Foundations of Higher Mathematics

Book Description

This text introduces students to basic techniques of writing proofs and acquaints them with some fundamental ideas. The authors assume that students using this text have already taken courses in which they developed the skill of using results and arguments that others have conceived. This text picks up where the others left off -- it develops the students' ability to think mathematically and to distinguish mathematical thinking from wishful thinking.

Customer Reviews:

A Pricey and Often Frustrating Introduction.......2005-07-17

I bought this book so I could self-teach myself a prerequisite course to an advanced mathematics requirement course. Therefore, I needed to learn the material as well as if I had taken a course using this book, although I did not have the requisite professor as a guide. That said, this book suffers from some obvious flaws.

As a transition book to higher mathematics courses, this book needs to accomplish a great deal. First and foremost, it needs to modify the apt math student from an elementary way of thinking about mathematics to a logical, adept, reflexive, adaptable way of thinking about mathematics. This is done through way of introducing the logic of certain mathematical foundations and then allowing the student to participate in proofs to verify the theorems. Most of the times the theorems are presented in an understandable way, and the authors do explain and illustrate many of them; however, the way they typed their proofs (and many of the practice problems use a demonstration proof as a guide), in clustered format, although often acceptable, was very hard to follow. Also, early in the book they explained the benefit of visual explanations of their concepts, so I found it frustrating when they sometimes (but almost seldom) visualize or explain the theorems in more mundane terms, especially when they do not even write the proof for the theorem in the book.

Maybe as an accompaniment to a course in foundational mathematics (what this book is designed for), this book can successfully supplement an instructor; however, as a guide to someone self-taught (especially for the expense of buying), this book could improve a great deal.

effective but not thorough.......2005-03-28

I am using this book as my text for my upper mathematics bridge class. The text is certainly useful, but often I am frustrated by the lack of clear explanations for theorems, relegating them as exercises for the reader. I am not saying that the author should spoon feed the reader, but what do we do when we get stuck? For what we pay for this book, it certainly has some improvements to make.

A great read to see what lays ahead.......2003-08-28

I read and worked the problems in this book during my break as I transferred from a community college to a 4-year university, and found it very helpful in introducing me to all the fancy terminology, notation and basic proof writing that I was intimidated by. I found the problems to be hard enough to be challenging, but also neither impossibly hard nor hinging on a silly trick.

If you are a eager HS student, or a curious college student, get this book and work the problems.

Great introduction to mathematics.......2001-09-21

I bought this book for a course in classical algebra. I found the book well explained and well done. It contains a lot of exercise and example of differents difficulty. It covers logic, set, relation, induction, function, combinatorial proofs, countable sets and uncountable sets, groups and some calculus. The book has a lot of subject in it and it make it very flexible. If you want to ontroduce yourself to mathematics, I would recommend this book if you want to spend some money.

An excellent introduction to mathematical logic!.......2000-06-01

This book provides an excellent introduction to mathematical logic, set theory, graph theory, number theory, and more -- everything which is "neat" in higher math.

I would strongly recommend this book before any proof-based math class. The authors explain methods of proofs very well, and give some principles universally important in mathematics -- Zermelo's thm., Dirichlet's prin., and such.

The exposition in this book is great. If this is your first exposure to, for instance, the proofs by induction, this probably provides an excellent description of what's going on and how it works, why it works.

Book Description

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets.

* Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses.
* Reduced mathematical rigour to fit the needs of undergraduate students

Customer Reviews:

John Wilson.......2007-07-24

Keen students may find if they study and parse both editions of Enderton's

Logic they may find much of interest. Getting to the root of a problem

can be of use in many situations. So best of luck.

Moderately difficult and very effective.......2006-07-19

This is the most clear book on intermediate level logic that is available. I have many of the logic books that are on its level, and this one is perfect. It covers the most important, difficult concepts in the easiest way possible. It is above all clear (though very terse). It is easier than Mendelson's text but, in my opinion, as it pertains to First Order Logic and Computability Theory, one learns no more through Mendelson's approach.

