Reviews (4)

All you need and more
Like the other books of the Schaum's Outline series, "Theory and Problems of Set Theory and Related Topics" provides you with hundreds of problems, which have been worked out step by step, to show you how the theory works.This is similar to a professor in college showing you the problems on the board, but these don't get erased and are much more legible.

In addition to the solved problems, there are supplementary problems to test your understanding of the material.At the end of each chapter, the answers are printed for you to check your work.

The discussions of set theory include definitions to ensure that you are familiar with the lingo.There are also plenty of examples to illustrate the author's meaning.With all the information on set theory, there are also chapters on quantifiers, boolean algebra, logical reasoning, and the algebra of propositions.

This book provides some good assistance in learning set theory.

Provides essential knowledge to e-commerce security pros
This is an excellent introductory text on set theory.It provides a fast and thorough coverage of the basics in an economical, if terse, style.If you diligently go through this short 200 page book you will have a solid grasp of set theory that you can immediately use on the job.

I am an IT consultant so, although this book has a much wider audience, I am focusing my review on how this book applies to IT.Because many e-commerce systems are being designed using role-based access controls that are independent of any specific application it is essential that the designers have a thorough understanding of set theory. As such, I think this book is an indispensable quick reference guide for IT security specialists, system architects and QA team members who need to brush op on the subject. These folks design, validate and verify systems that require strong security from within and outside of their organizations.This type of work requires a provably correct design that can only be approached with set theory.Moreover, the testing of the design, as implemented, also requires an understanding of this subject in order to develop a viable test strategy and produce test cases.Given the fact that most IT professionals are not versed in set theory, this book is one of the quickest ways to get up-to-speed.

Among the highlights of this book are: the paradoxes in set theory, which is "must reading" regardless of how you intend to use the knowledge and skills that this book provides, algebra of propositions and logical reasoning.

This book is clearly written, however, the material requires a strong dedication to learning because the subject is complex and dry. No author, regardless of his or her writing skills can change that.If you need to understand set theory and don't know where to begin, this book is one of the best starting points I have come across.

Isbn: 0486634159
Sales Rank: 308489
Subjects:  1. Geometry    2. Geometry - General    3. Mathematics    4. Science/Mathematics    5. Mathematics / General   

Reviews (2)

Handy reference
This book is useful for anyone doing maths at college. It summarizes just about everything and is good as a quick reference to refresh your memory on something. (It is also fun just to page through, and look at all the wierd things that mathematicians have come up with). Another advantage is that if you are studying a section in maths that you are finding a little confusing or hard to grasp, the entry in this dictionary will give the important points. But don't expect too much detail, which I think is the main drawback of the book.

Reviews (12)

too many wrong solutions
too many wrong solutions and typo~~~~
I don't recommend ppl buy this study guide!!!
This book confused ppl even more!!

Good for all science/Linear Algebra courses and exams
This study aid provides information about all possible content on linear algebra courses. I passed all algebra tests with the help of this and other nessecary books. Everyone should have material such as this piece. Not only do I reccomend this book, but it also works for everything possible. The guidelines in this book help you know the subject, and this book is easy and hard. For excellent knowledge of mathematics, natural sciences, computer science, etc. you can count on this.

Editorial Review

Reviews (8)

a weaker schaum's outline
Pitched to a weakish high school math background. Scant treatment of continuous random variables is one of several reasons this fails to achieve the comprehensiveness of other schaum's outlines. There may be courses where this is a good study aid (e.g. probability for poets), but it's a poor reference to basic probability for someone with a stronger math background.

Like other schaum's it is riddled with typographical errors and there is no errata. Eg. Appearing twice:

Correlation(X,Y) = Cov(X,Y) / E(X)E(Y).

Even at their worst, I still find the schaum's guides (this one included) better than a typical textbook with the same scope. This one, however, the problem is scope.

Better than most porbability texts
I believe this book is excellent, but I would supplement it with a book on probability (perhaps A First Course in Probability by Ross).It's introductory, easily-read, and offers theoretical foundations for most everything in the problems.

