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"Pure mathematics," Albert Einstein once remarked, "is, in its way, the poetry of logical ideas." In The Universe and the Teacup, Los Angeles Times science writer K. C. Cole discusses some of the ways this "poetry" can be used to look at science and other realms of experience.

Mathematics, Cole explains, enables us to "translate the complexity of the world into manageable patterns," whether we're trying to comprehend the risks of smoking or the usefulness of DNA matches in criminal investigations. Cole also looks at how mathematical principles apply in unexpected fields. One chapter, for example, vindicates the theories on voting rights that cost Lani Guinier her Justice Department nomination in 1993.

Without relying on a single equation, Cole's gently humorous prose helps make mathematics unthreatening to laypeople, enabling them to better understand the world in which they live. ... Read more

Reviews (33)

The Politics of Truth and Beauty
Despite the title, not once in this book is an actual mathematical problem presented coherently. Instead, Cole drones on about the virtues of cooperation, the importance of minorities, and other left-wing philosophical themes. I'm a liberal and would tend to agree with her politically, but that ignores the central problem with this book: Cole's failure to make the distinction between mathematics itself and beliefs that just happen to be justified by statistics or quasi-mathematical reasoning.

Perhaps The Universe and the Teacup is best described as a meta-popularization, since virtually all of Cole's sources are themselves popularizations. She hypes such familiar staples of popular science writing as fuzzy logic, chaos and complexity theory ("all the rage these days" -- I thought that's what they said back in the 80's), and Godel's theorem (both "a shattering blow" AND "a staggering blow to our sense of certainty"), without showing that she understands any of these things on more than a superficial level. (I don't claim to be an expert on these topics, either, but then again I didn't write a book about them.)

For general readers interested in how mathematics relates to everyday life, I'd recommend John Allen Paulos "Innumeracy"; for a survey of modern mathematics, both "From Here To Infinity" by Ian Stewart and "Archimedes' Revenge" by Paul Hoffman succeed where "The Universe and the Teacup" fails.

The Leonardo da Vinci of science writing!
That's a direct quote from Amazon, and boy, were they right. Only Cole would link the O.J. Simpson trial to the discovery of the top quark in order to explain various roads to truth. The best part is the relationship between beauty and truth, in which she explains the unexplainable--showing how Einstein's theories (and in fact, all modern physics) is based on the notion of symmetry. But there's also so much less etheral food for thought here: the geometry of fairness, for example!

what is truth exactly
Being disenchanted with religion, I picked up this and other books in search of some other kind of truth. I do feel as though after reading this book I have a much better understanding of what 'truth' is and what it's not. I think those who nit-pick about their claims of little discrepancies in the book are really missing out on the bigger picture. The book is full of interesting little facts and factoids but the interesting thing to me was to see how she's pulled together these common insights that are gained from so many fields of study. I think this was just about my favorite book ever.

So many better choices out there.
Chapter two, second paragraph: "The Milky Way galaxy contains 200 billion stars..."
Chapter two, a few pages later: "Fifteen billion is also more or less the number of stars in the galaxy." Obviously, the number of stars in the galaxy is not precisely known, but we do know that 15 billion and 200 billion are two different things. One of the author's "truths" is self-evidently not true. Purveyors of "truth and beauty", whether scientists, gurus, philosophers, spiritual leaders, or journalists, often regard their subject and their audience far too casually. Here we have a case in point. Perhaps most books contain 'typos' and the miscues inherent to humanity, but here it seems that both the author and the editor were asleep at the wheel, something that needs to be addressed if the book achieves a second printing (and I don't see why that would happen).
The subject is truly fascinating; or at least it should be -- the relationship of aesthetics, mathematics, and logic. At the deepest levels of the human intellect's inquiries, the answers are all about a mysterious mathematical beauty. The reality of this escapes most people, which is why the "National Bestseller" heading on the cover of Cole's book intrigued me. Apparently the book has enjoyed a larger readership than most such popularizations. Unfortunately the superficial, disjoined 'newspaper style' of science serves the material poorly. The writing rambles almost aimlessly. The books of many mathematicians and physicists have examined the relationship of reality, reason, mathematics, and aesthetics. Devlin's 'The Language of Mathematics' is very good. Fairly recent works by Penrose, Davies, Rucker, Berlinski, Greene, and others come to mind. Some of these books are far better than others. This volume is one of the others.

how to write a book in five minutes
Should it be that easy to write a book? Collect all the bits and pieces from newspapers' weekend-supplements and almost scientific coffeetable-talk and toss in some currently fashionable phrases concerning physics and mathematics, stir until the lumps have disappeared and do not bother with the spices of explanation and insight. If you love math and physics, stay off ! ... Read more

Book Description

The idea of a "category"--a sort of mathematical universe--has brought about a remarkable unification and simplification of mathematics. Written by two of the best-known names in categorical logic, Conceptual Mathematics is the first book to apply categories to the most elementary mathematics. It thus serves two purposes: first, to provide a key to mathematics for the general reader or beginning student; and second, to furnish an easy introduction to categories for computer scientists, logicians, physicists, and linguists who want to gain some familiarity with the categorical method without initially committing themselves to extended study. ... Read more

Reviews (8)

Objects and maps are everywhere
Excellent book for non-professional mathematicians, like me (I'm a software engineer), who wants to understand modern mathematics and apply its ideas in analysis of complex problems. Lots of pictures and diagrams (compared to terse wording in other mathematical books) really help to understand and master the subject. I think most of negative reviews come from professional mathematicians, but they don't need this book.