Perhaps its only problem is that it might be just a bit too difficult without an understanding, helpful instructor (or TA) to guide one through the exercises. At any rate an effective progression up to the book might entail: Patty's "Foundations of Higher Mathematics", to Klenk's "Understanding Symbolic Logic", to "Logic, Sets, and Recursion" by Causey. Only after equivalent material has been understood thoroughly can the more hardcore semantics and mathematics of Enderton's book be fully comprehended. And, gone at alone on one's free time such a progression might take up to 2.5 years, maybe more.

Readable but a bit rough.......2005-07-12

It tries to be a readable undergrad introduction and mostly succeeds. Explanations are generally not tight and memorable, proofs seem loose, there are sometimes gaps in the train of thought, and exercises often require a significant conceptual leap from the preceding text. It was particularly annoying the way he suddenly switched to Polish notation for a while and then just as suddenly dropped it, without any obvious benefit. However, it is more accessible than most mathematical logic texts. The main competition for this text would be Ebbinghaus, which I prefer. The benefits of Enderton over that book are that it covers a wider range of topics and has a lot more exercises.

Terrific Book.......2005-01-03

Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.

Still the best........2003-09-22

I review the classic FIRST EDITION. If you buy only one book on mathematical logic, get this one. It's by far the best logic book (see my other reviews) that is both 1)introductory and 2)sufficiently broad in scope and complete. The exposition is very clear and succinct- its suitable for beginners without getting wordy. Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading.

Besides being easy to learn from, it's also the most rigorous introductory book I've seen- a rare combination. The proofs are detailed and complete, instead of the usual hand-waving or leaving everything as an exercise for the reader. There are some weak points in it, but overall you're not going to find a better book. It requires a little more 'mathematical sophistication' than most intro books- but if you've had some logic in a computer science course, or a little combinatorics or abstract algebra you'll be more than ready. Familiarity with automata/computability theory will help you in a few of the sections. Although Enderton is very good, it always helps to get several books on a subject- I'd recommend you pick up cheap copies of Boolos & Jeffrey's _Computability and Logic_ and Smullyan's _First-order logic_ as supplements.

Here is the complete table of contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS . . . .1
Chapter One - SENTENTIAL LOGIC/ Informal Remarks on Formal Languages 14 /The Language of Sentential Logic 17/ Induction and Recursion 22/ Truth Assignments 30/ Unique Readability 39/ Sentential Connectives 44/ Switching Circuits 53/ Compactness and Effectiveness 58

Chapter Two - FIRST-ORDER LOGIC/ Preliminary Remarks 65/ First-Order Languages 67/ Truth and Models 79/ Unique Readability 97/ A Deductive Calculus 101/ Soundness and Completeness Theorems 124/ Models of Theories 140/ Interpretations between Theories 154/ Nonstandard Analysis 164

Chapter Three - UNDECIDABILITY/ Number Theory 174/ Natural Numbers with Successor 178/ Other Reducts of Number Theory 184/ A Subtheory of Number Theory 193/ Arithmetization of Syntax 217/ Incompleteness and Undecidability 227/ Applications to Set Theory 239/ Representing Exponentiation 245/ Recursive Functions 251

Chapter Four - SECOND-ORDER LOGIC/ Second-Order Languages 268/ Skolem Functions 274/ Many-Sorted Logic 277/ General Structures 281
Index 291

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions

Average customer rating:

no complaints
Intermediate Level
Short and to the point
Very well written book
Now I know how beautiful proofs can be

An Introduction to Mathematical Reasoning: Numbers, Sets and Functions
Peter J. Eccles
Manufacturer: Cambridge University Press
ProductGroup: Book
Binding: Paperback

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Proofs and Fundamentals: A First Course in Abstract Mathematics
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ASIN: 0521597188

Book Description

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

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Customer Reviews:

no complaints.......2007-09-30

In less than a week I had the book and I'm happy with my purchase

Intermediate Level.......2007-08-21

I'm biased. This is the sort of thing U.S. schools should require. Can't go wrong with this. This has 2X as much as "How to Read @ Do Proofs" (Daniel Solow). Wich is better? No Answr. I Luv em both. Former is more expository the other more mechanical. Go For It! Recomended.

Short and to the point.......2006-04-11

This book is excellent! It chapters are broken down into short sections and the content in each section is to the point! I also bought the book Proofs and Fundamentals by Ethan D. Bloch but found it to be long and drawn out. If you liked The Nuts and Bolts of Proofs by Antonella Cupillari then this book is for you!