Too basic - non standard
I picked this book without realizing that it is meant for high school students. Thus the book is very basic and really only goes into Gaussian distributions.

My real complaint with this book is the notations used to represent some equations are non-standard. As well it is quite heavy on the mathematical shortcuts like
S={x:x is N C Z C R} which really means S is equal to x, such that x is a natural/positive integer contained in all integers positive or negative which is contained in all real numbers. (Whew!) The book seems to do this often. (trying to impress rather than teach?)

I felt its definition of random variables was poor and quite vague, and what little time is spent on them is single discrete random variables. Also the mathematical shortcuts are a nuisance since the non-standard notation being used, I found myself going back and wonder what capital Gamma and Phi were defined as.

There are millions of (easy) examples in this book, and as a graduate student this book had little to offer me except some frustration. I gave it 3 stars since I think that when I was in high school, the mathematical notation would have intimidated me, but it does have some useful examples. (I most likely would have given it 1 star back then)

If you are looking for a better book (Schaums) than this, try
Schaum's Outline of Probability, Random Variables, and Random Processes. This book covers everything up to multiple R.V's, and random processes, moment gen functions etc etc.

Reviews (3)

Good text reference
If you need as few texts as possible that cover all the maths you need for engineering purposes, this book is a must buy.Adding Kreyzig's Advanced Engineering Mathematics and maybe a Statistics text to it and you should have no problems with 99% of the maths for the rest of your life (unless you want to do substantial computer program design, in which a discrete maths text may help).

A few problems though: a proper treatment on sequences and series is absent, l'Hopital's rule is not discussed either (both presumably because they are too pure maths for engineering, but some advanced engineering maths courses do use them), and there are some topics (like Simpson's rule) that are only discussed in problems sections but not in the main text.

Very sloppy textbook
One big disappointment about the book is that, not only are few solutions to problems provided, there are a relatively large number of errors in the solutions which are presented. This is very frustrating to students trying to learn this material. Given that the text is in its second edition, this is really not understandable.

The book is surprisingly inexpensive, however. Perhaps it cost less to publish since proofreaders were not utilized.

Remarkably Clear And Comprehensive Mathematics Primer
Whether your personal focus is upon mathematics, engineering, physics, or even computer science or the arts, this work provides a superb instructional foundation for applied mathematics.

Beginning withdifferentiation and integration, the text continues on its mathematicaljourney, taking the reader through complex numbers, linear algebra,differential equations, even LaPlace transforms and Fourier series. It thenends with overview chapters on such varied topics as graph theory, settheory, boolean algebra, probability and statistics. In addition there is asection devoted to using symbolic computing with applications such asMathematica, which are essential to anyone interested in learning or usingmathematics today.

The overall look of the book is exquisite. Thetypefaces, equations and graphs are a pleasure to the eye (even as theygrow substantially in complexity). The prose discusses the subject matterwith rigor, yet is easy to read and guides gradually andcarefully.

Anyone wishing to review the fundamentals of mathematics or tofurther the education started through school will find this book to be ajoy to go through. Solutions to many exercises are provided in anappendix.

The softcover edition is highly durable. Upon completion ofevery chapter the reader will have substantial expertise in or exposure tomajor branches and topics of mathematics. For this wealth of information tohave so low a cost is remarkable.

Reviews (2)

Schaum's Outline of Operations Research
i found this book to be a very helpfull tool a long with my text book.it excels in its simplicity and a wide varity of examples and solved problems written in plain english.
hope you like it too.
thank you.
M. Madain.

Reviews (4)

Not a 5 star!
I waited several weeks for this book to arrive, and was disappointed to see that its 'just another maths book', nothing special. I didn't find this book any easier to understand than any other maths book, I ended up refering to my other book more.

Too Simple
I bought this book to help me survive my Discrete Math class, and ended up thoroughly disappointed. It's just too simple to be of any use. The examples it goes over are the obvious ones I've never had any trouble with, but more involved problems or complex proofs are nowhere to be found in the text. In the end I ended up relying on my course text much more than this, and I already had that book to begin with.

