Book Description

Fierce competition in today's global market provides a powerful motivation for developing ever more sophisticated logistics systems. This book, written for the logistics manager and researcher, presents a survey of the modern theory and application of logistics. The goal of the book is to present the state-of-the-art in the science of logistics management.  As a result, the authors have written a timely and authoritative survey of this field that many practitioners and researchers will find makes an invaluable companion to their work. ... Read more

Reviews (3)

Rigorous overview of logistic modeling
Logistics has always been an integral part of industry and the military, and with the advent of the Internet, it has taken on major importance. This book gives a rigorous introduction to the formalism of logistics, and as such is fascinating reading for anyone interested in this area. Even individuals not into supply chain management and logistics engineering, and interested merely in the mathematics, will find this book interesting. After a short overview of logistics in the introduction, the authors discuss worst-case analysis of various algorithms for the bin-packing and traveling salesman problems. They define two performance metrics to measure the worst-case effectiveness: the absolute and asymptotic performance ratios. The First-Fit, Best-Fit, First-Fit Decreasing, and Best-Fit Decreasing heuristics are discussed in detail for the bin-packing problem. The authors show that a polynomial time heuristic cannot have an absolute performance ratio less than 3/2. They also show that finding a heuristic for the traveling salesman problem with a constant worst-case bound is as difficult as solving any NP-complete problem. The minimum spanning tree based, nearest insertion, Christofides', and local search heuristics are all discussed in great detail.

The next chapter considers the probabilistic analysis of algorithms via the characterization of the average performance of a given heuristic. The analysis is asymptotic with large problem sizes needed. Again, the bin-packing and traveling salesman problems are considered for studying this approach. This is followed by an approach to studying the efficacy of a particular heuristic by using mathematical programming in the next chapter. The strategy here is to cast the (NP-complete) problem as an integer problem, and then relax the constraint of integrality and solve the linear program. The authors showthat tight lower bounds can be found for these integer programs. The authors switch gears somewhat in the next two chapters, where vehicle routing problems are studied. In particular, the single-depot capacitated vehicle routing problem with equal and unequal demands is analyzed via worst-case and probabilistic analysis. The analysis is generalized in chapter 7 for the case where time constraints are present. An analytical solution of this problem, called the vehicle routing problem with time windows, is considered in detail by the authors. They back up their analysis with computational results at the end of the chapter. In chapter 8, a column generation approach is employed to solve the vehicle routing problem. No time constraints are put in, and the authors give in detail the steps behind this technique.

The study of inventory models is begun in chapter 9, with the economic lot size model leading off the discussion. This model illustrates effectively the tradeoffs between ordering and storage costs, and the optimal ordering policy is found. This model is generalized to the case where finite time horizons are included and the optimal policing found. Multi-item inventory models are then studied via worst-case analysis. The Wagner-Whitin model, which is an inventory model with varying demands, is formulated and solved in the next chapter. The techniques used, interestingly, involve dynamic programming. This model is generalized to the case where there is an upper bound on the amount that can be ordered or produced, and then the optimal solution found.

The case where the demand is a random variable is considered in the next chapter on stochastic inventory models. Single period and finite horizon models are considered using a dynamic programming algorithm to determine the optimal policy. The analysis makes heavy use of the properties of convex and quasiconvex functions.

Facility location models are the subject of the next chapter. The p-Median, single-source capacitated facility location (CFLP), and distribution system design problems are analyzed as warehouse location problems, with Lagrangian relaxation techniques used to find the solutions to these problems.

Logistics models that integrate inventory and routing strategies are considered in chapter 13, with the success of Wal-Mart given as an example of a firm whose success was generated by a reliance on an efficient logistical design and planning model called cross docking. Along with analyses of zero inventory ordering policies, the authors give an asymptotic analysis of cross-docking strategies.

The last two chapter of the book consider the implementation of logistic algorithms in practice. Although short, the chapters do give a fairly good overview of how these algorithms are used in the real world. The authors consider the routing and scheduling of New York City school buses and a decision support system for network configuration. Only one exercise is found in these chapters though unfortunately.

Get this book or spend a month in library
Professor Simchi-Levi dedicates his time as co-author of this book and I'd like to thank to his effort. The logic of Logistics is only "ONE' book in current academic text books that bravely delineates the theory and algorithm; while most other books spends many hundread pages for "words" and "case studies". The models are showed with algorithm and proving. Examples are included as necessary. The way to illustrate case study is different -but good different. For a researcher, consulting companies, professors, graduate students, you can spend a month in library for literature reviews or take few days to go through this book. If you think your time is worth, grasp this book and you won't be disappointed. If you want to see less mathematic issue, you may want to look at another book of Simchi-Levi. It's "Designing and Managing the Supply Chain : Concepts, Strategies, and Cases".

