A third edition is in the making which will include recent advances. But since the authors make new discoveries available on the web, the present edition will continue for a long time to hold a prominent place in the book shelves of many computer scientist.

5-0 out of 5 stars Excellent if you have the math...
to understand it. This book is intended for serious students of computer science or those who have some similar training - it is definitely set up as a textbook. However, that being said, if you have the background the authors' delivery is fist-class and very clear.

The reviews below give more than enough information so I won't belabour the Kolmogorov complexity here. Suffice it to say you won't find the subject detailed more fully in any other reference work in existence today.

However, this book does need to be revised and updated. There has been a lot of development in the field and the sections overviewing Solomonoff's work, in particular, could be expanded. Also, I found it hard to believe that nothing about the 'philosophical' importance of the whole induction question - this is at the core of many very important questions and should not be treated trivially.

There should also be some overview of two other areas that, in combination with the theory outlined in this text, are starting to form the nexus of a "new kind of science" (definitely not Wolfram's pathetic attempt). I refer to some information regarding non-classical logical systems as well as anticipatory computing systems. Both will, I predict, become core areas in addition to extensions to Kolmogorov/Chaitin complexity in the future.

All textbooks should be as clear and concise as this example.

5-0 out of 5 stars The only one of its kind....
The theory of Kolmogorov complexity attempts to define randomness in terms of the complexity of the program used to compute it. The authors give an excellent overview of this theory, and even discuss some of its philosophical ramifications, but they are always careful to distinguish between mathematical rigor and philosophical speculation. And, interestingly, the authors choose to discuss information theory in physics and the somewhat radical idea of reversible computation. The theory of Kolmogorov complexity is slowly making its way into applications, these being coding theory and computational intelligence, and network performance optimization, and this book serves as a fine reference for those readers interested in these applications. Some of the main points of the book I found interesting include: 1. A very condensed but effective discussion of Turing machines and effective computability. 2. The historical motivation for defining randomness and its defintiion using Kolmogorov complexity. 3. The discussion of coding theory and its relation to information theory. The Shannon-Fano code is discussed, along with prefix codes, Kraft's inequality, the noiseless coding theorem, and universal codes for infinite source word sets. 4. The treatment of algorithmic complexity. The authors stress that the information content of an object must be intrinsic and independent of the means of description. 5. The discussion of the explicit universal randomness test. 6. The discussion (in an exercise) of whether a probabilistic machine can perform a task that is impossible on a deterministic machine. 7. The notion of incompressibility of strings. 8. The discussion of randomness in the Diophantine equations; it is shown that the set of indices of the Diophantine equations with infinitely many different solutions is not recursively enumerable; with the initial segment of length n in the characteristic sequence having Kolmogorov complexity n. 9. The discussion on algorithmic probability, especially the test for randomness by martingales. 10. The Solomonoff theory of prediction and its ability to solve the problem of induction. 11. The treatment of Pac-learning and the resultant formalization of Occam's razor. 12. The discussion of compact routing; the optimal space to represent routing schemes in communication networks on the average for all static networks. 13. Computational complexity and its connection to resource-bounded complexity. 14. The notion of logical depth, i.e. the time required by a universal computer to compute the object from its compressed original description. 15. The connection between algorithmic complexity and Shannon's entropy. 16. The discussion on reversible computation, i.e. logically reversible computers that do not dissipate heat. 17. The treatment of information distance, i.e. for two strings, the minimal quantity of information sufficient to translate from one to the other.

5-0 out of 5 stars Comprehensive and Excellent
This is one of the best-written mathematical texts I've read. It builds up the theory from basic principles, and illustrates it with numerous examples and applications. A definitive work.

5-0 out of 5 stars Careful and clear introduction to a subtle and deep field
Li and Vitanyi have done an admirable job at clarifying some very subtle and deep issues in computational complexity. The organization is clear and natural, and the notation good. While a superb undergraduate might learn from it, I suspect the greatest benefits are to advanced students and practicing professionals. ... Read more


12. How to Solve It : A New Aspect of Mathematical Method (Princeton Science Library)
by G. Polya
list price: $16.95
our price: $11.53
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Asin: 069111966X
Catlog: Book (2004-04-05)
Publisher: Princeton University Press
Sales Rank: 46582
Average Customer Review: 4.74 out of 5 stars
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Book Description

A perennial bestseller by eminent mathematician G. Polya, How to Solve It will show anyone in any field how to think straight.