Very well written book.......2004-09-30

I have a mathematics degree. Like most math majors, I struggled with proofs all through college. This book really has help me understand the art of writing proofs. The book is very well written and easy to read. This is just an awesome book!!!

Now I know how beautiful proofs can be.......2002-12-13

This book provides a nice introduction to mathematical reasoning and proofs. My intention on purchasing this book was to learn how to perform mathematical proofs. I believe it has achieved that purpose. The text is easy to follow and the author presents the work clearly.

Average customer rating:

Perhaps the best written written elementary book of logic
TIMELESS CORE HOLDING IN ANY LOGIC LIBRARY
I will always keep it as a reference

Introduction to Logic
Alfred Tarski
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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Introduction to Symbolic Logic and Its Applications
The Principles of Mathematics
Godel's Proof
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Axiomatic Set Theory

ASIN: 048628462X

Book Description

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

Customer Reviews:

Perhaps the best written written elementary book of logic.......2006-07-11

I bought the book just because my teacher of elementary philosophy in the university respected Tarski as a master of formal logic. It took me 26 years to get this book in my hands. What makes Tarski unique is, that he was a great logician and a great teacher, too.

I belive that there still are no better guide for a student who wants to understand logic, not just try to remember basic rules of it. The beauty of logic has never been exposed in a better way.

The fifth star was spared to a new, annotated edition of this classic among the field of logic. I hope I can find one some day.

TIMELESS CORE HOLDING IN ANY LOGIC LIBRARY.......2004-03-14

This timeless classic by one of the five greatest logicians of all time should be owned by anyone who cares about logic - especially at this illogically low price. The Greek philosopher Aristotle (384-322 BCE), the English mathematician George Boole (1815-1864), the German mathematician Gottlob Frege (1848-1925), the Austrian-American mathematician Kurt Gödel and the Polish mathematician Alfred Tarski (1901-1983) are considered to be the five greatest logicians of history. Today it is difficult to appreciate the astounding permanence of what is accomplished in the works of Aristotle, Boole, and Frege without seeing their ideas surviving in the work of a modern master. Of the two modern master logicians Tarski is by far the most suitable for this purpose since he was by far the one most interested in the articulation of the conceptual basis of logic, he was by far the one most interested in history and philosophy of logic, and he was the only one to write an introductory book attempting to explain his perspective in accessible terms. This book, together with Aristotle's Prior Analytics and Boole's Laws of Thought, should form the core of any logic library. All three are still in print and available in inexpensive paperback editions. Hackett publishes an excellent up-to-date translation of Prior Analytics by Robin Smith and Prometheus recently reprinted Laws of Thought with an introduction by John Corcoran.- Frango Nabrasa.

I will always keep it as a reference.......2002-07-14

This is one of the classic introductory mathematics books. When I was learning logic, I relied on it heavily, although it was not the text for the course. Over my years as a teacher, I have consulted it often and when I was working on a recent book on logic, there were very few days when I did not open it in search of an idea or clarification.
All of the basics of logic are covered in one of the most readable texts I have ever opened. Exercises are given at the end of each chapter, although no solutions are included. This is one of those books that will always be on my key shelves of reference works and it will no doubt receive a great deal of use.

Foundations and Fundamental Concepts of Mathematics

Average customer rating:

No India China
Good Introduction to Mathematics, Historically and Philosophically
Fundamental yes.... but is a very demanding introduction
india and china
The author lacks basic knowledge about mathematical history

Foundations and Fundamental Concepts of Mathematics
Howard Eves
Manufacturer: Dover Publications
ProductGroup: Book
Binding: Paperback

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ASIN: 048669609X

Book Description

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

Customer Reviews:

No India China.......2007-05-18

The reviewer who said see Page 2 is really hitting below the belt. Page 2 says nothing can be said about indian and chinese maths as they wrote on perishable items. This is a 'mention' about india and china on maths? I am ok with all but surely not this much bias..not needed... euro centric wins hands down .. the giants of maths come from europe and no grudging it... but be fair to some extent....