Highly Technical, Mathmatical textbook
Very technical with many mathmatical equations, exapmles and theorms. Includes exercises, and case study information. There are however, no answers to the exercises, and few "worked out" math problems. The format is very much a text book. ... Read more

Book Description

Book Description

This is an up-to-date textbook of model theory taking the reader from first definitions to Morley's theorem and the elementary parts of stability theory. Besides standard results such as the compactness and omitting types theorems, it also describes various links with algebra, including the Skolem-Tarski method of quantifier elimination, model completeness, automorphism groups and omega-categoricity, ultraproducts, O-minimality and structures of finite Morley rank. The material on back-and-forth equivalences, interpretations and zero-one laws can serve as an introduction to applications of model theory in computer science. Each chapter finishes with a brief commentary on the literature and suggestions for further reading. This book will benefit graduate students with an interest in model theory. ... Read more

Reviews (3)

Excellent
This textbook is an efficient condensation of Hodges's bulkier and more leisurely "Model Theory". As such, it excludes some of the "fun" topics in the larger book as well as the nice bibliography. On the other hand, it is a truly excellent textbook for model theory and, in fact, for logic (for those with some idea of what *that* is.)

Personally, I would have liked to see the following topics at least mentioned: higher-order languages, typed languages, ultraproducts, game theory. Nevertheless, this book is still the best and clearest textbook for model theory.

Hodges' Model Theory
This book and its 1993 expanded version are in the field of Mathematical Logic, and Hodges of London University shows that model theory in particular has had a remarkable variety of applications to other branches of mathematics, including computers (Prolog, undecidability, etc. - see my reviews of Penrose, Ablamowitz et al, etc.), geometry (see my reviews of advanced geometry via Clifford algebras, including Chisholm and geometric physics including Misner et al, and elementary geometry including Schaum's Outlines), topology (see my review of Greene's Elegant Universe which uses string theory and topology and also Carlip's book), algebra (see my Review of Weinberg's Gravitation and Cosmology which uses algebraic, topological, and analytic methods in general relativity, and Nachtmann's book which uses algebra in quantum theory), analysis/advanced calculus (see my review of Clarke, Yu, Nedyaev et al, Zwillinger's books, etc.), and so on. This book is in the Cambridge Encyclopedia of Mathematics and its Applications series, volume 42. Like most books of the Cambridge Encyclopedia series, it is very thorough up to the date of its publication - probably the most thorough book on model theory, which is roughly what is sounds like: mathematical models (of the physical and mathematical worlds). British Universities are among the world's greatest Creative Genius universities in math and physics, and this book is no exception. Most people should hire a reputable consultant or tutor to help them understand and "translate" the book, which will be well worth the effort in almost every field.

Excellent
Wilfrid Hodges is an excellent expositor. I have found the book a pleasure to read. This is how all textbooks should be... ... Read more

Book Description

To most people, mathematics means working with numbers. But as Keith Devlin shows in Mathematics: The Science of Patterns, this definition has been out of date for nearly 2,500 years. Mathematicians now see their work as the study of patterns—real or imagined, visual or mental, arising from the natural world or from within the human mind.

Using this basic definition as his central theme, Devlin explores the patterns of counting, measuring, reasoning, motion, shape, position, and prediction, revealing the powerful influence mathematics has over our perception of reality. Interweaving historical highlights and current developments, and using a minimum of formulas, Devlin celebrates the precision, purity, and elegance of mathematics.
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Reviews (4)

A pleasure to read
The book is fun to read ,very informative and very clear.
It doesnt matter if you know some university mathematics or not ,anyhow you will find this book a pleasure to read ,and if you dont know nothing about mathematics it will change your perspective on the world. I suggest you will get a copy and read it.Excellent book!.

Imaginative, Engaging,Fascinating, Delightul
This marvelous book to explains to non-mathematicians the joy, beauty and power of mathematics. Each topic is presented in an original manner with alot of colorful illustrations to delight the eye and mind. Devlin shows how mathematical thinking is critical to our exploration of the world around us. This is one my top ten of all time list

Beautifully illustrated, clear and engaging
Keith Devlin is one of the best popular mathematics writers around, and this is one of his best works. The six chapters cover number theory, set theory, calculus, group theory and topology; but to state it baldly like this is to miss the main value of this seductively illustrated book. Devlin titles his chapters innocuously--"Shape", or "Position"--and the initial discussion, couched generally in English, not mathematics, is so clear that a math-phobic can understand it. By the end of each chapter a great deal of fascinating mathematics has been described, and in some cases the formal basis is sketched--but the emphasis is always on narration, and a lay reader who doesn't even want to understand mathematics can still read this and get a sense of the dramatic history of mathematics. And of the dramatis personae, too; one nice feature is the large number of good pictures of mathematicians, including several more recent figures such as Ribet and Thurston.

Devlin states at the end that he decided to exclude many areas of mathematics in order to focus more effectively on what he did cover. As a result there is little or no coverage of chaos theory, game theory, catastrophe theory, or a long list of other topics. The fact is there will always be holes in a book this size--mathematics has expanded so much in the last hundred years that even a book ten times this size could barely survey it. The decision to focus was a good one, and the subjects chosen are good: the truly exciting stories are here: Archimedes, Fermat, Gauss, Galois, Riemann, Wiles, and many more.

The illustrations deserve an extra comment. I've already mentioned the pictures of mathematicians. There are good diagrams, of the quality you'd expect from Scientific American. There are also plenty of pictures of the sort you see in every maths book of this kind--Escher tessellations, Kepler's nested Platonic solids, a Durer perspective drawing. But there are several more that I've never seen (and I've read a lot of these books). Two notable pictures: a cardboard model of an aperiodic tiling of space, by John Conway; and a picture of a set of tiles at a Dutch high school, designed by Escher: I'm an Escher fan and have never seen these before.