Reviews (2)

A clear exposition of Shannon's results by a great mathemati
A Y Khinchin was one of the great mathematicians of the first half of the twentieth century. His name is is already well-known to students of probability theory along with A N Kolmogorov and others from the host of important theorems, inequalites, constants named after them.He was also famous as a teacher and communicator. The books he wrote on Mathematical Foundations of InformationTheory, Statistical Mechanics and Quantum Statistics are still in print in English translations, published by Dover.Like William Feller and Richard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

In his "Mathematical Foundations" books Khinchin develops a sound mathematical structure for the subject under discussion based on the modern theory of probability. His primary reason for doing this is the lack of mathematically rigorous presentation in many textbooks on these subjects.

This book contains two papers written by Khinchin on the concept of entropy in probability theory and Shannon's first and secondtheorems in information theory - with detailed modern proofs. Like all Khinchin's books, this one is very readable. And unlike many recent books on this subject the price is very cheap. Two minor complaints are: lack of an index, and typesetting could beimproved.

Reviews (10)

For the history buff
There are at many academic pathways to Godel's Incompleteness theorems: philosophy, computer science, and mathematics, to name a few. What does this presentation of the results - among the many others available - have to offer the wide readership it is likely to attract?

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

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

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

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

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

First, the introduction does a poor job in explicating the theory. I suppose it gives you the basic idea, but this is hardly the first account of the theory one should read. Brathwaite does not connect all of the dots, and it will take a long time to figure out how the proof works from his intro, if you can do it all. (And that's not a challenge or insult; it simply isn't that well written.)

Second, forget about wading through Godel's proof on your own. The reviewer who claimed to do so with two years of algebra and a really good dictionary is simply lying. You do not wade through difficult theorems in mathematical logic without the appropriate tools. And the appropriate tools include having done similar but simpler proofs on your own and having a solid background in mathematical logic. Without this background, it doesn't matter whether you have the ability to be a mathematics professor at Princeton or place top five in the Putnam - you simply will not understand the proof in a rigorous manner. By all means, take a look at it to get a general feel for what's going on, but if you want a semi-technical account read Smullyan's "Godel's Incompleteness Theorems."

Third, as one reviewer pointed out, there are multiple errors in this printing of the proof. This makes what was a tall task virtually impossible.

So what did Godel do that was so interesting?
He proved that there were certain arithmetical statements about whole numbers that were not provable but true. (This was important because it shattered the widely held belief that if you stated a problem in mathematics clearly enough you would be able to determine whether it was true or false. Godel showed this isn't always the case. As an aside, simpler mathematical systems have been shown complete; that is to say, they can answer any well formed question.)
So, how can something be true but unprovable?
The sentence Godel constructed said this, more or less: I am not provable. This statement, if true, is not provable. If it is provable it's false, and correct systems (systems that do not prove false statements) cannot prove false statements. Therefore, it must not be provable. But then it's saying something true, and thus it's true but unprovable. Now, I'm simplifying and being sloppy, and you need to know about the difference between mathematical statements and metamathematical statements, but in a nutshell that's the thrust of his first theorem.

The other interesting aspect of his proof is that he constructed a statement that referred to itself indirectly. Russell, in Principia Mathematica - the work that contains the arithmetical system that served as the model for the arithmetical system in Godel's proof - created a "Theory of Types" which did not allow statements to mention themselves. But the sentence "I am not provable" references itself so it would seem that I've erred. But in fact I haven't; I just didn't fully explain how that sentence worked. (I know you were worried, if for just an instant.) Where was I . . . Godel created a sentence which referred to itself indirectly. The sentenced said, "Sentences with such and such characteristics are unprovable." It so happened that a sentence with such characteristics was itself. Thus, it referred to itself, but only indirectly and not in violation of the "Theory of Types."

All of my blathering, I hope, has impressed on you . . .
1) That this proof is worth understanding.
2) That you shouldn't believe anyone who tells you they worked through and understood the proof without having a signficant background in mathematical logic and the history of the proof. If you don't understand certain basic features of Principia Mathematica you're not going to grasp fully his proof.
3) That you should get an introductory account. Nagle's "Godel's Proof" is excellent and easy to understand. Smullyan's "Godel's Incompleteness Theorems" is more difficult, but not impossible and amounts to what would serve as the textbook of a solid mathematical logic course or two at an elite university.
4) That you shouldn't buy this work if you're hoping to work through his proof, unless of course you have the requisite training. Brain power is not enough.