In lucid and appealing prose, Polya reveals how the mathematical method of demonstrating a proof or finding an unknown can be of help in attacking any problem that can be "reasoned" out--from building a bridge to winning a game of anagrams. Generations of readers have relished Polya's deft--indeed, brilliant--instructions on stripping away irrelevancies and going straight to the heart of the problem.

... Read more

Reviews (19)

4-0 out of 5 stars Resolute favorite: How to Solve It
How does a teacher go about teaching? It is a hard trick. Written and published in the '40s, and then again subsequently Polya's "How to Solve It" is an attempt to describe the general paths to the student's Eureka! moments. As such it is also of interest to those who go about the task of discovery, and you must constantly rethink their strategies, in the face of a stubborn unknown.

Polya's consideration of the Various Approaches to problem solving hangs on several key structural bands that take the forms of a teacher's questions: Do you know any related problem? Do you know an analogous problem? [Parallelograms are considered.] Here is a problem related to yours and solved before. Can you use it? Should you introduce some auxiliary element in order to make its use possible?

These ring true to this recently mustered parental pedantic.

Polya's actual treatise is just 30 pages; the associated 'dictionary' definitions section is quite extended, actually, making up some 200 pages. He describes going back to first principles in problem solving. January 1, 2003 is a day perhaps to remember such back tracking is sometimes in order.

4-0 out of 5 stars Getting to Eureka
How does a teacher go about teaching? It is a hard trick. Written and published in the '40s, and then again subsequently Polya's "How to Solve It" is an attempt to describe the general paths to the student's Eureka! moments. As such it is also of interest to those who go about the task of discovery, and you must constantly rethink their strategies, in the face of a stubborn unknown.

Polya's consideration of the Various Approaches to problem solving hangs on several key structural bands that take the forms of a teacher's questions: Do you know any related problem? Do you know an analogous problem? [Parallelograms are considered.] Here is a problem related to yours and solved before. Can you use it? Should you introduce some auxiliary element in order to make its use possible?

These ring true to this recently mustered parental pedantic.

Polya's actual treatise is just 30 pages; the associated 'dictionary' definitions section is quite extended, actually, making up some 200 pages. He describes going back to first principles in problem solving. January 1, 2003 is a day perhaps to remember such back tracking is sometimes in order.

5-0 out of 5 stars Buy it!
The issue is that solving problems is not made interesting and fulfilling experience.

This book beautifully explains the process of problem-solving. It starts from simple problems, lays down the fundamentals and leads to more complex problems.

One of the gems is the simple formula:
1. Understand the problem
2. Devise a plan (seeing how various items connect
3. Carry out the plan
4. Look back at the completed solution, review and discuss it.

It is also a good reference to teach kids how to approach problems.

Buy it and it will be a very handy reference.

5-0 out of 5 stars Very helpful to my programming work
Polya prescribes different forms to approaching a problem through some guide questions that a solver should ask ("Is there a related problem"). The exposition is quite short, majority of the book is devoted to a glossary of heuristic terms which prove very helpful. Polya uses common problems in high school geometry to demonstrate his point which make it easily understandable.

I'm glad I have discovered an excellent book on problem solving which would prove indispensable in my programming career. Other programming books mainly demonstrate features of an OS or a computer language but this book goes into the heart of the computer science which is problem solving.

4-0 out of 5 stars For math thinkers maybe
There seems to be a cultish following for Polya's book, so I decided to pick it up even though I'm not a mathematician. I'm a philosophy PhD with an interest in "business strategy" (as they call it). The book's a little bit tough to move through, since he chose to write it as a glossary for the bulk of the text. That makes it boring. The more fundamental issue of course is that he's thinking about math when he gives his ideas for solving problems, and more specifically about TEACHING kids to solve math problems. Now, this is useful. And the general tenor is applicable to all kinds of problem solving. But I think it's not the holy grail it's meant to be -- there are other books on problem solving that make more practical sense if you are working on non-formal mathematical puzzles. ... Read more


13. An Introduction to Mathematical Reasoning : Numbers, Sets and Functions
by Peter J. Eccles
list price: $31.99
our price: $31.99
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Asin: 0521597188
Catlog: Book (1997-12-11)
Publisher: Cambridge University Press
Sales Rank: 134173
Average Customer Review: 4.8 out of 5 stars
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Book Description

This book eases students into the rigors of university mathematics. The emphasis is on understanding and constructing proofs and writing clear mathematics. The author achieves this by exploring set theory, combinatorics, and number theory, topics that include many fundamental ideas and may not be a part of a young mathematician's toolkit. This material illustrates how familiar ideas can be formulated rigorously, provides examples demonstrating a wide range of basic methods of proof, and includes some of the all-time-great classic proofs. The book presents mathematics as a continually developing subject. Material meeting the needs of readers from a wide range of backgrounds is included. The over 250 problems include questions to interest and challenge the most able student but also plenty of routine exercises to help familiarize the reader with the basic ideas. ... Read more

Reviews (5)

4-0 out of 5 stars Now I know how beautiful proofs can be
This book provides a nice introduction to mathematical reasoning and proofs. My intention on purchasing this book was to learn how to perform mathematical proofs. I believe it has achieved that purpose. The text is easy to follow and the author presents the work clearly.