Potential purchasers should note, by the way, that this book was reworked into Devlin's "Language of Mathematics". In Devlin's words (not from either book): "The Language of Mathematics is a restructuring of Science of Patterns that omits most of the color illustrations (a minus) but has two new chapters covering topics not in Science of Patterns (a plus). If you want lots of color, go for patterns; Language of Mathematics covers more ground."

This is a fine book. Strongly recommended.

Easy to Understand, Hard to Put Down
This book is a brilliant example of mathematics at it's best. It is from Scientific American, so you know you can trust it. And it is written at an understandable level, quite a feat for many very complex topics. The book features incredible illustrations, every concept is laid out in a colorful image. If you like the works of M.C. Escher, you will like this book. It has a lot of substance to it, and it will keep you busy thinking for a long time, and that's time well spent ... Read more

Book Description

Lattice-valued Logic aims at establishing the logical foundation for uncertain information processing routinely performed by humans and artificial intelligence systems. In this textbook for the first time a general introduction on lattice-valued logic is given. It systematically summarizes research from the basic notions up to recent results on lattice implication algebras, lattice-valued logic systems based on lattice implication algebras, as well as the corresponding reasoning theories and methods. The book provides the suitable theoretical logical background of lattice-valued logic systems and supports newly designed intelligent uncertain-information-processing systems and a wide spectrum of intelligent learning tasks. ... Read more

Reviews (4)

Boole's LT Breaks Ancient Mold , Founds Math Logic
. Aristotle and Boole are the two most original logicians before the era of modern logic. Aristotle presented the world's first system of logic. His system involves the standard three parts: first, a limited formalized predicational language; second, a formal method of step-by-step deductions for establishing validity of arguments having unlimited numbers of premisses; and third, an equally general method of countermodels for establishing invalidity. Boole's LAWS OF THOUGHT showed that logic is mathematical. Its stated aims were to refine, systematize, and complete the project started by Aristotle and, more ambitiously, to demonstrate the mathematical character of logic. His two-part system involves, first, a limited formalized equational language capable of expressing tautologies or "laws of thought", a breakthrough dramatically altering Aristotle's plan, and, second, a semi-formal method of derivation using equational reasoning totally absent from previous systematic logic. Boole's primary goals included construction of a method for generating solutions to sets of equations regarded as conditions on "unknowns", an unprecedented innovation with radical implications for the future development of logic. As for the third part of a system of logic, a method of establishing invalidity, surprisingly, Boole's book contains no systematic discussion of independence nor does it contain anything like a method of countermodels. Boole's LAWS OF THOUGHT set in motion forces that would lead to the ultimate fulfillment many of his goals including the establishment of mathematical logic.

NOW IS A GOOD TIME TO STUDY BOOLE.
The publication of The Laws of Thought in 1854 launched modern mathematical logic. The author George Boole (1815-1864) was already a celebrated mathematician specializing in what is known as analysis. If, as Aristotle (384-322 B.C.E.) tells us, we do not understand a thing until we see it growing from its beginning, then those who want to understand modern mathematical logic should begin with The Laws of Thought. There are many wonderful things about this book besides its historical importance. For one thing, the reader does not need to know any mathematical logic. There was none available to the audience for which it was written-even today a little basic algebra and a semester's worth of beginning logic is all that is required. For another thing, the book is exciting reading. The reader comes to feel through Boole's intense, serious, and sometimes labored writing that the birth of something very important is being witnessed. Of all the foundational writings concerning mathematical logic, this one is the most accessible to the nonexpert and it has the most to offer the nonexpert. The secondary literature on Boole is lively and growing, as can be seen from an excellent recent anthology (A BOOLE ANTHOLOGY by J.Gasser 2000) and a complete bibliography that is now available (Nambiar 2003). Boole's manuscripts on logic and philosophy, once nearly inaccessible, are now in print (Grattan-Guinness and Bornet 1997). This is a good time to start to study Boole.
It is true that Boole had written on logic before, but his earlier work did not attract much attention until after his reputation as a logician was established. Today he is known almost exclusively for his logic. In 1848 he published a short paper "The Calculus of Logic" (Boole 1848) and in 1847, at his own expense, he published a pamphlet The Mathematical Analysis of Logic (Boole1847). By the expression 'mathematical analysis of logic' Boole did not mean to suggest that he was analyzing logic mathematically or that he was using mathematics to analyze logic. Rather his meaning was that he had found logic to be a new form of mathematics, not a form of philosophy as had been previously thought. More specifically, his point was that he had found logic to be a form of the branch of mathematics known as mathematical analysis, which includes algebra and calculus. (For a short description of this branch of mathematics, see the article "Mathematical Analysis" in the 1999 Cambridge Dictionary of Philosophy (Audi 1999, 540-41).
Although this book begins mathematical logic, it does not begin logical theory. The construction of logical theory begins, of course, with Aristotle whose logical writings were known and admired by Boole. In fact, Boole explicitly accepted Aristotle's logic as "a collection of scientific truths" (1854, 241) and he regarded himself as following in Aristotle's footsteps. He thought that he was supplying a unifying foundation for Aristotle's logic and that he was at the same time expanding the range of propositions and deductions that were formally treatable in logic. Boole thought that Aristotle's logic was "not a science but a collection of scientific truths, too incomplete to form a system of themselves, and not sufficiently fundamental to serve as the foundation upon which a perfect system may rest" (Boole 1854, 241). As has been pointed out by Grattan-Guinness (2003; Grattan-Guinness and Bornet 1997), in 1854 Boole was less impressed with Aristotle's achievement than he was in 1847. In "The mathematical analysis of logic" (Boole 1847) Aristotle's logic plays the leading role, but in The Laws of Thought (Boole 1854) it occupies only one chapter of the fifteen on logic. Even though Boole's view of Aristotle's achievement waned as Boole's own achievement evolved, Boole never found fault with anything that Aristotle did in logic, with Aristotle's positive doctrine. Boole's criticisms were all directed at what Aristotle did not do, with what Aristotle omitted doing. Aristotle was already fully aware that later logicians would criticize his omissions, but unfortunately he did not reveal what he thought those omissions might be (Aristotle, Sophistical Refutations, Ch. 34).
The new 2003 edition by Prometheus Books(ISBN 1-59102-089-1, Paper ...)contains an accessible 25-page introduction by a modern logician.