Good Introduction to Mathematics, Historically and Philosophically.......2006-06-29

Though originally published in 1958, Howard Eves' book was a completely new find for me. Fortunately this classic text has found extended life through Dover Publications, which is making many great older volumes available for newer generations. I am not a mathematician by vocation or training and I am usually only interested in more philosophically focused books concerning logic or meta-logical issues. But I found this book extremely enlightening, showing the interrelations of (what had previously been to my mind) unrelated historical streams of thought. In the following I will give a brief summary and point out some of, what I consider, the highlights of Eves' volume.

In the first chapter Eves gives a brief but good historical overview of mathematics in ancient civilizations. He deals with the early Egyptians, Babylonians, and of course the Greeks. This approach naturally segues into an emphasis upon Euclid and his monumental Elements. Eves pays particular attention to Euclid's methodology, the material axiomatic, discussing its origin and ensuing problems.

Other texts that I have read on the subject of mathematical logic tend to give quite a bit of time to Euclid's fifth (or parallel) postulate. Not until reading Eves' book have I understood why though. Euclid's fifth postulate has the appearance of being quite different from the first four; any non-mathematician can perceive this fact from a mere browsing of the first several postulates. Euclid needed this fifth statement for his geometry; and since he could never prove it as a theorem, he made it a postulate in his system. Eves notes that a good deal of mathematical history is devoted to this same exact project that Euclid failed to accomplish. "It would be difficult to estimate the number of attempts that have been made, throughout the centuries to deduce Euclid's fifth postulate as a consequence of the other Euclidean assumptions, either explicitly stated or tacitly implied. All these attempts ended unsuccessfully, and most of them were sooner or later shown to rest on an assumption equivalent to the postulate itself" (53). Several notable mathematicians (Gauss for instance) suspected the fifth postulate was independent of Euclid's system. But these results were considered far too radical or ridiculous in their time. Eventually certain mathematicians (Saccheri, Lambert, and Legendre) did go forward with geometrical systems that excluded the fifth postulate, showing its independence; and thus non-Euclidean geometry was born. The far greater importance of this though was that geometry had been liberated from its traditional mold. What had been previously considered absolute and intuitive was shown not to be the case.

Eves notes that another shortcoming of Euclid's system, besides the independence of his fifth postulate, was that some of his basic definitions proved to be circular. This problem eventually showed that some terms in a mathematical system had to be conceived of as primitive or implicitly defined. Unlike Euclid whose use of diagrams pitted him towards unconsciously making numerous hidden assumptions, mathematicians such as Pasch, Peano, and Pieri are credited with trying to make geometry more formalistic and thus protecting it from any such intuitions. Eves claims: "Here we have the mathematician's ultimate cloak of protection from the pitfall of overfamiliarity with his subject matter" (81).

Eves spends a couple of chapters on the further formal axiomatizations of geometry and also of algebra as they progressed to modern times. Historically these developments are important in that they shape the modern axiomatic approach to mathematics in general. "The discovery of non-Euclidean geometry and, not long after, of non-commutative algebra led to a deeper study and refinement of axiomatic procedure; thus, from the material axiomatics of the ancient Greeks evolved the formal axiomatics of the twentieth century" (147). Eves makes an important observation here. The formalization of axiomatic systems brought a key distinction to the table that had never been made, at least explicitly, before. The abstract development of some branch of pure mathematics came to be recognized as formal axiomatics, whereas the concrete development of a given branch of applied mathematics came to be referred to as material axiomatics. "In the former case we think of the postulates as prior to any specification of the primitive terms, and in the latter we think of the objects and concepts that interpret the primitive terms as being prior to the postulates" (150). The former case is the newer idea of a postulate as a basic assumption about primitive terms. The latter case is the older Greek view of a postulate. The Greeks thought they were dealing with the unique structure of space and time. "But from the modern point of view, geometry is a purely abstract study devoid of any physical meaning or imagery" (150). But of course formalization brings with it its own set of issues that need to be addressed. Here Eves does an admirable job in Chapter 6 of explaining the important properties of axiomatic systems: equivalence, consistency, independence, completeness, and categoricalness. It is a lucid and crystal introduction.