5-0 out of 5 stars Just buy this
I needed a book that covered fundamental background information behind mathematical proof techniques for an undergraduate univeristy level linear algebra class.

With this book, I was able to truly learn and understand the major concepts behind mathematical logic and proof. This text brings a whole new meaning to teaching the reader about being precise; and I mean the author does an extremely terrific job of doing just that. Wow!

Seriously, the focus here is on content so you won't find any sexy graphs or anything. The content is so good that I often felt that just by reading it I was propelled into a quasi- pseudo-lecture meeting.

After following this text, I can say that I now appreciate the act of being precise to the point that is required for mathematical proof. If you want to extend the knowledge of your 'white board' then just buy this thing. I am so glad I did.

BTW, I only needed the content from the first five chapters, I can't say much about the rest of the text. However, taking an inductive approach, I must assume that the other chapters are also very excellent. Yess, see it worked!

5-0 out of 5 stars Fabulous So Far.
I'm at the end of my first discrete mathematics course and have struggled to find clear explainations of how to write a proof, meaning how to choose what method and how to choose what the next statement should be to lead to the desired conclusion. I'm only on chapter five and it is a breath of fresh air to read this. Rather than just showing the completed proof Eccles shows the "scratch" work that goes into making the proof, discusses the reasoning and alternative paths, and then has the final proof that is easily understood.

An excellent supplement for a typical college text.

5-0 out of 5 stars It's a Jewel
This book is a jewel for every Mathematics student! It has clear and understandable notation. It also help us understand some methonds for demonstration and how to write clear mathematics. It's a must-buy!

5-0 out of 5 stars User-Friendly! Almost makes learning analysis fun!
If you are struggling with a first analysis course or any course that uses proofs, this is the book for you! It introduces basic analysis topics like logic, sets, and the real numbers. And it is written in almost plain english! Moreover, the author focuses on teaching proof writing. ... Read more


14. Greek, Indian and Arabic Logic (Handbook of the History of Logic Series)
by Dov Gabbay, John Hayden Woods
list price: $145.00
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Asin: 0444504664
Catlog: Book (2004-02-06)
Publisher: North Holland
Sales Rank: 468861
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Book Description

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and Renaissance Logic and Logic: A History of its Central.

In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality.

Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world.

Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value.

Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas.
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15. Elements of the Theory of Computation (2nd Edition)
by Harry R. Lewis, Christos H. Papadimitriou
list price: $92.00
our price: $92.00
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Asin: 0132624788
Catlog: Book (1997-08-07)
Publisher: Prentice Hall
Sales Rank: 198504
Average Customer Review: 2.93 out of 5 stars
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Book Description

Lewis and Papadimitriou present this long awaited SecondEdition of their best-selling theory of computation. The authors arewell-known for their clear presentation that makes the material accessible to aa broad audience and requires no special previous mathematicalexperience.In this new edition, the authors incorporate asomewhat more informal, friendly writing style to present both classical andcontemporary theories of computation. Algorithms, complexity analysis, andalgorithmic ideas are introduced informally in Chapter 1, and are pursuedthroughout the book. Each section is followed by problems. ... Read more

Reviews (28)

5-0 out of 5 stars A classic text on the theory of computation.
Elements of the Theory of Computation, by Lewis and Papadimitriou, is something of a classic in the theory of computation. Of the many books I have used to teach the theory of computation, this is the one I have been most satisfied with. It covers all of the fundamental concepts one would expect in such a book (more on this below) but offers a bit more mathematical rigor than most other books I've seen on this topic. It also covers one topic that is rarely even mentioned in other textbooks: the composition of Turing machines.