Reviews (4)

If you actually like number theory...
...stay away from this book.This book was required for an upper-division number theory course I took, and the teacher used it for one of our eight assignments.I assume he did this because there are few worthwhile problems in the book.The author spends too much time explaining the history behind the theorems and too little time discussing applications.He proves the theorems but the proofs make no sense to someone learning just from the book.The teacher for my course explained all of the theorems in ways that actually made sense and were easily reproducable.When studying I never looked at this book and instead used Nathanson's GTM for number theory.If you only care about the history of number theory, this book has some redeeming quality.Otherwise, stay away.Dover math books tend to be inferior to texts like UTMs and GTMs, as evidenced by the latter costing literally ten times as much sometimes.

Good for self teaching
I am currently working through this book, and I really like the format, which is usually a small, digestible chapter followed by a set of exercises (usually 10 or so).Quite a wide variety of topics are covered including congruences, primitive roots, analysis of the number theoretic functions (e.g., the number of primes below x), and a little on diophantine approximations and continued fractions.Nothing post-calculus is used in the book except for some algebraic structures such as fields and rings, however, they are fully explained at the beginning of the book.(And some previous acquaintance with these would probably be good.)The exercises are especially good, being not too easy and not too hard.In response to the review below, to actually understand math like this you must be willing to do some work yourself.If you are looking to sit back in your easy chair and be entertained, then you should buy a book on the history of number theory, not a textbook.

Reviews (7)

It depends...
Visually, the quality of the print is not satisfactory. Formulas are not easy to read, especially that nowadays we're used to professional mathematical typesetting.

About the content, I must say that it may be appropriate for an engineers perspective. Myself, trained in the mathematical rigor, found that it doesn't explain things well. It's just a recopilation of formulas and recipes that only makes sense if you've seen this before.

If you want to learn this stuff from scratch, I would recomend finding another book.

An outstanding tutorial reference
I love this book. I've owned three copies of it over the years and I can honestly say that I would not have achieved the final class of degree in Physics that I did without it.

The learning curve is very gentle - really nothing is assumed about the reader's background beyond basic integral and differential calculus. The concepts of vectors are introduced one by one, and the book builds logically towards its final stages (introductory tensor analysis) via, inter alia, dot and cross products, partial differential operators on vector spaces (grad, div, curl, Laplacian etc.), line and surface integrals (along with vital allied therorems such as Stokes' and Green's theorems), and general theory of curvilinear coordinate systems (in which the differential operators are refined and generalised).

This book is absolutely ideal for an undergraduate course in Physics, Electronic Engineering or Vector Analysis.

Reviews (4)

Great Introduction
This is a text for Real Analysis at the Junior Level (American university level).It goes to extreme lengths to make analysis understandable to people who have no prior exposure.The organization is good.Completeness is introduced early as (the "piggy in the middle").Proofs are written in detail with fill-in-the-blank spots to force the reader to follow the argument.It has good exercises making it an easy book to teach out of.Excellent for the absolute beginner.Good candidate for the classroom.

Basic Real Analysis unleashed
Bryant builds the basic concepts of a first course in mathematical analysis upon the notion of numerical sequences. This approach gives an unified vision and amazing insights. Infinite series, limits, derivatives, Riemann integral are studied in an integrated vision. Clear ideas, illustrations and humor are found across all its pages. Good and illuminating exercises, too. An excellent introduction to basic real analysis.

Exposes Mathematical Analysis Without Set Theory Background
Mathematical analysis is a refinement of calculus, and a pathway into further branches of mathematics, including topology and advanced topics in algebra. Analysis, however, may not seem to be at all related to calculus at its initial stages. An introductory course on analysis can render an unprepared student, even with experience in other branches of mathematics, perplexed and challenged to an extreme. Only later in the analysis course are even the most basic topics of calculus introduced.

