Book Description

The ability to construct proofs is one of the most challenging aspects of the world of mathematics. It is, essentially, the defining moment for those testing the waters in a mathematical career. Instead of being submerged to the point of drowning, readers of Mathematical Thinking and Writing are given guidance and support while learning the language of proof construction and critical analysis.

Randall Maddox guides the reader with a warm, conversational style, through the task of gaining a thorough understanding of the proof process, and encourages inexperienced mathematicians to step up and learn how to think like a mathematician. A student's skills in critical analysis will develop and become more polished than previously conceived. Most significantly, Dr. Maddox has the unique approach of using analogy within his book to clarify abstract ideas and clearly demonstrate methods of mathematical precision. ... Read more

Reviews (1)

OK
Books for transitional mathematics courses are becoming more common and this book falls short of most of them.The writing style is nice and dry, but the topics in the book don't flow well.The author doesn't develop foundational material in the first three chapters (no axiomatic set theory or predicate calculus).Also, there isn't enough discussion of proof techniques (existencial and universal arguments, indirect proofs, etc.).The abstract algebra and analysis chapters aren't enough to give a full treatement, but are beyond the scope of good "bridge" course.

This book deserves its three stars because it does provide a very good selection of problems and appropriate hints.A student able to complete the problems does in fact demonstrate comprehension of the subject matter. ... Read more

Book Description

Book Description

Thisintroduction to rigorous mathematical logic is simple enough in both presentation and context for students of a wide range of ages and abilities. Starting with symbolizing sentences and sentential connectives, it proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables. Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Throughout the book, the authors emphasize the pervasive and important problem of translating English sentences into logical or mathematical symbolism. 1964 edition. Index.
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Reviews (1)

Good coverage, but lack of solutions weakens it.
Logic is one of the foundations of mathematics, making quality textbooks essential. This is one of the better ones, with descriptions followed by a large number of exercises. The basic strategy is to present the material in small sections, most of which are two pages or less in length. One unusual aspect of the book is that the authors chose to wait until chapter four to present truth tables. Traditionally the first topic in logic textbooks, leaving truth tables until later forces the reader to learn the operational meanings of the connectives. I approve of this pedagogical technique, undue reliance on truth tables can lead to the masking of some of the concepts of logic. Propositions and predicates are covered, with more ink spent on propositions that predicates and a simple set of axioms for addition is also presented
One negative point is that no solutions to the exercises are included. In my opinion, any book without solutions to some of the exercises is of reduced value as a textbook. Students work problems on their own and it is very important for them to get immediate feedback. This also reduces its' value as a text for self study.
I can recommend this book, the quality of the explanations and the number and detail of the exercises make that easy. However, the lack of solutions means that it can only receive my second highest recommendation.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Book Description

This is a softcover reprint of the English translation of 1968 of N. Bourbaki's, Théorie des Ensembles (1970).

Book Description

This is an introduction to, and survey of, the constructive approaches to pure mathematics. The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism. Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate. Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists. This account for non-specialists in these and other disciplines. ... Read more

Book Description

Reviews (1)

Not for beginners
Logic has an elegant simplicity to it that this author tries valiantly to capture, but fails. If one is already acquainted with propositional and predicate calculus, this is a joy to read for its elegance (something often attributed to Mates, but I think erroneously). This is a second-course, not a first encounter, text. But once the foundations of logical theory are laid, this is worthy of beholding the same concepts from a purely mathematical perspective. ... Read more

Book Description

Automata and natural language theory are topics lying at the heart of computer science. Both are linked to computational complexity and together, these disciplines help define the parameters of what constitutes a computer, the structure of programs, which problems are solvable by computers, and a range of other crucial aspects of the practice of computer science. In this important volume, two respected authors/editors in the field offer accessible, practice-oriented coverage of these issues with an emphasis on refining core problem solving skills. ... Read more

Reviews (1)

Pile of crap!
This book was terrible! It was very boring, overly-technical, and had spelling and grammar errors all over the place! Not very professional... If you are a self-teaching programmer I would recommend that you find a different book because you won't learn much from this one! ... Read more