The book begins with a brief introduction to the relevant discrete mathematics (such as set theory and cardinality) and proof techniques, then introduces the concepts of finite automata, regular expressions, and regular languages, describing their interrelationships. It proceeds to context-free languages, pushdown automata, parse trees, pumping lemmas, Turing machines, undecidability, computational complexity, and the theory of NP-completeness. (These are all standard topics.) Along the way, Lewis and Papadimitriou also introduce random access Turing machines and recursive functions, and do a nice job of explaining the halting problem and how this translates into undecidable problems involving grammars, various questions about Turing machines, and even two-dimensional tiling problems. All of these topics are covered with an appropriate mix of formalism and intuition.

Perhaps the feature I like best is the discussion of composing simple Turing machines to obtain more complex and interesting machines. The authors even introduce a convenient graphical notation for combining Turing machines and spell out specific rules for composition. While most authors are forced to immediately employ heuristics in reasoning about complex Turing machines (lest the notation become overwhelming), Lewis and Papadimitriou are able to keep the discussion more formal and structured by virtue of their Turing machine "schema". I believe this makes their arguments more rigorous and even easier to follow.

This is clearly one of the best books on the theory of computation. However, be aware that there have been very significant changes from the first edition, which was lengthier and more thorough. I confess that I actually prefer the first edition, as it contains nice sections on logic and predicate calculus (which have been removed from the 2nd edition), and is a bit more formal (albeit with some fairly awful notation). The 2nd edition is definitely crisper, with much cleaner notation; it is clearly more student-friendly, which was presumably the aim of the new edition.

If you wish to teach an introduction to theoretical computer science, or wish to learn it on your own, this would be a fine book to use. It's hard to go wrong with this classic.

5-0 out of 5 stars You'll love it or hate it.
I discuss the first edition, I havent read the updated version. People have strong opinions about this classic book. Many students have it forced upon them for some class and they absolutely despise it. But a small number of people like me loved it, in fact its one of the best textbooks I own. To get through it you need to enjoy mathematics and careful, rigorous definitions and proofs- rather than viewing these things as pointless obscuring or pedantic arrogance. Engineering students tend to find the book tedious, boring, and too difficult. Some people are intimidated by the sheer volume of special notation used. But if you're inclined towards mathematics or theoretical work I think you'll enjoy the extra rigor and precision (compared to most computation theory books). There are a few rough spots in it but overall a great book that will give you the foundation to begin studying computational complexity theory, recursive function theory, or mathematical logic. Note that the 2cd edition has unfortunately removed the chapters on logic, and I've heard its a little watered down, so be careful choosing which one you want.

2-0 out of 5 stars A reference at best, a textbook from hell
I took a Theory of Computation class with Harry Lewis, one of the book's author this last semester at Harvard. Lewis may be a gifted professor, but if you are looking for a textbook, look for something else (Sipser would be a much better idea). It is impossible to learn from this book; the examples are too complex, the questions are outlandishly difficult. I got my A but it was not thanks to this book. Steer clear.

5-0 out of 5 stars Recommended for some...
Concerning the FIRST EDITION: (I haven't read the second edition but on flipping through it briefly it appeared watered down) One of the best undergraduate textbooks I had. I would recommend it to anyone interested in math or in approaching computer science from a mathematics perspective. If you dont like abstract mathematics and proofs, or are just interested in writing code for actual, physical computers then you'll hate it. The level of rigor in the book is high for an introductory text, and while there are no specific mathematical prerequisites, you'll need a certain amount of mathematical experience to be able to process it quickly. The style is dry- no history, philosophical musings, or isnt-this-cool hullabaloo -just the facts and a minimum of exposition. Chapter 1 gives the basic discrete math stuff: sets, relations, strings, induction and such. Chapter 2 introduces finite automata (regular languages), a great way to begin careful analysis of computation. Chap 3 covers context-free languages, and is perhaps too long (skim the later sections if you get bored). Chap 4 covers Turing machine basics. Chap 5 gives two alternative formalisms of computation (unrestricted grammars and mu-recursive functions), shows their equivalence with Turing machines, and a construction of a universal turing machine. Chapter 6 is uncomputability and the halting problem. If the earlier chapters seemed a little dull and unmotivated, then things suddenly become very interesting here. Includes an unsolvable tiling problem. Chap 7 is computational complexity and np-completeness. Chap 8 is a solid intro to basic propositional logic. Chap 9 is first-order logic (without equality) and is the weakest chapter in the book. Formal proof systems are not introduced (forgivable considering the length of the book), but the presentation of the Herbrand expansion theorem is hurried, awkward, and confusing.

Overall a great book, but you should supplement it with other inexpensive computation theory and logic books. After you've mastered this, check out Papadimitriou's "Computational Complexity" and Hartley Rogers' "Theory of Recursive Functions and Effective Computability", two fantastic intermediate-level books.