One of the most important considerations prior to taking an analysis course is the level of background and understanding of mathematical logic. Set theory, a branch of mathematical logic, is in fact the basis of calculus as well. Due to an emphasis upon computations, however, the highest grades in calculus are possible without understanding, or even knowing of, this underlying foundation.

This work is unique among those introducing analysis, in that it does not require a background in set theory. It in fact teaches numerous fundamental concepts of set theory, without stating that it is doing so. Examples provided are based on daily concrete experience, yet are altered for purposes of mathematical instruction. These descriptions are sufficiently general as to prepare the reader for when formal set theory is introduced in more rigorous textbooks.

In addition to being an extremely readable and accessible work,solutions and hints are provided for every review question for every section of the book. This is in stark contrast to textbooks on the subject, which, while costing several times more, are typically designed for a classroom setting, and so leave all questions unanswered. This self-testing of the understanding of each section is crucial for subject matter requiring such attention to detail and precision.

The numerous illustrations throughout the book are rendered clearly and with instructional purpose, yet are often drawn by hand, adding to the sense of familiarity with the author. All of the basic subject matter for a course on analysis is provided, yet has been specifically tailored for a reader in the stages of preparation, of review after completion, or one who is simply inquisitive as to what is required to comprehend analysis successfully.

The softcover edition is durable and portable, and the book remains in excellent condition through numerous readings, which it will almost certainly go through.

Reviews (5)

mm, not quite what I was hoping for...
This installment in the Schaum's outline series doesn't do it for me. I had reason to pick this one up simply out of curiousity (and out of habit, by usingother titles in this series for other lower-level undergraduate courses).It hardly stimulates any interest on the part ofthe reader; and the presentation is dry; not to mentionthat the selection of solved problems wasn't carefully thought out (most, I found, were proofs of some pretty standard results, which I would have rather not seen all over again). And the bare-hands computations weren't all that exciting,either.

Ploughing through a course text proper would better serve the serious student of mathematics. There areother well-written books devoted to solved problems in algebra (group/ring theory, for instance). It's just amatter of scoping them out carefully, and dishing out the money (for photocopies, even).

Not Great
Gallian's Contemporary Abstract Algebra is my required textbook for this course.I bought Schaum's Outline to supplement it and help me with some of the proofs.Instead, I found exactly the same worked-out examples from Gallian.My homework problems I was having trouble with were also under the "extra" problems section which does not have solutions.Basically this book was useless to me.If you aren't already using Gallian, that's the book you should get;this one would be okay if you want to save a few bucks.

Very well explained
I think this book would be good by itself in picking up algebraic theories and methods.Each section is well explained and the sample problems take you through the process step-by-step.

Reviews (12)

an extra dimension
This book is mainly concerned with exactly what the title says and I have been searching for a book like this for a very long time. because for one, it provides a very detailed explanation of topics that are intersting in the realm of physics. Such as the fourth dimension. it is very visual and explains things in a way that I can understand. I also like this book because it doesn't spend half the book telling you about which scientist hated the other scientist, Or the entire biography of Dr. Planck before they tell me what the planck length is

Good but Missing a Few Things
I haven't completely read this book, but I've read several like it. I want to point out some things that other reviewers haven't touched on. There is no index to the Dover edition. Maybe the original one had an index. That automatically knocks off one star in any book rating I give. It has some pretty sturdy exercises at the end of each chapter. There are no answers in the book. That's OK though. One can get some additional sense of the subject by looking at the questions. There is a very good annotated bibliography at the end of the book. It is not tied into page numbers, but I get the feeling the order of the list and their reference in the book are in the same order. There's good and bad news about the list. He makes many of these books sound very appealing, but many are long out of print. Rucker's book was produced around 1975.

There are times when I wish the author would have pressed a little harder one some seemingly simple points. Maybe by giving an alternative view. For example, early on in the book he talks about a flatlander being inside a balloon as he expands the balloon from the inside. Suddenly the flatlander is on the outside. Maybe it's me, but how that happens is not clear. I've found other such passages. However, a studious reader will find the topics interesting. The price is certainly right.

