Reviews (6)

Would someone please explain this to me
On page 42 Mazur writes:
Exploring Math through Fables
I should start by saying that I didn't read this book to learn anything about mathematics; I already knew all the math in it.But I have long been interested in ways to present math ideas to people who aren't strong in math.Mazur's approach of putting the math into stories sounded interesting.
Adventure and Math.What's not to like?
I purchased this book a couple of weeks ago and can hardly put it down.I work in the area of operation research for the Navy and have recently decided to become a highschool math teacher.There are so many interesting examples of math in this book and great stories to boot.This book is an excelent example of how math can be entertaining.I would recomend it to anyone who wants to have a better understanding of math. ... Read more

Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Science/Mathematics    5. Mathematics / Logic    6. Popular science   

Reviews (2)

Visual justification has a role in mathematics
The first mathematical proofs were no doubt primarily diagrammatic in structure, and we all should appreciate the role they have played in the development of mathematics. Unfortunately, the figure is now somewhat maligned as a tool in mathematics. A symbol used in a proof is a representative of an abstract concept, and if a diagram is also considered in that way, then it should be just as acceptable. The proofs in this book are not truly without words, as most of the time there is a formula as well. However, they are easy to understand and cannot fail to be appreciated. Engaging exercises to train your intuition
Famous mathematicians have often emphasized the role of visual intuition;e.g., Hilbert: "Who does not always use along with the doubleinequality a > b > c the picture of three points following oneanother on a straight line as the geometrical picture of the idea"between"? Who does not make use of drawings of segments andrectangles enclosed in one another, when it is required to prove withperfect rigor a difficult theorem on the continuity of functions or theexistence of points of condensation?" (from his famous address at the1900 International Congress). This book is a collection of well over 100one-page proofs, collected from various sources. The topics range fromnumber theory to calculus, and most of them require no advancedmathematics. Typically there is a statement of a result, with a labelleddiagram showing how it is "proved"; in some cases there are a fewauxiliary equations along with the picture. These are not simple, oftenrequiring quite a bit of thought before the "Aha!" moment.Working through them is a valuable exercise for the student ofmathematics--having seen, e.g., six different visual proofs of thePythagorean theorem, one comes to really *understand* the result, not just"follow the logic". I have not encountered any better way thanthis book to "see" how mathematical truth is discovered andproved. It can be valuable as a supplement to courses through precalculusand elementary calculus. Perhaps one of its best uses is to inspireteachers to present results in a more lively way then"definition-theorem-proof" or "just memorize it". ... Read more

Subjects:  1. Charts, diagrams, etc    2. General    3. Mathematics    4. Mathematics (General)    5. Science/Mathematics    6. Mathematical logic    7. Mathematics / General   

Subjects:  1. Artificial Intelligence - General    2. Logic    3. Logic Design    4. Mathematics    5. Science/Mathematics    6. AI logics    7. Mathematical theory of computation    8. Mathematics / Logic    9. Programming languages    10. classical logic    11. complexity    12. computational logic    13. finite models    14. higher order logic    15. linear logic    16. logic programming    17. mathematical logic    18. model checking    19. nonclassical logic    20. permutative logic    21. program logic    22. proof theory    23. term rewriting    24. verification   

Reviews (2)

The best reference for Proofs
This is an advanced book, with a lot of information on every page.I use it as a reference book, since it has hundreds of wonderful proofs and problems, along with thorough and concise definitions for just about everymajor branch of mathematics. Very good introductory text
With a disposition towards problem solving, the book was very helpful forsomeone like myself who needs the mathematical tools but lacks thebackground. Occasionally the book lapses into the silly notion that you'veseen some of the material and concepts before. ... Read more

Subjects:  1. Applied    2. Calculus    3. Logic    4. Mathematics    5. Mathematics (General)    6. Problem solving    7. Science/Mathematics    8. Mathematics / Calculus    9. Philosophy of mathematics   