1-0 out of 5 stars Just plain boring.
When I first signed up for the theory of computation I expected it to be a great class. I had looked over some of the subject material before the course started and it found it very interesting. I mean, who can't find this subject interesting? Learning about not only how computers function but what their limitations are. I'll tell you who, anyone who has had the misfortune of reading this book. I don't know how or why two obviously educated people can write a book that is so lackluster. This book turned my course into an absolute nightmare. I instead turned to another book, which ended up saving me. Do yourself a favor and run from this book as fast as you can. ... Read more


16. Foundations and Fundamental Concepts of Mathematics
by Howard Whitley Eves, Howard Eves
list price: $16.95
our price: $11.87
(price subject to change: see help)
Asin: 048669609X
Catlog: Book (1997-05-01)
Publisher: Dover Publications
Sales Rank: 21372
Average Customer Review: 5 out of 5 stars
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Book Description

Third edition of popular undergraduate-level text offers overview of historical roots and evolution of several areas of mathematics. Topics include mathematics before Euclid, Euclid’s Elements, non-Euclidean geometry, algebraic structure, formal axiomatics, sets, more. Emphasis on axiomatic procedures. Problems. Solution Suggestions for Selected Problems. Bibliography.
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Reviews (3)

5-0 out of 5 stars 'Swiss Army Knife' of Upper Level Mathematics
I totally agree with the previous two reviewers on what they had to say about this wonderful book. However, I did want to briefly note that -- beyond merely being a fascinating overview of the development of beyond-calculus mathematics -- it is also a great resource for people needing to look up or review topics in advanced mathematics (especially mathematical logic). Again, to repeat what the others have said, buy this book if you have ANY interest in mathematics. You won't regret it.

5-0 out of 5 stars Excellent Overview. Belongs on Your Bookshelf.
Howard Eves presents this five-star story of mathematics as two intertwined threads: one describes the growing content of mathematics and the other the changing nature of mathematics. In exploring these two elements, Eves has created a great book for the layman. I find myself returning to his book again and again.

My few semesters of calculus, differential equations, and other applied math failed to formally introduce me to abstract algebras, non-Euclidian geometries, projective geometry, symbolic logic, and mathematical philosophy. I generally considered algebra and geometry to be singular nouns. Howard Eves corrected my grammar.

"Foundations and Fundamental Concepts" is not a traditional history of mathematics, but an investigation of the philosophical context in which new developments emerged. Eves paints a clear picture of the critical ideas and turning points in mathematics and he does so without requiring substantial mathematics by the reader. Calculus is not required.

The first two chapters, titled "Mathematics Before Euclid" and "Euclid's Elements", consider the origin of mathematics and the remarkable development of the Greek axiomatic method that dominated mathematics for nearly 2000 years.

In chapter three Eves introduces non-Euclidian geometry. Mathematics is transformed from an empirical method focused on describing our real, three-dimensional world to a creative endeavor that manufactures new, abstract geometries.

This discussion of geometries, as opposed to geometry, continues in chapter four. The key topics include Hilbert's highly influential work that placed Euclidian geometry on a firm (but more abstract) postulational basis, Poincaire's model and the consistency of Lobachevskian geometry, the principle of duality in projective geometry, and Decartes development of analytic geometry. For the non-initiated these topics may seem daunting, but Eves' approach is clear and quite fascinating.

Chapter five, which might have been titled "The Liberation of Algebra", may at first be a bit overwhelming to those unaware of algebraic structures like groups, rings, and fields. But take solace as even mathematicians in the early nineteenth century still considered algera to be little more than symbolized arithmetic. As Eves says, non-Euclidian geometry released the "invisible shackles of Euclidian geometry". Likewise, abstract algebra created a parallel revolution. (Again, don't be intimidated by the terminology. Eves is quite good.)

The remaining four chapters look at the axiomatic foundation of modern mathematics, the real number system, set theory, and finally mathematical logic and philosophy. Eves concludes with the surprising discovery of contradictions within Cantor's set theory as well as Hilbert's unsuccessful effort to define procedures to avoid inconsistencies or contradictions within an axiomatic system.

Eves mentions Godel's fundamental contribution to mathematical logic, but stops short of delving into Godel's Proof. For additional reading I highly recommend "Godel's Proof" by Ernest Nagel and James R. Newman.

I also highly recommend Richard Courant's and Herbert Robbins' classic, "What is Mathematics?", a more detailed examination of the development of fundamental ideas and methods underlying mathematics. I would suggest that most readers, particularly non-math majors, first read Eves and later tackle Courant and Robbins.