difficult, but a classic--and worth the effort.
Yes, this is the Boole of Boolean algebra. No, this is not a primer. But if you have any interest at all in intellectual history or where the tools of computer science came from, then you will find this book worth the effort.

a period curiosity only
This is Boole of boolean algebra. The book, now 150 years old, is a long winded philosophical curiosity. Do NOT expect to learn boolean algebra! ... Read more

Book Description

This lively introduction to mathematical logic, easily accessible to non-mathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursivefunctions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.
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Reviews (2)

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.

A marvel of brevity while not watering anything down. ... Read more

Book Description

The book is a short, concise and complete presentation of constraint programming and reasoning. The use of constraints had its scientific and commercial breakthrough in the 1990s. Programming with constraints makes it possible to model and specify problems with uncertain, incomplete information and to solve combinatorial problems, as they are abundant in industry and commerce, such as scheduling, planning, transportation, resource allocation, layout, design and analysis. The theoretically well-founded presentation includes exercises with solutions and application examples from real life. It is ideally suited as a textbook for graduate students and as a resource for researchers and practitioners. The Internet support includes teaching material, software, latest news and updates. ... Read more

Book Description

This textbook explains the basic principles of categorical type theory and the techniques used to derive categorical semantics for specific type theories. It introduces the reader to ordered set theory, lattices and domains, and this material provides plenty of examples for an introduction to category theory, which covers categories, functors, natural transformations, the Yoneda lemma, cartesian closed categories, limits, adjunctions and indexed categories. Four kinds of formal system are considered in detail, namely algebraic, functional, polymorphic functional, and higher order polymorphic functional type theory. For each of these the categorical semantics are derived and results about the type systems are proved categorically. Issues of soundness and completeness are also considered. Aimed at advanced undergraduates and beginning graduates, this book will be of interest to theoretical computer scientists, logiciansand mathematicians specializing in category theory. ... Read more

Reviews (1)

Excellent introduction to categories for computer scientists
The book gives you all of the cateogry theory you need to study type theory. The examples are from domains that are comfortable for computer scientists. The difficult proofs are given in great detail, while other books often gloss over the details. ... Read more

Book Description

Computability theory originated with the seminal work of Gödel, Church, Turing, Kleene and Post in the 1930s. This theory includes a wide spectrum of topics, such as the theory of reducibilities and their degree structures, computably enumerable sets and their automorphisms, and subrecursive hierarchy classifications. Recent work in computability theory has focused on Turing definability and promises to have far-reaching mathematical, scientific, and philosophical consequences. Written by a leading researcher, Computability Theory provides a concise, comprehensive, and authoritative introduction to contemporary computability theory, techniques, and results. The basic concepts and techniques of computability theory are placed in their historical, philosophical and logical context. This presentation is characterized by an unusual breadth of coverage and the inclusion of advanced topics not to be found elsewhere in the literature at this level.The book includes both the standard material for a first course in computability and more advanced looks at degree structures, forcing, priority methods, and determinacy. The final chapter explores a variety of computability applications to mathematics and science.Computability Theory is an invaluable text, reference, and guide to the direction of current research in the field. Nowhere else will you find the techniques and results of this beautiful and basic subject brought alive in such an approachable and lively way. ... Read more

Reviews (1)

A Unique Introduction to Computability
This book is an introduction to computability theory. It is organized in three parts, starting with basic computability theory and moving up to advanced topics, some of which cannot be found in textbooks today.

In the first part the reader is introduced to basic concepts and results of computability like models of computation, coding, universal machines, enumerability, fixed point theorem. The author also discusses the historical context in which various notions appeared (not only in this part but throughout the book) like Hilbert's programme and makes connections with logic (language, theories, Peano Arithmetic, Godel incompleteness theorem). Computability and Unsolvability in the real world is also discussed, along with the search for natural examples of incomputable sets, a topic which is currently more interesting than ever. Most of the content of Part I can be found in other good text books (like Odiffreddi's or Roger's) but the way it is presented is unique: the arguments and proofs are given in an informal yet accurate way (according to the modern mode for doing computability) and the whole arrangement is very schematic, often assisted by diagrams, figures, tables and boxes. This is especially helpful in a text book in computability theory, a subject that makes understanding rely so much on intuition and visual images.