Subjects:  1. Logic    2. Mathematical And Symbolic Logic    3. Mathematics    4. Philosophy Of Mathematics    5. Proof theory    6. Science/Mathematics    7. Set Theory    8. Mathematics / Advanced   

Reviews (1)

Great Resource For The Seriously Interested
This book is written, chapter by chapter, by many of the most important researchers in this field- Since it is an advanced text, the content is a bit uneven and varied in writing style- It is essentially a compendium of review articles by experienced researches covering the various facets of the DL community. The choppy design is a small tradeoff for being able to get some very up-to-date information on Description Logics.
Read more

Subjects:  1. Computer Books: General    2. Computer Books: Languages    3. Computers    4. Computers - General Information    5. Description logics    6. Expert Systems    7. Handbooks, manuals, etc    8. Programming - Software Development    9. Programming Languages - General    10. Artificial intelligence    11. Computers / Programming / Software Development    12. Mathematical logic    13. Mathematical theory of computation   

Reviews (11)

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

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

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

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

Reviews (5)

Fast entry to notation
I purchased this book to gain enough information to read a technical paper.Fuzzy logic is new to my industry (petroleum) and my 1970's education did not provide any background.The book took less than an evening to absorb and provided more than enough understanding of the notation and basic operations that I was able to read my paper and start building an interest in deploying Fuzzy logic in my daily work.Other books are needed to fully apply the methodology.However, sufficient demonstration of basic fuzzy arithmetic was provided to know that 2 times 3 divided by 2 is not necessarily 3.
Breezy Intro to Fuzzy Logic
I was impressed by the fact that most symobology used in the book is defined during its first use.There are a few symbols like 'sup' on page 38 that I'm still unclear about.
Excellent starting point
This book gives you an excellent introduction to Fuzzy Logic Concepts, Reasoning and Applications.
Read more

Subjects:  1. Artificial Intelligence - Fuzzy Logic    2. Computers - General Information    3. Fuzzy Sets    4. Fuzzy logic    5. Logic    6. Mathematical And Symbolic Logic    7. Mathematics    8. Science/Mathematics    9. Set Theory    10. Automatic control engineering    11. Computers / Artificial Intelligence    12. Fuzzy set theory    13. Mathematics-Logic   

Reviews (6)

The very best book I have read on Boolean
I'm a Computer Engineering student and have read many books on the subject.Until I read this book, I was constantly struggling with complex Boolean equations. Fortunately, this book has helped me master the subject -with a lot of hard work. I highly recommend this book for any student in the Computer Engineering field.

Excellent Introduction
This is an extremely clear introduction to Boolean Algebra.It is very well written and edited.Plenty of exercises will answers to cement the concepts in your brain.
Great books for boolean algebra
Incredible book for understanding boolean algebra. What I like about the book is the simple and clear language that the author uses and a lot of very helpful examples throughout the book to help understand the concept. ... Read more

Subjects:  1. Algebra - General    2. Algebra, Boolean    3. Boolean Algebra    4. Circuits    5. Computer Engineering    6. Computer Logic    7. Computers    8. Design    9. Electronic digital computers    10. Electronics - Circuits - Logic    11. Electronics - Digital    12. Engineering - Electrical & Electronic    13. Logic, Symbolic and mathematic    14. Logic, Symbolic and mathematical    15. Science/Mathematics    16. Technology & Industrial Arts    17. Computer architecture & logic design    18. Mathematical logic    19. Science: General Issues    20. Technology / Electronics / Circuits / General   

Subjects:  1. Algebra - Abstract    2. Categories (Mathematics)    3. Logic    4. Mathematics    5. Science/Mathematics    6. Algebra    7. Mathematics / Logic    8. Mathematics | Logic   

Reviews (1)

Great introduction to the theory of toposes.
When Goldblatt's book first hit the stands, it was blasted by reviewerswho had a geometric predilection. They claimed that Goldblatt hadtrivialized this essentially rich geometric subject by giving aset-theoretic treatment.Read more

Subjects:  1. Logic    2. Mathematics    3. Number Theory    4. Science/Mathematics    5. Toposes    6. Mathematics / Logic   