Eves spend another whole chapter on the subject of the real number system and its importance for the foundation of analysis. This issue became important with the initial development of calculus by Leibniz and Newton. This newly developed branch of mathematics became an astounding tool for scientific use. But it was a tool, though powerful, that had not been thoroughly examined before its use. Eves claims, "It was more exciting to apply the marvelous new tool than to examine its logical soundness, for, after all, the processes employed justified themselves to the researchers in view of the fact that they worked" (175). So, "Attracted by the powerful applicability of the subject, and lacking a real understanding of the foundations on which the subject must rest, mathematicians manipulated analytical processes in an almost blind manner, often being guided only by a native intuition of what was felt must be valid" (176). So attempts at rigorizing calculus were begun by many notable mathematicians such as Euler and Lagrange. The most successful of these attempts was by Karl Weierstrass. Weierstrass realized that the number of problems that were being found in trying to establish certain branches of mathematics, like calculus, were endemic to properties belonging to the real number system. "Accordingly, Weierstrass advocated a program wherein the real number system itself should first be rigorized; then all the basic concepts of analysis should be derived from this number system" (178). Weierstrass and his followers eventually realized the so-called arithmetization of analysis. So "today it can be fairly said that classical analysis has been firmly established on the real number system as a foundation" (178). But the consistency of the real number system ends up depending on a more fundamental system, the natural numbers. Eves shows how the real number system is obtained from the natural number system in a purely definitional way. For the not-so-mathematically-inclined this section is highly formal. But Eves does a good job of making the formalization as clear as possible. Also notable in this chapter, Eves shows how the rational and complex numbers are extensions of the natural number system.

The next chapter deals with what most books that I have read start with when addressing the issues of axiomatic systems and formalizations: set theory. Though Mary Tiles and Stephen Pollard do much better book-length treatments of the subject, it is a good introduction to set theory if you are unfamiliar with it. In the last chapter Eves looks at logic and philosophy showing how symbolic logic has benefited from the axiomatizations that mathematics has experienced in its developmental history. He addresses the big three philosophies of mathematics: logicism, formalism, and intuitionism, showing a surprising attraction, at least for a mathematician, to the last one. This is a good introductory chapter as well but better book-length treatments, like Stewart Shapiro and Stephan Körner, have also been given to this subject. Eves' book also has an excellent appendix that contains some explicit proofs and has some smaller discussions that Eves evidently felt were not directly needed for the general text of his volume. In addition each chapter ends with exercises where one can become more proficient and familiar with the ideas presented in each section. There are some answers, though not all, provided in the back of the book. As a non-mathematician I found the exercises extremely difficult with the exception of the logic chapter.

Like the title says, Eves' text is a historical analysis and introduction to the foundations and fundamental concepts of mathematics. If you are interested in an historical overview of how modern mathematics (or logic in my case) has developed, I cannot recommend this book enough.

Fundamental yes.... but is a very demanding introduction.......2006-06-18

Back in the days when I thought that mathematics could be understood by a through understanding of fundamental logic, I thought this book would help me on that.. well it did, by showing me that is not the path to take... the book is a good historical developments of fundamental concepts with many exercises which makes you review your calculus again.

india and china.......2006-04-11

in answer to the reviewer who stated:

"The author reviews mathematical history but mentions no India nor China."

this is plainly false as any reader of this review can attest to simply by clicking on the 'look inside this book' link and reading page 2 of the book.

The author lacks basic knowledge about mathematical history.......2006-03-22

The author reviews mathematical history but mentions no India nor China. He presented a biased view of mathematical history.
The books is misleading in that regard.

Collected Papers on Mathematics, Logic, and Philosophy

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Collected Papers on Mathematics, Logic, and Philosophy
Gottlob Frege
Manufacturer: Blackwell Publishers
ProductGroup: Book
Binding: Hardcover

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ASIN: 0631127283

Basic Category Theory for Computer Scientists (Foundations of Computing)

Average customer rating:

Good Introduction
Basic crib sheet for category theory
Really expensive for a set of notes...
Too terse
the best understaning of categories you can get

Basic Category Theory for Computer Scientists (Foundations of Computing)
Benjamin C. Pierce
Manufacturer: The MIT Press
ProductGroup: Book
Binding: Paperback

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ASIN: 0262660717

Book Description

Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Benjamin C. Pierce received his doctoral degree from Carnegie Mellon University.

Contents: Tutorial. Applications. Further Reading.