The second part is concerned with oracle computation (a core part of computability), Turing degrees, Enumeration degrees, and many other related and complementary topics like polynomial bounds, P=?NP, the Scott model for Lamda calculus and others. The author here tries to give a general idea of the subject by discussing interesting topics (like the ones mentioned above) which don¡¯t necessarily lie on the core of computability theory. This is pretty much the spirit of the whole book: to give the non-expert reader access to the most exciting (and sometimes apparently inaccessible at this level) topics in the subject and motivate him/her to further study towards the direction that looks and feels more appealing.

The third and last part discusses advanced topics like approximation constructions, priority injury, Sack¡¯s theorems, maximal sets, even the 0¡¯¡¯¡¯-priority method. This is the longest part of the book and the choice its contents (along with the approachable and attractive way they are presented despite their advanced nature) is just another feature which makes this book unique. The construction of maximal sets is remarkable since it uses a tree argument (with infinitary activity of the nodes but without injury) thus making it more intuitive and understandable, in contrast to the usual e-maximal state method which was introduced by the original paper (with the first proof that maximal sets exist) and followed by most text books I am aware of, without many changes. The proof of the existence of a noncuppable c.e. noncomputable degree also deserves to be mentioned as it is not something that one finds in text books. Also, it is different than the original pinball argument one finds in papers (with the restraints tending to infinity, often mentioned as an example of this bizarre feature) as it is done on a tree. Finally, computability in mathematics (structures, combinatorics, Analysis) and science is discussed along with randomness and computable models.

In the end of the book there is a bibliography for further reading. This is very personal (and, of course, by no means complete) but very helpful as it ranges over a wide range of computability related topics and it matches the spirit of the book very well.

To sum up, this introduction achieves the aims set by the author (a leading specialist in computability) in the preface and the epilogue: it deals with the subject in a very wide context, discusses it from its most hardcore features (priority, forcing) to its most distant echoes (incomputability in science) and most importantly it relates these two, showing how technical work is motivated and inspired by more general concerns. It is intended as a text book for undergraduate and early postgraduate students but is also suitable for any non-specialist. The features discussed above along with the modern style of presentation make the subject look as attractive as it really is and the book unique over the other computability text books available today. I wish this book had been in my library when I first started reading computability. ... Read more

Book Description

This first volume of the Handbook of Formal Languages gives a comprehensive authoritative exposition on the core of language theory. Grammars, codes, power series, L systems, and combinatorics on words are all discussed in a thorough, yet self-contained manner. This is perhaps the most informative single volume in the history of theoretical computer science. ... Read more

Book Description

This book leads readers through a progressive explanation of what mathematical proofs are, why they are important, and how they work, along with a presentation of basic techniques used to construct proofs.The Second Edition presents more examples, more exercises, a more complete treatment of mathematical induction and set theory, and it incorporates suggestions from students and colleagues.Since the mathematical concepts used are relatively elementary, the book can be used as a supplement in any post-calculus course.

This title has been successfully class-tested for years.There is an index for easier reference, a more extensive list of definitions and concepts, and an updated bibliography.An extensive collection of exercises with complete answers are provided, enabling students to practice on their own.Additionally, there is a set of problems without solutions to make it easier for instructors to prepare homework assignments.

* Successfully class-tested over a number of years
* Index for easy reference
* Extensive list of definitions and concepts
* Updated biblography ... Read more

Book Description

Lucas presents a comprehensive study of the foundation of mathematics, showing us that it is actually at the heart of the study of

epistemology and metaphysics. ... Read more

Book Description

Coherent, well organized text familiarizes readers with complete theory of logical inference and its applications to math and the empirical sciences. Part I deals with formal principles of inference and definition; Part II explores elementary intuitive set theory, with separate chapters on sets, relations, and functions. Last section introduces numerous examples of axiomatically formulated theories in both discussion and exercises. Ideal for undergraduates; no background in math or philosophy required.
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Book Description

The traditional debate among philosophers of mathematics is whether there is an external mathematical reality, something out there to be discovered, or whether mathematics is the product of the human mind. This provocative book, now available in a revised and expanded paperback edition, goes beyond foundationalist questions to offer what has been called a "postmodern" assessment of the philosophy of mathematics--one that addresses issues of theoretical importance in terms of mathematical experience. By bringing together essays of leading philosophers, mathematicians, logicians, and computer scientists, Thomas Tymoczko reveals an evolving effort to account for the nature of mathematics in relation to other human activities. These accounts include such topics as the history of mathematics as a field of study, predictions about how computers will influence the future organization of mathematics, and what processes a proof undergoes before it reaches publishable form.

This expanded edition now contains essays by Penelope Maddy, Michael D. Resnik, and William P. Thurston that address the nature of mathematical proofs. The editor has provided a new afterword and a supplemental bibliography of recent work. ... Read more

Reviews (3)

provides insight into how mathematicians think
To most people, "philosophy of mathematics" probably sounds like the driest subject in the world. I admit that a typical person in the street would probably never want to read this book, but many people who would be put off by the title would find it fascinating.