Subjects:  1. Essays    2. History & Philosophy    3. Logic    4. Mathematical And Symbolic Logic    5. Mathematicians And Their Works    6. Mathematics    7. Science/Mathematics    8. Mathematical logic    9. Mathematics / Logic    10. Mathematics | Logic   

Subjects:  1. Arithmetic    2. Foundations Of Mathematics    3. Logic    4. Mathematics    5. Predicate calculus    6. Science/Mathematics    7. Set Theory    8. Mathematical logic    9. Mathematical theory of computation   

Reviews (7)

Great introduction to game theory
The first half of the book was absolutely terrific and I am glad I book, however, it gets a little scattered toward the second half.

Interesting book
The book is divided into three parts: Great for learning about Game Theory
This book is fascinating in how it demonstrates what we do in politics, science, and even meditation can affect our rational (and irrational) decision making processes, both individualy and in community.Read more

Subjects:  1. Applied Psychology    2. Game Theory    3. Logic    4. Mathematics    5. Reasoning    6. Science/Mathematics    7. Thought and thinking    8. Mathematics / Game Theory    9. Philosophy & theory of psychology    10. Psychology & Psychiatry / General    11. Science: General Issues    12. golden rule    13. human frailty    14. rational behaviour   

Reviews (4)

Proofs
Interesting book for who need some rules about mathematical proofs.
A Good Introduction, but Stilted Prose
I will not disagree with the previous reviews, except to say that the author while pretty clear and straight-forward in explaining things, has a stilted style that doesn't always flow or read well. I suspect though for many people, particularly many novice, and some professional, mathematicians will never really notice it or find it problematic.

Outstanding math book, and great intro to proofs
This is a great book, and one of my favorite math books. Like the other reviewer, I also wanted to learn how to read and write proofs.I am an engineer, (many years ago), and not a mathematician, (but really enjoy math). The author communicates clearly, and provides lots of good examples.But the heart of the book is the problem sets for each chapter.Most books on proofs spend way too much time on Logic, (or geometry), and not enough on "math" proofs.The book provides problems from a wide variety of math areas. The latest edition added a lot of new material.I struggled at times, since I went through the whole book without an instructor, and worked on all of the problems. So having most of the possible answers in the back of the book, or on the internet helped as a check on my understanding. This book would make a great gift. ... Read more

Subjects:  1. Logic    2. Logic, Symbolical and mathemat    3. Logic, Symbolical and mathematical    4. Mathematics    5. Proof theory    6. Science/Mathematics    7. Textbooks    8. Mathematical logic    9. Mathematics / Logic   

Subjects:  1. Computer Bks - Languages / Programming    2. Computer Books: Languages    3. Computer Engineering    4. Computers    5. Computers - Languages / Programming    6. Programming - General    7. Programming - Software Development    8. Programming - Systems Analysis & Design    9. Calculus of Constructions    10. Computers / Programming / Software Development    11. Coq    12. Dependant Types    13. Formal Languages    14. Formal Methods    15. Inductive Construction    16. Inductive Inference    17. Inductive Types    18. Mathematical Logic    19. Program Extraction    20. Program Validation    21. Program Verification    22. Programm Construction    23. Proof Assistants    24. Proof Construction    25. Proof Theory    26. Software Engineering    27. Theorem Proving   

Reviews (3)

Most Excellent Logic Text
Logic: Techniques of Formal Reasoning, 2nd Edition, by Donald Kalish, Richard Montague, and Gary Mar 1980 (1964)
Best Textbook
With some saddness, I noted the death of Dr Kalish this month.As an undergrad at UCLA during the mid 60's, I was fortunate to take Dr Kalish's class in Symbolic Logic.Over the years, I realized that his course usingthis textbook was the most valuable class I took, either as an undergrad ora graduate student.The text offers disciplined procedural logic thatclarifies thought processes.At various times, I worked as a mathematicalprogrammer in several higher level languages without ever taking a formalcourse in any of them.I attribute my success in this area to theTechniques of Formal Reasoning.