Customer Reviews:

Good Introduction.......2007-02-21

I have been reading several different category theory texts recently, and this one was very succinct and accessible. Particularly useful for understanding functional programming.

Basic crib sheet for category theory.......2006-04-03

Anyone coming to this book from Pierce's "Types and Programming Languages" will be disappointed. While his "Types ..." book is a model of clear exposition, this book reads like a set of notes jotted down on the back on an envelope. The extensive bibliographic sections are more than fifteen years out of date. Much of the material referenced is no longer in print, and recent developments are, of course, not mentioned. Those seeking a very gentle introduction to category theory would do better with the book by Lawvere and Schanuel, who cover more of category theory than Pierce. Mathematically mature computer science readers will find everything they need to know about the subject in Mac Lane's book.

Really expensive for a set of notes..........2005-12-07

You can find better introductions to category theory available on the net for free. And I'm not talking about P2P! Try searching for Lambert Meertens, Marten Fokkinga, and Jaap Van Oosten, for example.

If you have some money to spend, get Barr and Wells, Category Theory for Computing Science. It's a great book, *way* better than this!

Too terse.......2004-03-28

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

the best understaning of categories you can get.......2002-05-06

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

Architecture of Systems Problem Solving (IFSR International Series on Systems Science and Engineering)

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Architecture of Systems Problem Solving (IFSR International Series on Systems Science and Engineering)
George J. Klir , and Doug Elias
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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Facets of Systems Science (IFSR International Series on Systems Science and Engineering)
Uncertainty and Information: Foundations of Generalized Information Theory

ASIN: 0306473577

Book Description

This is the definitive text for one of the major schools of thought in systems science. It presents both a comprehensive framework for characterizing all forms of systems problems, and a set of specific methodologies for some key problems. These methodologies are based on a combination of classical and fuzzy set theories, probability and possibility theories, graph and hypergraph theories, and information theory, among others. The hardcopy text contains a revised, updated and condensed version of the first edition, accompanied by a CD containing supplementary material including additional chapters on related topics, explanatory material drawn from many years of class presentations and lectures, exercises, and fully worked out examples showing both the framework and methodology in operation on actual real-world problems. Fully operational software is made available on an associated website. The material is suitable for upper-level undergraduates and first-year graduate students with a modest background in discrete math, probability and statistics.

2000 Solved Problems in Discrete Mathematics

Average customer rating:

Discrete Math, why?
Good book: Depends on your needs
Lacks current and expanded problems:
Reviews the field and Enforces problem solving abilities
It's helpful!

2000 Solved Problems in Discrete Mathematics
Seymour Lipschutz
Manufacturer: McGraw-Hill
ProductGroup: Book
Binding: Paperback

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Schaum's Outline of Discrete Mathematics (Schaum's)
Schaum's Easy Outline of Discrete Mathematics
3,000 Solved Problems in Linear Algebra
Discrete Mathematics and its Applications
Essence of Discrete Mathematics

ASIN: 0070380317

Book Description

This powerful problem-solver gives you 2,000 problems in discrete mathematics, fully solved step-by-step! From SchaumÕs, the originator of the solved-problem guide, and studentsÕ favorite with over 30 million study guides soldÑthis solution-packed timesaver helps you master every type of problem you will face on your tests, from simple questions on set theory to complex Boolean algebra, logic gates, and the use of propositional calculus. Go directly to the answers you need with a complete index. Compatible with any classroom text, SchaumÕs 2000 Solved Problems in Discrete Mathematics is so complete itÕs the perfect tool for graduate or professional exam prep!

Customer Reviews:

Discrete Math, why?.......2007-09-07

For a subject as useless as discrete math, this book helps with learning how to solve homework problems.

Good book: Depends on your needs.......2007-01-10

The use of this book depends entirely on how you define discrete math. The discrete math course I was hoping this book would supplement included just about even thirds combinatorics, number theory and graph theory. The combinatorics is adequate. There was hardly any number theory. The Graph theory was quite extensive and very helpful in my opinion. This book also goes into some algebra, (groups, etc.), which is of course very exciting, but I had little use for it at this time.