Weird in all the right ways
I really enjoy Rudy Rucker's nonfiction, and some of his fiction too (_White Light_ is great). He's very good at presenting mind-blowingly cool ideas in accessible expository prose, and he knows _just_ when to throw in the bombs.

This particular book is published by Dover, and it's not one of their usual reprints; it was _originally_ published by Dover. (In 1977, but the geometry of spacetime hasn't changed much since then.) It's an exploration of just what the title says: the geometry of the four-dimensional spacetime that the theory of relativity says is Really Out There.

Well, this is a good book on the subject, but you can get others (although one of the best -- Cornelius Lanczos's delightful _Space Through the Ages_ -- has long been out of print). What's coolest about this one is that Rudy Rucker wrote it.

Which means you get those little bombs thrown in at all the right places. Of course Rucker gives you what any competent mathematician will give you -- a sound introductory presentation of the mathematics of 4D spacetime and relativity theory, which are weird enough if you haven't encountered them before (and maybe even if you have) -- but he doesn't stop there. You also get an argument that the apparent passage of time is an illusion, and a little speculation about how this might tie in with the Many-Worlds Interpretation of quantum mechanics. And even that isn't all: you get a suggestion that it's possible to _develop a spacetime consciousness_ via some sort of meditation techniques or mystical insight, together with an entry in the annotated bibliography referring you (cautiously) to Robert A. Monroe's _Journeys Out of the Body_, whose experiments Rucker himself has tried.

It's like Raymond Smullyan on acid, if you know what I mean. But honest, it really does make sense. And it really will knock your mind loose from your brain even without the use of chemical aids.

Reviews (13)

21st Century Science
Flakes brings together some of the best science of the late 20th Century that will have tremendous impacts far into this century.

Interesting Topics
Good book, [X] bucks is a bargain, it's worth twice that easily.

Favorite things about this book
Covers L-systems and also gives the rules for how to make some interesting plants. Also this book touches on some aspects of AI like game-trees and neural nets. The author discuses "boids" and self-organization with autonomous agents that act together, and shows simulations of ants and a flock of birds using this concept.

Real "How Nature Works". Already is "Legend in the Making."
I recently became interested a lot in Nature. Especially, being someone in the field of Computer Science, the computational aspect. And this book is by far one of my favourite among all the "How Nature Works" kind of books I've read.

This Computational Beauty of Nature (CBofN) covered a lot of topics. Ranged from brief introduction to Computation Theory, Fractals, Chaos, Complexity, Adaptation. (See the Table of Content for more details).

All topics are written in surprisingly clear and very understandable manner. With as little Math as possible. (From my opinion, these topics cannot be completely understood without Mathematics -- The Language of Nature). Therefore, it is also accessible to layperson.

This book does not, however, go so deep into each subject. (You won't expect it to do that with its less-than 500 pages, don't you? :-) Instead, it does give nice backgrounds, fundamental knowledge, and important ideas for each. So, if you are interesting in any of the subjects presented here, you can go on to the more specialized books on your own.

One of the nicest feature of this book, which can hardly be found in other text, is that the it does show how things work together, where and why. For example, natural phenomena like adaptation, evolution, computation, and some other things else related to each other. How can one view this from that perspective, and vice versa. etc.

One other nice feature of this book is, you can really play with almost all concepts using a number of computer programs. All the programs are downloadable (with source code, under GNU license) from the book's homepage. So, you can reproduce almost all the figures from the book.

However, for one thing, the homepage address given in the book, in the edition/printing I have is incorrect. Maybe MIT Press had changed the structure of their website or something...

...you can still search for it using your favourite web-search engine.

About the website, all the good things are there as well, including errata. (Of course, Perfect things are very rare in Nature... So, books with some errors are ok. The thing that matter is the authors know it/admit it and tell the readers or not).

Conclusion: If you want to understand "How Nature Works" from the computational point of view. If you interested in Chaos theory, Fractals and Complexity. Then, make no mistake, you can't go wrong with this one. (And, get the hardcover edition, because you will read it, read it, read it again, and keep refering to it. So the paperback edition probably can't endure that :-)

I want to give it more stars if I only could. This book will always get the highest rating possible from me wherever and whenever I review it.