Book Description

George Boole (1815-1864) is renowned as the first logician to apply algebraic methods to logic successfully. His Mathematical Analysis of Logic, first published in 1847, was the ground-breaking work that laid the foundations for what is known today as Boolean algebra and the propositional calculus. Written in response to the altercation between Sr. William Hamilton and Augustus de Morgan over the quantification of the predicate within syllogistic theory, its remarkable innovations led other logicians, among them William Stanley Jevons, John Venn, Charles Sanders Peirce, and Ernst Schrder, to refine and develop Boole's system. In turn, their efforts were incorporated by Alfred North Whitehead and Bertrand Russell into the monumental system of Principia Mathematica. In short, modern symbolic logic was founded in the pages of this book. ... Read more

Reviews (1)

What sould I do?
I'm 16 years old and I'm a boy and I'm from Switzerland, that's it ... Read more

Book Description

There are several approaches to attack hard problems. All have their merits, but also their limitations, and need a large body of theory as their basis. A number of books for each one exist: books on complexity theory, others on approximation algorithms, heuristic approaches, parametrized complexity, and yet others on randomized algorithms. This book discusses thoroughly all of the above approaches. And, amazingly, at the same time, does this in a style that makes the book accessible not only to theoreticians, but also to the non-specialist, to the student or teacher, and to the programmer. Do you think that mathematical rigor and accessibility contradict? Look at this book to find out that they do not, due to the admirable talent of the author to present his material in a clear and concise way, with the idea behind the approach spelled out explicitly, often with a revealing example. Reading this book is a beautiful experience and I can highly recommend it to anyone interested in learning how to solve hard problems. It is not just a condensed union of material from other books. Because it discusses the different approaches in depth, it has the chance to compare them in detail, and, most importantly, to highlight under what circumstances which approach might be worth exploring. No book on a single type of solution can do that, but this book does it in an absolutely fascinating way that can serve as a pattern for theory textbooks with a high level of generality. (Peter Widmayer) The second edition extends the part on the method of relaxation to linear programming with an emphasis on rounding, LP-duality, and primal-dual schema, and provides a self-contained and transparent presentation of the design of randomized algorithms for primality testing. ... Read more

Reviews (1)

Good Book for Advanced Computing
Algorithmics for Hard Problems addresses an important (if somewhat small) group of technical people who work exclusively on intractable problems. It takes a great deal of creativity and a large set of intellectual tools to make even modest progress on these problems. Prof. Hromkovic's book presents and organizes a big package of ideas and tools for approaching these problems. Even the last section, entitled "A Guide to Solving Hard Problems" addresses some of the practical issues of working on problems for which a completely satisfactory answer may be out of reach.

This book is especially helpful for software professionals solving difficult problems because it helps one to categorize and understand where the "pain points" can be found. In many software applications very hard problems are hidden within perfectly reasonable appearing and seemingly benign systems -- it's critical in these cases to correctly project the details of the problem onto a well defined set of basis tasks. ... Read more

Book Description

Carefully written introduction covers all areas of mainstream formal language theory, including such topics as operations on languages, context-sensitive languages, automata, decidability, syntax analysis, derivation languages (first complete discussion of this topic), and more. Numerous worked examples, problem exercises and elegant mathematical proofs for almost all theorems. References.
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Reviews (1)

theory of languages in math form
An introduction to theory of formal languages, with a lot of mathematics an no programming.
There are also some chapter on automata, decidability and complexity of computation, but not algorithms on how to parse a program with a computer.
Interesting, concise but I recommend to complete it with a book with some computer program. ... Read more

Book Description

Categories for the Working Mathematician provides an array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. The book then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterized by Beck's theorem. After considering a variety of applications, the book continues with the construction and expoitation of Kan extensions. This second edition includes a number of revisions and additions, including two new chapters on topics of active interest. One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories. ... Read more

Reviews (7)