The basic question is how we should think about mathematics. When we do mathematics, are we describing an independent reality, following arbitrary rules, building a social construct? One can ultimately say only so much about this particular question, but it leads off in many wonderful directions. To me, the highlight of this book is the article by Thurston, which provides a beautiful description of how mathematicians actually think about and do mathematics. It really rings true to me (I'm a mathematician too), and is much better than any other account I've ever seen.

In general, whenever people seriously discuss the philosophy of mathematics, they are likely to make revealing comments about their approach to the field. People who are curious about this (e.g., students considering studying mathematics, or anyone who has heard about the results of mathematics and wonders about the mindset behind them) should read the book. As a bonus, once they start reading the essays they'll rapidly start caring about the philosophical issues as well, even if they've never thought about them before.

Philosophy of mathematics reconsidered
After almost a century in which the attempt was made to reduce philosophy of mathematics to set theory, philosophers have begun to reconsider the traditional approaches. The Tymoczko volume provides a solid intro to these new approaches, that is both readable and insightful. A background in formal logic, or traditional philosophy of mathematics, is not presupposed, as there are basic essays that should get the reader up to speed on the terminology. Nor does one have to be a mathematician to appreciate the thoughts presented. At the same time, I don't believe any mathematician would be offended or alarmed by the presentations of this book.

Ultimately, there is no final consensus offered. Rather, the topic is reinvigorated with a collection of fresh approaches that do not falsify the experience of mathematics by trying to reduce it to something else.

I suggest a closely related book
Hi, I'm one of the contributors to Tymoczko's anthology, and I would like to suggest a related book on the quasi-empirical view of mathematics. That's my book "The Limits of Mathematics" just published by Springer Verlag. Together these two books make a nice set. Greg Chaitin, IBM Research ... Read more

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Reviews (1)

Good Unifying View
Although this book may be somewhat outdated (published in the mid-70's), it does provide a cohesive view of the developments in logic up until that point. One gets a very strong sense of the status of logical development, while at the same time receiving a historical motivation for the methods employed in developing the theory. Many proofs are shortened or synopsized, however the integrity and technical level of the work is never compromised. In my opinion, the sections on Model Theory, Set Theory, Proof Theory and Recursion Theory provided the reader with a good sense of the major results in those areas. The section on computers (and their limitations) was a hoot to read, because of the limited view provided by the author, but otherwise, Wang has a strong intuition as to where modern developments could have led. Recommended for anyone trying to get a unifying view of the major developments in logic. ... Read more

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Comprehensive account of constructive theory of first-order predicate calculus. Covers formal methods including algorithms and epi-theory, brief treatment of Markov’s approach to algorithms, elementary facts about lattices and similar algebraic systems, more. Philosophical and reflective as well as mathematical. Graduate-level course. 1963 ed. Exercises.
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Still an interesting read....
Those interested in mathematical logic will appreciate this book written by one of the main contributors to the field in the twentieth century. The technique of "currying" in higher order logic is named after the author, wherein unary functions can be used to emulate functions with many parameters. The book was first published in 1963, reprinted in 1977, and so is not a up-to-date treatment of mathematical logic, but it could still be used as an historical supplement to a course in this subject. The reader should be aware though the terminology employed by the author is very idiosyncratic and therefore it may not reflect what is currently used in the literature.

The first chapter of the book could be considered an introduction to the philosophy of logic and mathematics. The author though views "philosophical logic" as the study of the principles of valid reasoning, and this is to be distinguished from "mathematical logic", wherein mathematical systems are constructed to study (formally) the principles of valid reasoning. One can also according to the author view logic as a theory in itself, and many "models" of it can be studied, in much the same way as many different models of geometry can be considered. The author also discusses very succinctly the logical paradoxes, and the different schools of thought in mathematics, such as Platonism, intuitionism, and formalism. The author clearly advocates the formalist school of thought in this book.

In chapter 2, the author gets more into the details of formal reasoning, the field of semiotics is outlined, and the author first begins defining the grammar and symbols for the upcoming discussion. A theory is defined as a class of statements, and consistency and decidability of theories is defined. The idea of a deductive theory is also defined, and the author defines the notion of such a theory being complete. The notions of consistency, decidability, and completeness are the familiar ones now entrenched in current textbooks on mathematical logic. A formal system, according to the author, is a theory in which the parameters of the statements of the theory are introduced as unspecified objects, and the statements of the theory make assertions on the properties of the parameters and their relations. The author considers syntactical systems, wherein the formal objects are taken from some object language, and what he calls Ob systems, which are essentially the systems considered in modern mathematical logic.The author employs the familiar Godel numbering scheme to numerically represent formal objects. The notion of algorithm is brought in here as an effective procedure to manipulate the formal objects of a system.

The next chapter is basically an introduction to the analysis of what would now be called the metalanguage of a formal system. This analysis is done in terms of what the author calls epistatements and epitheorems. Examples of these epitheorems include the Godel incompleteness theorem and the Skolem-Lowenheim theorem. The author introduces and classifies variables, and defines free and bound variables. A brief introduction to the lambda calculus and combinatory logic is given.

Then in chapter 4, the author discusses logical systems which are relational but with no bound variables. These are called logical algebras by the author, and the reader will encounter the famous truth tables and lattices in this chapter. A discussion of the Heyting algebra is given in the notes to the chapter. The reader interested in the more exotic types of algebraic logic, such as quantum logic, could benefit greatly from the reading of this chapter.