Reviews (4)

A real bargain by a true master
Rudolf Carnap was one of the greatest philosophers of the 20th century, and the only student of Frege's worth thinking about. But what a student!

This is his intro text, a doubtful first text, but full of insight for those who already know some logic. Carnap trained as a mathematician; surprisingly, his text is of value mainly for philosophers. For instance, this is the ONLY undergrad logic text I know that grapples with the intension-extension dichotomy, with the Carnap-Morris syntax-semantics-pragmatics trichotomy. Metatheory is nonexistent, and Carnap's notion of proof is emphatically too casual for my taste.
The book is dated. In its treatment of first order logic, Carnap is a bit too loyal to Principia Mathematica. His axioms are a bit pedantic, a bit inelegant for my taste. (I should admit here that I have devised a radically simpler axiom for the truth functors, and am working on a simplification of quantification theory. Even if my theory doesn't pan out, I still believe that UG and UI suffice for quantifiers.) You won't learn any natural deduction, truth trees, or Gentzen sequents here. You most definitely won't learn anything about recursion. But the exposition incorporates thoughout Carnap's greatest discovery: his formal theory of semantics. You will also learn more about the logic of relations than you will in any other undergrad text. You will be given an idea of the mathematical power of logic (infinity, continuity, numbers). You will even be introduced to the lambda calculus, Alonzo Church's greatest discovery. Carnap was comfortable with the notion of predicate like few logicians since.

Part II of the book is without parallel anywhere: an introduction to a very wide range of axiomatic theories, presented as interesting applications of modern formal logic. This is a wonderful reference for ZF set theory, Peano axioms, Tarski's axioms for the reals, the Hausdorff- Bohnenblust axioms for topology, axioms for geometry, space-time, and mirabile dictu, even mereology. Other texts present at most the first 2 items on this list.

good books
It ismy experience as a reader that good books are always books thathave lots of examples because they make our understanding easier.Therefore, if this math book has examples in it,then it must mean it isvery good.So I recommend that whenever you have a chance to see it that youbuy it.

Reviews (3)

No-nonsense survey of logic
This is an introduction to the main ideas and results of mathematical logic. It is primarily a text for non-logicians but it is still very serious. Practically everything is proved, and the proofs are carefully crafted and not too technical. It is heavy reading at times, but for a reader with a bit of mathematical background this is very valuable in contrast to the more typical logic-for-casual-readers books, such as for instance Gödel's Proof by Nagel & Newman, which are too chatty and trivial and don't really prove anything. On the other hand, this book is perhaps not chatty enough. The clear proofs and discussions of the main results are nicely done, but the discussions of historical background, motivation and context are very sketchy. As a complement for these aspects I would recommend Morris Kline's Mathematics - The Loss of Certainty.

Six Rigorous Lectures - Not for the Faint-Hearted
Although this book - What is Mathematical Logic? - is written in an informal and entertaining style, it is unlikely to appeal to a reader not familiar with predicate calculus, recursive functions, and set theory. Despite its innocuous title, this little book is surprisingly rigorous.

The six chapters are derived from a series of lectures given by the five authors - J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams - at Monash University and University of Melbourne in 1971.The lectures were substantially revised for publication.

Only the first chapter, a detailed historical survey of mathematical logic, can be readily appreciated by the non-mathematician. The remaining five chapters examine advanced topics in mathematical logic including the Godel-Henkin Completeness Theorem, Model Theory, Turing machines and recursive functions, Godel's Incompleteness Theorem, and advanced set theory.

Chapter 2 introduces the Godel-Henkin Completeness Theorem, a proof that predicate calculus is complete. Chapter 2 is not easy, but it is essential to acquire a reasonable familiarity with predicate calculus before moving forward.

Chapter 3 offers a detailed look at model theory, the study of relations between formal languages and the interpretation of formal languages. Topics include Predicate Calculus with Identity, the Compactness Theorem, and the Lowenheim-Skolem Theorems. I had substantial difficulty with the details, but I did gain a general understanding and appreciation for model theory.