A Classic
Well, let us think about this a little bit...You want to learn Category theory, whether for some course or just for the fun of it, and now where do you turn in order to learn the necessary concepts. If you are a mathematician and have some experience, then you turn to the masters, the originators of the given subject and read their work. Sure, being the founder of a given subject does not imply that you are a good expositor and hence are capable of revealing the necessary concepts for the beginner-allow me to inform that Mac Lane is indeed as good as an expositor as he was a mathematician. For any doubters, I point you to the only other text you should read on Category theory, namely, "Category Theory" by Horst Herrlich and compare this text with Mac Lane's. Aside from that, and with respect to the text, for most beginners or interested readers I would suggest the following outline: Read 1.1-6; 2.1-3 & 8 possibly 2.4; all of 3; as for 4 skip section 3; 5.1-5; all of 8. Then, dependent upon your desires and or focus as well as your mathematical ability, it should become obvious which of the remaining topics should be read. Finally, the only other source I would recommend for learning Category theory can be found on-line using the keyword 'Awodey'. Anyways, Enjoy and good luck.

You may not need this unless you major in category theory.
I entirely agree with the reviewer Lucas Wilman.
As a book by the creator of category theory, it has extensively incorpoated relevant items.
However I don't think this is a *must read" unless you major in the subject: you will seldom need more than what is covered in a typical homological algebra course.
My inmpression is this book should be entitled "Categories for the starting/working category theorists".

Classic and worth it
It is difficult to make understand what "is" category theory. Is it a foundational discipline? Is it a discipline studying homomorphisms between algebras? Is it nonsense? Well, in my opinion this book does not help in gaining this kind of understanding. But all the stuff I read which have been written with that purpose in mind did not have any success - perhaps because I am not a mathematician, or perhaps because some concepts in category theory are really too abstract for anyone to give "an intuition" of them (you still can with functors and natural transformations, but try with adjointness...). This said, I found the book wonderful: Every concept is presented neatly. I use it as a reference each time I want a clear and rigorous definition of a concept. Sometimes this rigour helped me in gaining the famous intuition behind the concept.

OK, but not great
This book is a fairly good introduction to the ideas of category theory by one of the creators of the field. Unfortunately, the book is sometimes sort of confusing, and doesn't give many as many examples as I would like. Category theory (while it has become a field in its own right), is really a way of thinking about mathematics. The way you learn a way of thinking is by working out examples & doing excersizes, but this book doesn't provide as many connections to other areas of math as it should.

I don't think that this book was really intended "for the working mathematician," but rather for someone with some independent interest in category theory.

Definitely a grad text
This book is extraordinarily well written. It covers the necessary topics in a concise, orderly manner. HOWEVER, it presumes a substantial amount of knowledges concerning various algebraic/abstract structures in the field of mathematics. If you already have had experience with such structures, and are simply looking to understand them from a different perspective - this is the book for you. However, if you have limited knowledge with regards to advanced math (ie - grad level math) then try the book 'Arrows, Structures and Functors: The Categorical Imperative' by Manes and Arbib. This introduces the reader gradually to simple algebraic structures, monoids, groups, metric spaces, topological spaces, and the categories that can be built around them. ... Read more

Book Description

Most probably written by a cloistered monk sometime during the 14th century, this Christian classic ponders the significance of contemplation and prayer. More than a discussion of the author's own spiritual experiences, it is a revelation of how all can share in them. One of the most realistic of mystical writers, he draws his illustrations of divine mercy from the homeliest incidents of friendship and parental love. Benedictine monk Laurence Freeman provides an enlightening introduction.
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Reviews (5)

Good writing, lousy edition
Although the work itself is magnificent, this particular edition of it is very disappointing. It looks nothing like what's pictured here (and I checked the link to make sure I ordered the right one). The one I received is an oversized purple volume. It appears that someone just xeroxed an earlier edition of the work and bound it together. Also this edition has almost no footnotes or translation, and some of the medieval text is a bit difficult to understand. I'd recommend the penguin edition.