The logic of propositional calculus in terms of algebraic logic is discussed in chapter 5. Called propositional algebras by the author, the author proves the deduction theorem for such systems in this chapter. Interestingly, the L systems introduced by Gentzen are also discussed in this chapter. Although there are much better overviews of Gentzen's work in the current literature, a reader may still profit from a perusing of this chapter. L-systems where negation is added is then the subject of the next chapter.

Quantification in formal systems is taken up in chapter 7, considered both in the usual predicate calculus and in L systems. Prenex normal forms, the Herbrand-Gentzen theorem, and the completeness theorem are discussed in fairly good detail, albeit with old-fashioned notation.

The last chapter covers the interesting concept of modal logic. First considered by Aristotle, the author discusses it in the context of L systems, with the presentation being the shortest in the book. ... Read more

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This book is the final version of a course on algorithmic information theory and the epistemology of mathematics and physics. It discusses Einstein and Goedel's views on the nature of mathematics in the light of information theory, and sustains the thesis that mathematics is quasi-empirical. There is a foreword by Cris Calude of the University of Auckland, and supplementary material is available at the author's web site. The special feature of this book is that it presents a new "hands on" didatic approach using LISP and Mathematica software. The reader will be able to derive an understanding of the close relationship between mathematics and physics. "The Limits of Mathematics is a very personal and idiosyncratic account of Greg Chaitin's entire career in developing algorithmic information theory. The combination of the edited transcripts of his three introductory lectures maintains all the energy and content of the oral presentations, while the material on AIT itself gives a full explanation of how to implement Greg's ideas on real computers for those who want to try their hand at furthering the theory." ... Read more

Reviews (3)

This book is a piece of junk. Do not buy it.
Years ago I read a fascinating article in Scientific American
"Randomness and Mathematical Proof" by G Chaitin. (Of course
that was back in the good old days before SciAm switched to the
kindergarten market). Anyway, the article was fascinating and
well written. It was followed up by Martin Gardiner in his
"Mathematical Recreations" column, I think. This left me
with a desire to learn more. On the basis of the article and
the excellent reputation of Springer. I bought this book 2
years ago sight unseen.

What a big mistake. I feel cheated every time I see it on my
book shelf! The worst thing is that I simply cannot believe
that a reputable publisher like Springer actually published
this piece of junk.

I also cannot believe the same person wrote the SciAm article
and this book. Chaitin comes across in his book completely
self-obsessed and full of his own importance: comparing himself
(favourably) to K Godel and Einstein.

The book claims to be "a course in information theory". False.
It is not a course in anything and is written by an illiterate.
It has no value as a text book - indeed its value is negative
becasue it will turn readers away.

The subject of algorithmic information theory is worth learning
about. But please do look for another book to learn from - any
book except this one.

Belongs up there with Godel and Jantsch...
Chaitin has spent his life working on translating Godel's work (and Turing's too) into a version that is much more generally applicable: the limits of mathematical reasoning. I'm not sure why this book is not more generally known (probably because we don't like to have our balloon of arrogant rationality popped so easily) and read but it should be required for anyone contemplating further education in science.

Over the years the message from mathematics, physics and cybernetics (and AI for that matter) has simply been one of showing all formal forms of Aristotle's logic cannot ever capture the TRUTH.

Chaitin's easy delivery (this book is based on taped lectures) makes the book a gentle read even for those without the math background to fully understand the few equations. I highly reccommed buying a copy and reading it.

Beautiful attempt to understand the nature of mathematics
Greg Chaitin has written a beautiful and entertaining book about his work to understand the nature of mathematics. He describes his results in a highly informal and entertaining way which is still rigorous. Clearly he sincerly wants to _know_ what mathematics is. I wish that all math books were written this way.

Chaitin shows us the fairly simple computer programs he has written to demonstrate theorems about the limits of the axiomatic method. He shows results which are similar to Go"del's and Turing's results and in fact imply them. He has a simple and striking method of arriving at his results. It is all done with both brains and heart.

Chaitin defines a number which represents the probablility that a computer program halts. He shows how this number cannot be computed with a computer program which contains fewer bits than the number itself. Moreover, no set of mathematical axioms can compute this number with more precision than there are information bits in the axioms.

Since no set of axioms can enable us to fully compute the halting probability, and since axioms enable us to write proofs, and since proofs give us the reason why a mathematical statement is true, then there are some mathematical truths which are true for no reason, i.e. they are random.

But I cannot agree with this conclusion. Chaitin, and many others who say similar things, assume that a proof is a reason. But a proof is only a chain of implication. Our faith in the statement, in the light of the proof, rests on our faith in the axioms and logic.

Imagine a culture somewhere that has a mathematics like ours, except they don't have the distibutive property. They could prove some things that we know, but not others. They play around with their computers and discover some facts which they could easily show if they only had the distributive property, but they can't prove them.