I have read "Foundations and Fundamentals of Mathematics" at least twice. I gave my son a copy for Christmas. He says that the book is great and he even claims to be reading it as he walks across his campus between classes. The price is great. It belongs in your book collection.

5-0 out of 5 stars Ecellent description of the history of mathematical thinking
There are several books available on the history of mathematics. Some give an account on the development of a certain area, others focus on a group of persons and some do hardly more than story telling. I was looking for one that tells the story of the development of the main ideas and the understanding of what mathematics and science in general is (or what people thought it is and should be). Howard Eves' book is the first book I bought that gives me the answers I was looking for. Starting with pre-Euclidean fragments, going on with Euclid, Aristotle and the Pythagoreans, straight to non-Euclidean geometry it focuses on the axiomatic method of geometry. What pleased me most here is that the author really takes each epoch for serious. He quotes longer (and well chosen) passages from Euclid, Aristotle and Proclus to demonstrate their approaches. Each chapter ends with a Problems section. I was surprised to see how much these problems reveal of the epoch, its problems and thinking.

The book goes on with chapters on Hilbert's Grundlagen, Algebraic Structure etc, always showing not only the substance of these periods but also the shift in the way of thinking and the development towards rigor. The last chapter is titled Logic and Philosophy. Eves divides "contemporary" philosophies of mathematics into three schools: logistic (Russel/Whitehead), intuitionist (Brouwer) and the formalist (Hilbert).

The book ends with some interesting appendices on specific problems like the first propositions of Euclid, nonstandard analysis and even Gödel's incompleteness theorem. Bibliography, solutions to selected problems and an index are carefully prepared to round up an excellent book.

Should you buy this book ? Yes. What kind of mistake can you make in spending US$ 12.95 on a book that has withstood the test of time through three editions (each with a different publisher). I havent completed reading the book yet, but I dont regret having bought it. ... Read more


17. Attacking Faulty Reasoning: A Practical Guide to Fallacy-Free Arguments
by T. Edward Damer
list price: $47.95
our price: $47.95
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Asin: 0534551335
Catlog: Book (2000-06-14)
Publisher: Wadsworth Publishing
Sales Rank: 212823
Average Customer Review: 4.3 out of 5 stars
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Book Description

ATTACKING FAULTY REASONING is the most comprehensive, readable, and theoretically sound book on the common fallacies. It is designed to help one construct and evaluate arguments. The overriding purpose of the text is to help the students recognize when they construct or encounter a good or successful argument of a particular action or belief. This one skill is reinforced on every page of the text, from the first three chapters that focus on the criteria for a good argument, through the four major chapters on the fallacies or ways that arguments can go wrong. The emphasis is on resolving issues rather than pointing out flaws in arguments. ... Read more

Reviews (10)

5-0 out of 5 stars The antidote for contradiction and controversy.
Damer pulls off a next to impossible task-naming, describing, exampling, and attacking 60 fallacies while structuring them neatly within four criteria of a good argument: relevance, acceptability, sufficient grounds and rebuttal. The last chapter discusses the specifics of "A Code of Conduct for Effective Rational Discussion." I used this test as a key element of my Ph.D. research and continue to use it in my later work. This should be required study for every politician and philosopher. A simpler version should be required study for every middle school and high school student. Discovering what is true would be so much easier with good arguments absence of fallacy. Be the first to rid your "neighborhood" of polemics. Study this book.

5-0 out of 5 stars Best critical thinking book out there
That about sums it up. I have read MANY books on informal logic and critical thinking, but this one is by far the most clearly written and accessible.

I particularly liked the author's focus on WHY fallacies are bad instead of just rattling them off.

This is a book I can recommend to anyone, even the old veterans of argumentation.

3-0 out of 5 stars Argument Karate
I found this book to be well written, but it is more a book of argument karate. It is written with the idea of both ends of an issue following a logic and rules. "Attacking Faulty Reasoning" is certianly thorough, however academic. I found Nicholas Capaldi's "Art of Deception" much more practical- critical thinking street fighting.

4-0 out of 5 stars Liberal bias?
To say that this is a good book except that the examples are all about conservative views and that this makes the book stilted in any way is itself logically fallacious. Bad logic is bad logic period. It is unfortunate that the conservative cause is so riddled with bad reasoning like this.

4-0 out of 5 stars Excellent overall, but some examples are weak
This book is readable and thorough, and probably the best introduction to critical thinking around.