Very uneven, but still useful
As a topic in itself, category theory should need not to wait until grad-level to be described just because that may be when category theory's power can really begin to be exploited, but unfortunately, most of the category theory books I have looked at presume that level of mathematics.

Similar to what other reviewers noted, I would also say that this book demonstrates the potential of creating a good high-school/undergrad level intro to category theory. But unfortunately, that potential is not quite realized here.

There are hokey intermittent "conversations with students", as a tool to describe ideas, that are more distraction than aid. Some of the examples given are rather condescending in their simplicity. Yet, at other times the authors seem to breeze through more difficult topics with little or no examples. And the organization seems erratic - there is no clear sense of a gameplan as to where they are leading the reader or how all the concepts fit together.

Functors are surprisingly almost glossed over, as if they were relatively unimportant. There are exercises throughout the book, but with no answers provided, they are not really very helpful.

Having said all that, with some focused effort on the reader's part, the ideas do come forth, and admittedly, the authors do cover a fairly broad spectrum of aspects of category theory. This is certainly a non-trivial topic to try and teach, and an introductory book cannot be faulted for not carrying every notion to the nth-degree of either breadth or depth.

Category Theory is one of those topics that (to me) appears 'ho-hum' until you see it actually applied to various topics. The authors have necessarily had to perform a balancing act between describing concepts while not getting caught up in excessively complex examples. I think this will leave many readers less than satisfied, but realistically, the book would have been twice as long had they really delved deeper into examples (or they would have had to be very terse in the actual descriptions of category theory, which is the choice most authors writing for a more mathematically-inclined audience seem to make - e.g., _Mathematical Physics_ by Geroch (good book!) or _Basic Category Theory for Computer Scientists_ by Pierce).

If you are mathematically astute, you probably will find this book tedious. But if you are not a grad+ math major, then this book may well be worth the effort as a way to begin to learn a very profound and powerful set of tools and concepts.

Heavy Hitter Strikes Out
I sure hope Schanuel wrote this book and the publisher simply tacked on
Lawvere's name for marketing purposes. This text is a fantastic
example of why research mathematicians should not write for John Q.
Public. The random, pointless examples scattered throughout the book
remind me of the "word problems" that were so popular in high school
algebra texts written after the Chicago School hijacked the educational
textbook market.

After teasing the reader with examples of real mathematics, e.g.
Pick's Formula, the authors stop short of actually proving a theorem
and scurry back to their shelter of objects and arrows where they can
safely field trivial questions by ersatz students with politically
correct names.

Perhaps Category Theory is just not something that is accessible to the
general public? High school math teachers (I assume one intended
audience for the text) that can achieve even the slightest appreciation
of why Eilenberg and Mac Lane invented Category Theory are surely as
rare as rocking-horse poop.

What I would really like to see from someone as eminent as Lawvere write a
first year graduate level book that covers elementary set theory and/or
logic using Category Theory. Translating Model Theory and Topoi(1.) to
this level would be a good start. College math professors are really
the only people in a position to understand and transmit this beautiful
theory to aspiring mathematicians.

1. Model Theory and Topoi, Lecture Notes in Mathematics 445,
Springer-Verlag 1975

Keith A. Lewis ...

A retract in search of a section
There is a wonderful course in category theory for high school students, just begging to be excavated from this multi-layered book.
Please don't be put off by the disjointed and uneasy combination of materials that cluster around certain themes. You know you will have a lot of work to do when the same definition (of monomorphism) is presented both on page 52 and also on page 336.
With all the elementary themes covered in many varying ways, it would be best to consider this book as having been structured as a retract for which your job will be to construct the appropriate section.

A Good Introduction
As a first introduction to Categories, this book is well written, clever, simple and very clear. However, I was disappointed with it. From the notoriety of the authors and the, yes, cool illustrations I assumed it would be a gem. However, it fell short. I've been toying with Category Theory for a few years, and every time I try to get into a book on Categories I get stumped at the notions of Functors and Natural Transformations. This book, however, dealt with neither at length, despite the fact that Category Theory originated around the notion of Natural Transformations in the first place. (As I understand it at least.) That said, there are many very cool passages in the book, including a functional analysis of a Chinese restaurant and an elegent exposition of Brouwer's Fixed Point Theorem.