Would it be right for these people to claim that these facts are random, that they are true for no reason? I don't think so. Instead, I think that these people should say, and we should too, that there are some things which are true, they are not random, but they are beyond our ability to prove. They are true for a reason, but we just don't see it yet. ... Read more

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From the Reviews:

"...He (the author) uses the language and notation of ordinary informal mathematics to state the basic set-theoretic facts which a beginning student of advanced mathematics needs to know. ...Because of the informal method of presentation, the book is eminently suited for use as a textbook or for self-study. The reader should derive from this volume a maximum of understanding of the theorems of set theory and of their basic importance in the study of mathematics."Philosophy and Phenomenological Research ... Read more

Reviews (6)

The essential essence of set theory in 100 pages
This book is an excellent primer on the basics of set theory that all graduate students need, but are not necessarily obtained in the general undergraduate curriculum. Halmos writes in an abbreviated, yet effective style that imparts the necessary details without an excess of words. Theorems and exercises are very few, so it really cannot be used as a textbook. If you need a great deal of explanations, then it is not for you. However, if your need is for a book that distills the essence of set theory down to the shortest possible size, then this book should be yours, either in your college or personal library.

Mercy Sought
My previous review of Halmos' book stands. Exceptional book, but ... As an example of a question in the book to whet some appetites and in seeking someone's kind mercy in actually answering it for me and putting me out of my misery, consider p.59 on the Axiom of Choice. Quote: if {X (sub)i} is a finite sequence of sets, for i in n say, then a necessary and sufficient condition that their Cartesian product be empty is that at least one of them be empty ... (The case n=0 leads to a slippery argument about the empty function; the uninterested reader may start his induction at 1 instead of 0). Unquote. Induction from 1 is no problem. The slippery argument stuff (and other similar pearls thoughout the book) sends me wild. What is the slippery argument. Please. Anyone. With thanks to Paul Halmos for making my life 'miserably interesting' (sic)!!

Insightful
An exceptional book. The book, however, has little pedagogical value. I would not recommend those starting out in mathematics to purchase it. It is definitely for the mathematically mature. Indeed, it is the type of book that may force some to consider abandoning mathematics if it is read without assistance too early in their development. The lack of answers to exercises amplifies these considerations when the book is used for self study as there are few means to understand whether one is on the right track, especially when the less natural approach of recursion is required to answer some questions. If you want to maximise your understanding of set theory, however, this is an essential book to get. In a sense, when read in conjunction with Paul Halmos' background and some quotes attributed to him found elsewhere on the Internet, the book is almost autobiographical.

Mathematical writing at its best
Oh, to be able to write like Paul Halmos!

This is, quite simply, a beautiful book. Halmos has taken a field, wrapped his deep understanding around it, and brought the field forth into light in a way that it is accessible to any reader willing to invest the requisite effort, regardless of mathematical background.

Each word is carefully chosen; Halmos has a knack for qualifying his statements gently and subtly so that on a first reading, the qualifications and limitations placed on the main results don't slow one down. On a second reading, the qualifications actually shed light on the intricacies of the subject. "Why does he qualify this?", one asks oneself, and in discovering the answer, comes to a better understanding of the field. Similarly, the small number of exercises posed for the reader have been very carefully chosen to she light on the subject itself. Unlike the rote busywork included with many mathematics texts, each problem posed by Halmos is, I would argue, essential to the book.

The book is not easy going in that it can be read quickly. I have a reasonable mathematical background, I use mathematics daily in my professional life, and yet (taking time to work the exercises) I read this book at a pace of about four to six pages an hour. On the other hand, this is not so bad - the entire book is only 102 pages, and in those 102 pages Halmos manages to present a full semester's course in set theory.

Finally, I should mention that anyone who has spent more time with applied mathematics than with the foundations of mathematics is likely to find this a fascinating read. When I read this book, it was not only the most interesting mathematics book I had read in at least a year, but also the most interesting philosophy book. Just to give a few examples, I never REALLY understood Russell's paradox until I read Halmos' explanation (which he presents on page 6 of the book). By page 30, Halmos offers an explanation of what a function really is, and by page 42, he tackles the question of what we really mean when we talk about the number "2" or the number "6" or any other number, for that matter.

This book takes some work on the part of the reader, but the effort is repaid handsomely. The effort would have been worth my while purely to the learn the mathematics, purely for the philosophical issues raised, or purely as an example of how one can aspire to write about mathematics. Of course, for my effort, I was able to enjoy all three aspects of this marvellous text.

Very thorough yet too compact
This book is fascinating. Halmos proceeds to construct the most relavant concepts of set theory independantly of any other mathematics. For instance never once does he use numbers until he has constructed them out of sets. The level of rigor is not that of axiomatic set theory, so the book is accessible.

Unfortunately, as seems to be Halmos style (definitly evident in his 'Finite Vector Spaces' which I do NOT recommend unless you are far more gifted than I), he is quite compact. He compresses a wealth of information into a very short space, and most of the 25 topics are covered in under 4 full pages. The exercises are sparse and difficult.

This book could definitly have benefited from much more explanation and exercises. For the reader who possess the talent, though, this book is strongly recommended. Even for those (like me) who failed to grasp every detail, it is still a very worthwhile read. I fully intend to return to this when I have a more firm grounding in the thought patterns of abstract mathematics. ... Read more

Book Description

Kurt Godel was one of the most outstanding logicians of the twentieth century, famous for hsi work on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuim hypothesis. He is also noted for his work on constructivity, the decision problem and the foundations of computation theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory, permitting time travel into the past.The book is the fourth part of a five volume set, which is the first to maek available all of Godels writings in one place. The collected Works of Kurt Godel is designed to be useful and accessible to a wide audience without sacrificing scientific or historical accuracy. ... Read more