With such a large number of fallacies demanding multiple examples, the author must be forgiven if some of them seem a little off the mark, even while being technically correct. For example, the proposition (I'm paraphrasing) "Our baseball team was 1 and 11 this year, but with a new coach we'll do better next year." is in fact false. However, if the proposition were that "we'll probably do better" it would be true, because the probability is that we would get an average coach and an average coach has a record of 0.500, while assuming that coaching has a positive effect. Another example has former Predident Bush answering the question, "Did Dan Quayle's parents help him get into the national guard?" with words to the effect that "At least he served patrioticly and didn't run to Canada or burn the flag." The answer while technically irrelevant is a politician's way of saying, "Whether his parents helped or not is unimportant, at least ...blah, blah." Such an answer invites a rejoinder along the lines of "It really is important, because ..." The fault of the example is that it implies it is OK to rest on the technicalities even when you have a very good idea of what your opponent is really saying.

So if some of the example are a little off-base, perhaps that is all to the good as a learning experience. The small bits of uneasiness are left to the student as an exercise to resolve. The author provides the tools for doing so. ... Read more


18. Languages and Machines: An Introduction to the Theory of Computer Science (2nd Edition)
by Thomas A. Sudkamp
list price: $110.40
our price: $110.40
(price subject to change: see help)
Asin: 0201821362
Catlog: Book (1996-11-04)
Publisher: Addison Wesley
Sales Rank: 142658
Average Customer Review: 3.14 out of 5 stars
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Amazon.com

Languages and Machines is a user-friendly text that covers the key ideas of the theory of computation clearly and thoroughly. Examples and numerous diagrams, including diagrams that illustrate the principle of induction, aid in the understanding of the material. Relative to other books containing similar information, this text contains in-depth coverage of languages and parsing. ... Read more

Reviews (7)

4-0 out of 5 stars Good Book for those who have had discrete math
Some chapters of the book seem to be out of logical order for easy learning of automata. Overall a well written book, with good examples and organization. A fundamental understanding of discrete mathematics is a must to understand all of this book. enjoy!

1-0 out of 5 stars Great Read (For those of us with IQs higher than 180)
If you find yourself alone on a Friday night this may be the book for you. You'll learn plenty of notation with which to impress your friends. Learn new tricks, such as depth first parsing, construction if context free grammars, and even the highly controversial Greibach Normal Form (GNF).

3-0 out of 5 stars Difficult stuff, but not in the way you might think
First of all, let me assure you that books of this kind are rare. They're simply a pain to write, and few have the expertise in the area to write them. It's just a fact that discrete math isn't the most popular topic of study.

That said, this isn't too bad a textbook. It does a pretty decent job explaining the subject matter, which isn't very hard to pick up...

IF you can stay awake! I get the feeling Dr. Sudkamp was falling asleep as he finished off each paragraph, and he sure managed to do the same to me when reading it. Though the chapters seem short, you'll find it takes you up to an hour to read through them. Why? Because you'll read the first part, go on to the next... realize you've forgotten everything in the first part, and have to re-read it. Repeat as necessary, ad infinitum.

The exercises have no solutions whatsoever, which can be a pain for this sort of material. Examples don't really help you do them because only the very general solving method is similar, and each problem really is a completely new thing.

The bottom line is, if you have to learn this material, then you're probably going to have to read it. But I wouldn't advise getting this book for some light reading at night.

5-0 out of 5 stars Excellent Book, A Must have.
This is one of the better books that I read on languages and machines. This book is great for someone who is interested in parsing, compilers or pattern matching. The book covers a lot of theory on computation and is not for a beginner. I would recommend that one be well grounded in set theory, recursion and mathematical induction before attempting to read this book. I did not read all the chapters; I only read those that were relevant to my project and I had not seen before in other texts. The 1st chapter get you upto speed with a good review of set theory followed by a quick review of induction and recursion. The 2nd chapter gives an excellent introduction to strings, languages and regular expressions along with relations on regular expressions. Chapter 3 is where the rubber hits the road. It covers context-free and regular grammars. I feel this chapter covers the subjects very well. Chapter 4 gives a good description of parsing and methods of parsing. Chapter 6 covers Finite Automata. This chapter describes deterministic finite state machines, nondeterministic finite state matchines and nondeterministic finite state matchines with lambda transitions. The presentation of the subject in this chapter was excellent. Chapter 7 presents Regular Languages and Sets. This chapter gives a good presentation of how to put together different types of machines from different languages and build languages from machines. I found it best not to read the chapters in orders, instead I read them in the following order which helped to understand the material better; 1,2,6,7,3,4,11,12

My only complaint: It would have helped if the author could have gave answers to some of the problems at the end of the chapters.