Lacks current and expanded problems:.......2002-12-21

I recently purchased this book for a discrete math class I was attending at UMD. This was no help in practicing iteration, induction, or summation. The only good point about this book is that it gives plenty of examples for visuals (i.e. set theory, graphs, and trees).

Reviews the field and Enforces problem solving abilities.......2001-01-24

Computer science has been the primary catalyst transforming the status of this once-obscure field of discrete mathematics into state-of-the-art information, and required knowledge. Switching circuits, programming languages, expert systems and their underlying logic, to name but a few computer-related topics, each rely upon the results from this field as a foundation. There is a proliferation of books on graph theory, discrete mathematics, and even set theory and mathematical logic, for this reason.

A paucity exists, however, of texts facilitating a newcomer in verifying the lessons presumably learned through formal education or through self-study. Schaum's solved problems books are invaluable for this purpose. The more recent n'000 solved problem books are beneficial to any willing to invest the effort and time required to attempt the numerous problems posed.

The discrete mathematics book reviews subjects ranging from fundamentals of logic and set theory, to lattices, combinatorics, as well as abstract algebra and the basics of the theory of languages. There are a variety of questions, from recalling definitions, to number-crunching, to providing sketches or fragments of proofs. These are typically in a progressive problem-solving form, which lends itself to becoming a programed course. The book easily and quietly supplements any standard, classic, or assigned works from the field.

Gradually the questions in this work will be perceived as increasingly basic, perhaps some sections too basic now - however that is only because the book has served its purpose, as a foundation for other more practical, detailed, or complex assignments or problems.

While reading through other works in this field, make note and observe how most have numerous exercises suffixed to each chapter, and yet how few actually provide comprehensible solutions to them. The true value of Schaum's series of solved problems books then becomes more evident.

It's helpful!.......1999-12-07

The problems in this book were arranged the way that leads readers understand simple concepts first, then bring you to problems and applications. In addtion, my professor Dr. Mott (he wrote one of the top selling discrete book) recommanded this book to me as well. I found it VERY helpful.

Yes, I think it servers as an aid to textbooks. You cannot teach yourself discrete math by reading it.

Book Description

This volume represents the first philosophically sound discussion of the concept of number in Western civilization. (Mathematics)

Customer Reviews:

The first escape from the Elencus..........2005-10-17

You know how _frustrating_ it is, reading a platonic dialog? Some question like "What is virtue?" or "What is justice" is asked, and Socretes goes on for pages showing that the so-called "experts" don't have a clue about what it really is?

But what's _really_ frustrating is that you're all expecting, at the end of the dialog, after following a hard line of argument, that you'll be rewarded with THE definitivie definition of 'virtue' or 'justice' or whatever--only to be disapointed. All you get in the end is a new appreciation of your own hopeless ignorance...

...well, imagine a platonic dialog which started the same as any other platonic dialog, but with the question "What is a number?" Only this time, at the end of the dialog, you actually get an answer to the question?

In retrospect, its pretty amazing that Plato didn't write a Socratic dialog concerned with the question "What is number?' After all, Plato considered numbers more real than physical objects, and people like the Pythagorians were going around claiming that everything _was_ made out of numbers. But what the heck _is_ a number, anyways?

Perhaps the reason was that everybody thought they already understood what numbers were. But Frege, like Socretes before him, realized that this so-called knowledge was really just a collective ignorance. So Frege starts out this book with a thorough, merciless review of what his coleages and predicessors were saying about what numbers were, showing that they ranged from cocksure to confused, from pompously-wrongheaded to just plain silly.

But then Frege does something really amazing--for the first time in history, he goes on give a real answer to the question "what are numbers?" Building on the work of Hume, he gives a sustained argument now known as "Frege's theorem" which shows how numbers can be grounded on an understanding of one-to-one correspondence.

Unfortunately, this work had to wait almost a century for the rest of us to really catch up to its significance. Russell found a contradiction in the arguments presented here, and for the next 80 years attention shifted elsewhere. But first Charles Parsons, in 1964, and then Crispen Wright and others in the 80's and 90's begain to realize that Frege's theorem could be reconstructed without the paradox. This sparked a whole flurry of neo-Fregean studies which is one of the most active branches of analytic philosophy today.