Gem of Western Spirituality
A very interesting description of a contemplative discipline conveyed intimately from an unknown teacher to an unknown student. The text is a mystic exploration of the divine that rejects the use of the intellect and the imagination, preferring an inner knowing, like a numinous experience based on feeling, a gnosis of the heart. An important theme is the difference between the active and the contemplative personality; others include the awareness of self, death of the ego, and advice on leading the contemplative life. Although this writing is deeply embedded in Christian tradition and I disagree with many of these beliefs, I still admire the author's gentleness and sense of humor. All spiritually-minded people will benefit from a study of this almost poetic text. The most beautiful sentiment is this: "For it is not what you are nor what you have been that God regards with his most merciful eyes, but what you would like to be."

Breathtaking
This is a beautiful translation of one of the most exceptional surviving mystical works.

What words can capture the breadth of a mystical work that, five minutes into reading it, brings forth tears of ecstasy running down the reader's cheeks?

Whoever wrote these words was most certainly one close to God, and Underhill's translation is incredible.

Read it. The scope is breathtaking.

Oh Heavenly Father, I beg you: fill me with your Being!

Blessings,

Stephan Vrudny

A Mystic in the Mists
This very gently edited version of The Cloud of Unknowing is a must read for aspiring mystics or those who seem to be becoming mystics through little intent of their own. The language of this edition captivates by retaining many of the 14th century original phrases and is easily understood, despite antiquity, by any reader who already feels to be within God's mists.

The Essence of Spirituality
This beautiful, extraordinary and timeless book by an anonymous 14th century author is one of the greatest mystical treatises of any time in any religion. It is to be most warmly recommended to all true and sincere students of mysticism. It radiates the warmth of St. Francis de Sales, touches in a uniquely loving and gentle way on the sufferings on the soul immersed in the dark night of the spirit, offers guidance on ways of contemplation and the attainment of true humility, which, as the author asserts, is the prerequisite of self-knowledge preparing the way for the love of God which the purified soul must enter through the cloud of unknowing. This book is my steady companion along with Jean-Pierre de Caussade, Rumi and The Hidden Words of Baha'u'llah. ... Read more

Book Description

A growing interest in many-valued logic has developed over recent years, which to a large extent is based on applications, intended as well as already realised ones. These applications range from the field of computer science, e.g. in the areas of automated theorem proving, approximate reasoning, multi-agent systems, switching theory, and program verification, through the field of pure mathematics, e.g. in independence of consistency proofs, in generalised set theories, or in the theory of particular algebraic structures, into the fields of humanities, linguistics and philosophy.

This book provides coverage of many multi-valued logic from theoretical foundations to latest results and applications: including theorem proving, approximate reasoning, program verification and generalised set theories.It covers the main areas of actual research interest, surveys the most important results and indicates the lines of actual research tendencies.This is a unique study of many different fields of applications for multi-valued logic, ranging from applications in computer science, to mathematics, logic, and linguistics.The text is suitable for those with limited or extensive knowledge of classical logic, set theory and algebraic structures. ... Read more

Reviews (11)

Better approach to the subject for an intro
Typically books of this nature are shrouded in cryptic and terse (albeit mathematically precise) theory. This book reworks the presentation in a more understandable way (e.g. non-determinism is presented early and used often; the term automata is dropped for the more descriptive term finite machine). I used this text last fall (fall '99) for an intro course on automata and formal languages and I plan on using it again.

Worth every ounce of effort you put in
LOM is or ought to be a classic. For my money, it's the "Dragon Book" for undergrad/graduate teaching of a first course in automata theory.

This book was used as the textbook in the automata theory course that I took post-baccalaureate. It is probably the book most responsible for my decision to go to grad school. I was working almost full time during the term I took this class, and I remember getting up early just so I would have a chance to read the optional material, since evenings were for the required parts.

And yes, it was my first exposure to the subject; indeed to all of theoretical computer science, since I'd not yet taken an algorithms, discrete math, or programming languages course. Contrary to the frustration expressed in some other 1-star reviews (many from Chicago -- same class? same person?), I found the material clearly and systematically presented. The one change I think would benefit the book is a slightly more terse formalism, although what's given seems designed to aid the intuition of the learner.