Chapter 4 addressed in considerable detail a more familiar topic, Turing machines and recursive functions. The discussion concludes with a key proof: there is no algorithm which will enable us to decide, given any particular formula of predicate calculus, whether or not this particular formula is deducible from the axioms of predicate calculus.

Chapter 5 was a detailed examination of Godel's Incompleteness Theorem for formal systems that include arithmetic of the natural numbers. I had less difficulty with this topic as I had previously read Godel's Proof by E. Nagel and J. R. Newman. This chapter would very likely be tough going for a reader entirely new to Godel's exceeding complex and abstruse proof.

Chapter 6, titled Set Theory, might be better named Advanced Set Theory. I was entirely new to the Axiom of Choice and the Generalized Continuum Hypothesis.

I highly recommend this intriguing and lively look at mathematical logic to readers with some familiarity with this rather formidable subject. For readers new to mathematical logic, I suggest that the following books might be better starting points.

Foundations and Fundamental Concepts of Mathematics by Howard Eves is outstanding. The chapter titled Logic and Philosophy is an excellent introduction to mathematical logic.

The Advent of the Algorithm by David Berlinski is an eclectic, rather bizarre introduction to a complex mathematical topic. Although many reader reviewers aggressively criticize this book, I enjoyed puzzling my way through Berlinski's discursive discussions.

Godel's Proof by Ernest Nagel and James R. Newman offers a fascinating look at a mind boggling, incredibly complex, inventive mathematical proof.

Dense but readable
After a 10-page historical survey of logic from the 1850s through the 1960s, similarly brief chapters on Completeness, Model Theory, Recursion Theory, the Incompleteness Theorems, and Set Theory give an idea of what might be covered in an undergraduate course and the first several graduate courses in mathematical logic. (The last 5 pages of the book are an introduction to forcing arguments and a fairly detailed sketch of the consistency of not-GCH.)

Results are clearly and carefully stated; and while sketches of proofs have a hard time staying nontechnical and still meaningful, most such attempts are admirable.

Reviews (2)

Another Dover classic reprint at a bargain price.
Another classic reprintrom Dover at a reasonable price. Martin Davis is a very well-known worker in the area of logical foundations of computing.This book covers muchfascinating material and provides answers to some deep questions relating to the limits of computations. The material can be a little dry but worth the effort. The book is worth the price for the appendix which is a reprint of an article by Davis on the proof of the unsolvability of Hilbert's Tenth Problem.

Mapping the Outer Limits of Computation
The book introduces the theory of computability and non-computability tothe mathematically-comfortable.The theory of recursive functions providesentry to that theoretical territory at the limits of what is computable andwhat is solvable.The theory is relevant to important philosophicalquestions and also in the theory of computing and what is possible (andnever possible) by use of computing machines.

The result for philosophyis establishment of absolutely unsolvable problems and undecidablequestions, even ones that can be completely and precisely formulated usingrigorous logic.The result for computing is problems that are absolutelyunsolvable by use of a computer program.

So what problems aretheoretically solvable by a computer program?First, the Universal TuringMachine (UTM) is presented along with the famous demonstration that alluniversal computers are equivalent in the sense that any one of them can bemade to simulate any of the others, using a suitable representation.

So,if we establish that the computer we have at hand is a universal computer,we can be confident that, in principle, anything that any computer cancompute, this one can also.

The book goes on to address what evenuniversal computers can't do.The most well-known result incomputer-science circles is the unsolvability of the halting problem.Thatis, if the computer is powerful enough to be universal, one of itslimitations is the impossibility of an algorithm that will determinewhether any program for that machine will always terminate for all inputs. It is as if the price of universality is the inevitability of programs thatwon't finish, along with having no absolute way of telling whetherarbitrary given programs will finish or not.

Davis maps the boundarybetween the impossible (the unsolvable) and the merely inhumanly difficult(the computable).With that foundation, one can move on to other work thatintroduces what has been learned about computational complexity and how toapply the analysis of algorithms to finding computational methods that arepractical and no more complex than absolutely necessary.