This revival means that Frege's importance, and the importance of reading and comming to grips with the arguments presented by Frege in this book, are going to continue to grow. Although tragically Frege didn't live to see the day, we now realize that the line of reasoning he followed in this book was one of those signature moments in human history, every bit as profound as the invention of the wheel or the discovery of the pythagorian theorem--it was the moment where, for the first time ever, the question "what the heck _are_ numbers, anyways?" got a real answer.

Frege, You're Not Supposed To Have..........2004-03-24

*The Foundations of Arithmetic*, one of the most durable works of philosophy of mathematics ever produced, is something of a curiosity as presented by J.L. Austin (who translated the work for the use of an Oxford undergraduate course); and perhaps Frege's platonism got the best of Austin, and this work is really just as , well, Kantian as it appears, "a good sight" more Kantian than "standard" Frege is typically allowed to be. Frege's definition of number in terms of equipollence (one-one correspondence of sets) is legendary: that is to say, it is traditionally understood to do a great deal more work than the "thin" version allowed by mathematical logic as reconstructed to avoid Russell's paradox.

But here Frege's work-up of the concept for a general readership is so "genteel" as to suggest that this may not in fact be the case, and that Frege actually partook more heavily of Neo-Kantian bromides than his *theory of arithmetic* suggests; to wit, that this theory was always intended to be situated within a general philosophy of mathematics obeying the strictures of reasoning involving Kantian "intuition" (as is typically said of Frege's last efforts in the field). As such, it would be unfortunate that we cannot effectively read this book (formerly available *en face*, and unfortunately much the worse for the original's omission) in conjunction with its contemporary geometrical counterpart: long out of print, rarely making its way into the philosophical Frege literature, and perhaps in all parts an *anticipatory* if "crochety" rebuke to Hilbertian formalism.

Perhaps Frege was to a certain extent wholly other than the mathematics of his time; perhaps we are not well-served by a Frege "out of time"; we certainly have one of the great prose stylists of English on hand here, and perhaps it would actually do to consider his aptitude for "gold" extraction here as a clue to puzzling out the rest of Frege -- a figure supremely unconcerned with sameness of meaning, and already owing a certain debt to those para-philosophical figures all his work is at cross-purposes with (the German '70s having been quite a time indeed). A great help to understanding number theory, a marvelous thing for a library to have.

Excellent work.......2003-06-14

His conclusion (p.99e) is that the laws of arithmetic are analytic judgements and consequently a priori.

Note that he is very consistently hard on Mill.

Some interesting quotes: p. 115e #106. "...number is neither a collection of things nor a property of such, yet at the same time is not a subjective product of mental processes either, we concluded that a statement of number asserts something objective of a concept.

... (p. 116e) We next laid down the fundamental principle that we must never try to define the meaning of a word in isolation, but only as it is used in the context of a proposition: only by adhering to this can we, as I believe, avoid a physical view of it.

#107. (p.117e) "A recognition statement must always have a sense."

great work.......2001-11-28

possibly one of the greatest works in history of philosophy and the founding book of 20th century analytic philosophy... I read it only once and a better appraisal will be coming shortly..I can say right away this is not simply a 'technical' work in philosophy of mathematics but a broad although short philosophical investigation in notions of truth, meaning and identity - although it expressly deals with defining numbers in purely logical terms. continental philosophers who read this work might change some of their negative ideas about where analytic philosophy is coming from.

A Must for Any Philosopher of Mathematics.......2000-09-24

This book written by Gottlob Frege is one of the most influential books of the 20th century philosophy of mathematics. In here Frege establishes the nature of arithmetics as founded in logic, which is his logicist proposal. For that, he refutes the assertion that logic as such is founded on psychology.

Sometimes he distorts a little bit what others say about logic, so he argues against those thinkers more effectively. In here he establishes the anti-psycology difference between concept and object; though he has not made a difference yet between sense and reference. He also refers to a principle called the contextual principle, in which the word makes reference to something depending on the context. Afterwards after he wrote the book, he would reject this principle, because of his doctrine of sense and reference: the sense of the words determine the sense of the sentence; and the reference of the words determine the reference of the sentence.

This is a great philosophical work, and I would suggest it to anyone who is starting to study Analytic philosophy (philosophy of mathematics, logic and language), and also those who want to consider the platonist proposal.

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