Yes the exercises were hard. But most of the problems are also very beautiful, and the insight I gained by doing them fully justified the effort. Even several years past the last of my grad course work, my memory of the course that used this book remains one of the highpoints of my education.

As a basic reference work, I think LOM compares well to many other related texts. Sipser's more terse style is probably quite usable in the hands of a lecturer who can fill in the missing pieces, but I suspect it would be much harder to use on your own. In addition to the standard chapters on automata, formal languages, and computability (all extensive), there is a chapter each on recursion theory and intractability. Chapter 0, giving mathematical preliminaries, is *60 pages* long, and is the material I used to learn the prerequisite discrete mathematics.

refreshing
The distinctive feature of this text is that wherever possible, it develops the subject from the standpoint of [theoretic] machines rather than purely from algebraic constructions. This almost always makes the concepts more intuitive, just as the Turing machine is often considered more intuitive than studying first order logics and their models. I highly recommend this book, but in general the reader should be aware that the subject is so rich and complex that no single book can suffice. Books on automata, mathematical logic, and modern/general/universal algebra can come in handy.

It looks as if this book is already out of print; it is surprising that the publisher did not give this book an extended life given that Floyd is one of the authors and that this was his only book. This is Floyd of Floyd-Warshall shortest paths, Floyd-Steinberg dithering, Floyd-Hoare style verification, and though not named after him, he more or less can be considered the founder of parsing and backtracking algorithms.

not for weaklings
This is the most hardcore CS book I have ever set eyes upon. Topics covered in this book are postgraduate-level discussions of the type that allow us to make decisions about the computability of the most difficult problems that computers solve. If you can gain a clear understanding of concepts in this book you should easily be paid in excess of 100,000 annually. Learn and apply the techniques in this book and you might be in posession of the kind of mind that wins a Nobel prize.

Too bad to read this book
If I took a class using this book as textbook, I will... ... Read more

Book Description

The book addresses ways and means of organizing computations, highlighting the relationship between algorithms and the basic mechanisms and runtime structures necessary to execute them using machines. It completely abstracts from concrete programming languages and machine architectures, taking instead the lambda calculus as the basic programming and program execution model to design various abstract machines for its correct implementation.

The emphasis is on fully normalizing machines based on a full-fledged beta-reduction as an essential prerequisite for symbolic computations that treat functions and variables truly as first-class objects. Their weakly normalizing counterparts are shown to be functional abstract machines that sacrifice the flavors of full beta-reductions for decidedly simpler runtime structures and improved runtime efficiency. Further downgrading of the lambda calculus leads to classical imperative (von Neumann) machines that permit side-effecting operations on the runtime environment.

Book Description

Here's something to do at your next dinner party after the guests have wined and dined, tell them you're auctioning off a dollar--a plain U.S. one-dollar bill. But set the rules as follows: while the winner gets the dollar, the second-highest bidder must also pay the amount of his last bid, even though he gets nothing.

If you can get a single one-cent bid, chances are the auction will go to fifty cents. If someone bids fifty-one cents, bidding will almost certainly reach a dollar. If it goes to one dollar and one cent, the sky's the limit. No one wants to be the second-highest bidder. Afterwards the bidders say they spent so much because their opponent "went crazy," which is a little like saying "the fight started when he hit me back."

Everyone has gotten into situations that resembled the dollar auction--where they devoted way too much effort to something not worth the commitment, because the cost of giving up was worse.

Why do we act this way? How can we recognize these traps before we're in too deep? Is there such a thing as rational behavior, and if so, how do we use it to our advantage?

Hungarian mathematician Laszlo Mero introduces us to the basics of John von Neumann's game theory and shows how it illuminates such aspects of human psychology as altruism, competition, and politics. Mero covers such concepts as zero-sum games; Prisoner's Dilemma; the game of Chicken (played with cars in Rebel Without A Cause), where logic proves that the rational strategy is to be irrational; how to be kind to your lover through game theory; and when the Golden Rule works and when it leads to disaster.