Kalish
If taught by the right person, this book will reveal all of the issues of contemporary logic. It is best supplemented on first reading with Schuman's "logic" guide. As Kalish was influenced by Russell, I found theTheorems in the Principia matched perfectly those found in this book. Thelatest edition of this book is much improved, in my opinion. It offers moreguidance to understanding how to solve problems, and offers a great manyuseful hints and tips. Kalish is/was (?) the Chair of the department ofUCLA. He is a modern master and authority on this subject. In my opinion,if you wish to spend time on understanding the strengths ofcontemporarysymbolic logic, there is no better book to buy. ... Read more

Subjects:  1. Christianity - Theology - Ethics    2. Logic    3. Philosophy    4. Religion    5. Philosophy / Logic    6. Philosophy | Logic & Mathematics    7. Social & political philosophy   

Subjects:  1. Earth Sciences - Geology    2. Logic    3. Material Science    4. Science    5. Science/Mathematics    6. Geology & the lithosphere    7. Mathematics and Science    8. Science / Geology   

Reviews (28)

It's ok
The book was required for a course, and I found it to be pretty straightforward, if a little dull.It's written at a simpler level than Sipser or Hopcroft and Ullman, which might make it appropriate for someonewho is a little apprehensive about the topic (although I think both of those are better books). My biggest complaint was that after finding the first two errors in the text (in an exercise solution and an example), I wasn't comfortable trusting the book to tell me what it meant.The errorswere listed in the errata, so if you're using the book, print out a copy of the errata, and mark your book up.Finding these errors for yourself is a good test of your understanding, but it's also more pain than necessary.Use with caution.

Too advanced for most CS students
Many of the other reviews are negative. I have a nagging feeling that the book was simply too advanced for several, though not necessarily all, of the reviewers.
It's a sleeper
I haven't found any reason for someone to buy this book. The writing sytle is dry. The examples are complex and poorly explained. The concepts are covered adequately, but often with a wordiness that leaves the reader bewildered (if still awake). As an "Introduction" manual, this text fails miserably. I'd have given it zero stars, if possible. It just does not do anything well, and does too many things poorly. There's too many well written texts in the world to waste time with this one. ... Read more

Subjects:  1. Computer Books: General    2. Computers    3. Formal languages    4. Logic    5. Machine Theory    6. Mathematics    7. Programming - General    8. Programming Languages - General    9. Programming languages   

Reviews (3)

Covers a lot, but not all that well
Bell & Machover is meant for a one-year graduate course and is comparable to the more well-known text by Shoenfield. B&M is larger than Shoenfield, having additional chapters on Boolean algebra, intuitionist logic, and nonstandard analysis. There are a few newer things in B&M and they use the tableau method, which Shoenfield doesn't. Otherwise, they have about the same coverage, trading off as to which has more detail on this or that item. The main difference is in the writing style. The delivery in B&M is less articulate, proofs are terse and schematic, and problems have little setup. I find it harder to follow the train of thought in B&M than in Shoenfield and would not want to try learning anything for the first time from it.

This is the book you need as a logic primer
When I was in my third year of graduate school and was deciding to specialize in set theory, I realized that it was time to get some formal training in first-order logic and model theory. At our school, there were no courses in foundations at all, so I had to find the right book and map out a course of study for myself. I sat in on a seminar for more advanced students that was going through Chang and Keisler (model theory); although I now use this as a basic reference, at the time I needed a more systematic treatment of first-order logic before getting into the details of model theory. Probably the most comprehensive Course in Mathematical Logic
This is probably one of the most oustanding textbooks on advanced mathematical logic written this side of the century.Read more

Subjects:  1. Information Theory    2. Logic    3. Logic, Symbolic and mathematic    4. Logic, Symbolic and mathematical    5. Mathematics    6. Science/Mathematics    7. Computers / Information Theory    8. General Theory of Computing    9. Mathematical foundations