4-0 out of 5 stars A good introduction, bent towards formal languages
I have a mathematical background and wished to acquaint myself with the basics of theoretical computer science. This book didn't disappoint me.

The book stresses formal languages and parsing, and is therefore best suited for persons interested in creating languages, compiler technology and parsing. However, it covers also Turing machines, computability and complexity issues, among others, and is therefore reasonably comprehensive.

Exercises range from easy to moderate, and many of them are stimulating. Another reviewer complained about the lack of drill exercises (see below). I can understand the anguish of students; some of the exercises, as well as parts of the text, may be difficult if one doesn't have much experience in formal reasoning and abstract problem solving. However, all exercises I have taken a look at are solvable with the knowledge provided in the text, and are therefore suitable for readers with at least a fair mathematical background.

My main complaint is the small number of applications. In chapter 3, there is a nice example: the arithmetic expressions of Pascal; in chapter 15, good examples of NP-complete problems. However, these are exceptions. In my opinion the text would greatly benefit from e.g. end-of-chapter exercises related to programming mini-languages which could be defined on the spot. Also examples of finite state machines (copier machines, services in a mobile phone etc.) would add flesh to exercises.

All in all, this is a good entry point to theoretical computer science for a person trained in mathematics or a related field, but may partly be too challenging to a first-year student. ... Read more


19. Friendly Introduction to Mathematical Logic, A
by Christopher C. Leary
list price: $68.00
our price: $68.00
(price subject to change: see help)
Asin: 0130107050
Catlog: Book (1999-12-08)
Publisher: Prentice Hall
Sales Rank: 726793
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Book Description

This user-friendly introduction to the key concepts of mathematical logic focuses on concepts that are used by mathematicians in every branch of the subject. Using an assessible, conversational style, it approaches the subject mathematically (with precise statements of theorems and correct proofs), exposing readers to the strength and power of mathematics, as well as its limitations, as they work through challenging and technical results. KEY TOPICS: Structures and Languages. Deductions. Comnpleteness and Compactness. Incompleteness--Groundwork. The Incompleteness Theorems. Set Theory. : For readers in mathematics or related fields who want to learn about the key concepts and main results of mathematical logic that are central to the understanding of mathematics as a whole. ... Read more


20. Basic Category Theory for Computer Scientists (Foundations of Computing)
by Benjamin C. Pierce
list price: $25.00
our price: $25.00
(price subject to change: see help)
Asin: 0262660717
Catlog: Book (1991-08-07)
Publisher: The MIT Press
Sales Rank: 232540
Average Customer Review: 4.5 out of 5 stars
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Book Description

Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. ... Read more

Reviews (4)

3-0 out of 5 stars Too terse
This is a very short book: 70 pages of text + a bibliography. The first 50 pages are about general category theory, and the last 20 pages are specifically for computer scientists. My interest is in general category theory, and I bought this because I have a BS in CS and thought I'd find plenty of familiar examples. Unfortunately this book doesn't have nearly enough examples. I found it easier to skim some undergrad abstract algebra books in the library (groups, rings, vector spaces) and then continuing with category theory intros written for math students.

5-0 out of 5 stars the best understaning of categories you can get
This book is tiny in volume but large in contents. It does not only provide the definitions of the fundamental concepts but also clear explanations and motivations of why must everything be defined that way, which are not always found in other texts. Plenty of the right examples help you build the right intuitions. The case studies at the end put everything into context and prepare you for CS texts on semantics, type theory, etc.
If you want to UNDERSTAND this wonderful theory read this book!

5-0 out of 5 stars Clear and concise
This is an excellent introduction to category theory, not just for computer scientists, but for mathematicians as well. The author has a very clear writing style--it's evident that he writes to help people to understand the subject, and not to show off his knowledge. The examples illustrating various principles are easy to understand, especially the ones used to illustrate adjoints, arguably one of the more difficult concepts in category theory. This book also comes with a very valuable annotated bibliography, enabling one to intelligently choose from the many books and articles in this burgeoning field.

Read this book before you tackle Mac Lane.

5-0 out of 5 stars This book is a CCC.
Which stands for "Compact, Complete, and Comprehensible".
It is fairly easy to read, has every basic aspects of Category Theory, and has a lot of good examples.
If you would like to know the first step of Category Theory and you are in CS realm, this book is the one you have to try. ... Read more


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