Mero shows how game theory is applicable to fields ranging from physics to evolutionary biology, and explores the role of rational thinking in the context of real-life situations ranging from doorway etiquette to the nuclear arms race. He also explains how moral dilemmas arise; how to act rationally and ethically when they do; and how the intersection of rationality and irrationality inevitably becomes what we call "wisdom." This fascinating, urbane book shows us how we can better understand ethical behavior. ... Read more

Reviews (6)

Interesting book
The book is divided into three parts:

I. Introduces some concepts from game theory, psychology
II. Gives examples from fields of economics, physics, biology
III. Discusses rationality and irrationality.

I found the first two parts both helpful and easy to read. They weren't technical at all, there isn't a single formula in the book. However, the last part is very hard to read and understand. I had to re-read some sentences several times and am not sure if I followed the author correctly.

Overall, I would suggest reading the book - it won't take much time and you can learn such concepts as evolutionary stable strategies, mixed strategies. Also, presentation of some concepts from physics and economics is interesting.

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.

I actually got referred to this book by a website that discussed Game Theory and when I went to Amazon to purchase it, I was amused by the reviewers on Amazon that were horrified that the last of the book covered Meditation/Mysticism. Thats what absolutely convinced me to buy it. I am a scientist and I am practicing meditation, I see both sides.

Mero does a great job of showing how Game Theory can explain the Rational and also show how irrational we are. He practically predicts why the reviewers that didn't like the last parts of his book are the way they are, which of course made them say the things they say in the review!

As the famous ad says, Just do it.

For crying out loud, NEVER play the dollar auction!
In this book, Mero adds some flesh to game theory, explaining why so many people become entranced with how it sets up economic and social choices as toy puzzles, in which people are trying to get the best possible outcome.

In particular, he explains some of the more infamous non-zero-sum games: the dollar auction, in which a dollar bill is auctioned off, but with a twist - both the highest bidder and the second-highest bidder must pay their bids, but only the highest bidder gets the dollar; and the prisoner's dilemma (and its variants), in which cooperation is the best result for the 2 players combined, but there's the danger of being undermined by the other player and individually losing everything. The dollar auction is likened to an arms race (indeed, the U.S. won =that= particular dollar auction against the Soviets), where what is being bought is military supremacy between superpowers. The prisoner's dilemma can be likened to a situation like a lane closure on a busy highway: if one merges in turn, and everybody else does, traffic keeps flowing somewhat; however, if only one person zooms ahead and merges ahead of where they should in fairness, the traffic can still go on fine and that one zoomer gets a benefit over the other drivers. But if =everybody= tries to cut in line... traffic clog.

It is true that trying to extend game theory to morality is a tricky business, and as another reviewer has put it, the book gets downright embarrassing towards the end. However, I am a math teacher, and have used ideas from the book to put more =oomph= in my classes on game theory to gifted teenagers. There isn't much in the way of math in here, but plenty of rational thinking and can lead a little light on to why game theory research has led to the winning of a Nobel Prize in Economics (or two).

An interesting book gets very, very strange.
Mero's book is an interesting read, and is a very good non-mathematical review of the basics of game theory and its relationship to human ethics. I suspect that its translation from Hungarian led to its sometimes strange choices of words, but on the whole, it is a good way to get a sense of the surprising generality and importance of the subject.

Unfortunately, the book is marred by Mero's expansion of game theory into a "theory of everything". By the last chapter, he has gone from straightforward applied mathematics into fuzzy-headed mysticism. At this point, the book has become more embarassing than interesting. My suggestion is to read the first half of tbe book and forget the rest!

A very well written and important book
I found this book compelling: an understandable, non-dogmatic, open-minded, non-technical yet scientific look at how mathematical game theory might be applied by someone trying to behave morally in a universe of limited resources. The concepts are clearly stated without jargon in such a way that people with little aptitude or patience for mathematics will remain interested throughout. An important book for anyone interested in morality at any level. ... Read more