Still, for my purposes, I prefer Robert Goldblatt's "Topoi: The Categorical Analysis of Logig" and Michael Barr's "Category Theory for Computing Science". As both are intended for non Category Theorists, both build their presentations of Category Theory from sratch. Sadly, I think both are out of print. Not for the faint of heart, I'm told Saunders Mac Lane's "Categories for the Working Mathematician" is the classic. (It's on my wish list.) ... Read more

Book Description

Algebraic Geometry is the study of systems of polynomial equations in one or more variables, asking such questions as: Does the system have finitely many solutions, and if so how can one find them? And if there are infinitely many solutions, how can they be described and manipulated? The solutions of a system of polynomial equations form a geometric object called a variety; the corresponding algebraic object is an ideal. There is a close relationship between ideals and varieties which reveals the intimate link between algebra and geometry.Written at a level appropriate to undergraduates, this book covers such topics as the Hilbert Basis Theorem, the Nullstellensatz, invariant theory, projective geometry, and dimension theory.

The algorithms to answer questions such as those posed above are an important part of algebraic geometry.This book bases its discussion of algorithms on a generalization of the division algorithm for polynomials in one variable that was only discovered it the 1960's. Although the algorithmic roots of algebraic geometry are old, the computational aspects were neglected earlier in this century.This has changed in recent years, and new algorithms, coupled with the power of fast computers, have let to some interesting applications, for example in robotics and in geometric Theorem proving.

In preparing a new edition of "Ideals, Varieties and Algorithms" the authors present an improved proof of the Buchberger Criterion as well as a proof of Bezout's Theorem. Appendix C contains a new section on Axiom and an update about Maple, Mathematica and REDUCE. ... Read more

Reviews (6)

Symbolic computation
This book explains and illustrates the algorithms used by symbolic math packages such as Mathematica, Maple, CoCoA, MatLab, MuPAD,... to solve problems involving polynomials in many variables, and along the way teaches the elements of real algebraic geometry-- most mathematics texts concentrate on the complex-variable version. It is not just for undergraduates; electrical engineers, for instance, should see it. Lots of pictures!

Easiest introduction to Algebraic Geometry
This is the easiest introduction to algebraic geometry and commutative algebra, the authors had done a great job in writing a book that assume very little from the readers. To learn some algebraic geometry, you can either start with this book, or you can spend a year to read a lot of background materials in algebra and then go to a Graduate Text like Harris' book. Of course, if you want to be an expert in algebra, you eventually need a lot of background, what this book can help you is to offer you a quick start, much quicker than you would ever imagine.

Straightforward and lucidly written
Having just finished using this text in the course of an undergraduate seminar, I can attest to the fact that the authors' style is outstanding - they are able to synthesize an enormous amount of material in this volume and present it in a manner that is highly accessible to almost all students of mathematics. The presentation of important theorems (for example, Hilbert's Nullstellensatz and Basis Theorem) along with just the right amount of copncrete examples makes for a book of superb quality. All-around, I highly recommend this volume to anyone who has an interest in learning about Algebraic Geometry.

Good book
I don't have the second edition of this book but did read the first, and the authors do a fine job of introducing the reader to the computational side of algebraic geometry. I will forego a chapter by chapter review therefore, but no doubt the second edition (which I do not own) is as well-written as the first. I would recommend it to anyone interested in the many applications of algebraic geometry and to those who need to understand how to compute things in algebraic geometry. The good thing about this book is that it gives a concrete flavor to a highly abstract subject. Algebraic geometry, through its applications to coding theory, cryptography, and computer graphics, is fast becoming the subject to learn. It is no longer just an esoteric, high-brow subject but one that is taking on major importance in the information age. Even without applications though it is a fascinating subject, and readers will get a taste of this in this book.

The best book on the topic
I learned the basics of Groebner bases from this book and its the best introductory book on this topic. Authors have explained all concepts with the help of examples which makes it readable for people from other fields also. It also talks about applications of Groebner bases to other fields. The book gives lot of exercises which help in understanding the contents more. I recommend that if you wish to learn Algebraic Geometry and Groebner bases then this is the book to start with. ... Read more

Book Description

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics.He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics is about things that really exist. ... Read more

Reviews (1)

Very Readable Textbook
This book is designed to prepare students for upper division math courses-like abstract algebra and advanced calculus-in which mathematical rigor and proofs are emphasized. The authors have made a serious effort to present the material with clarity and sufficient details to make it accessible to students who have completed two courses in calculus. Much of the material covered is fairly standard for such a textbook. Chapters 1-9 are devoted to basic topics from set theory and logic (including four proof techniques: direct proof, proof by contrapositive, proof by contradiction, and mathematical induction), equivalence relations, and functions, as well as a special chapter under the heading, "Prove or Disprove." Chapters 10-13 cover cardinalities of sets and proof techniques applied to results from number theory, calculus, and group theory. In addition, the authors have a web site which includes three additional chapters (Chapters 14-16) dealing with proofs from ring theory, linear algebra, and topology. Thus instructors using this book will have a wide choice of options in selecting the material they want to include after the basic concepts are covered.

The emphasis throughout the book is on proofs and proof techniques--how to recognize proofs, understand them and, above all, how to create and write them. The presentation is leisurely and thorough. Many examples are given, and discussions are always presented with all the details that students at this level would need to follow the argument. There are ample exercises at the end of each chapter (including those in the web site) that range in difficulty from routine to moderately challenging. The book also contains answers and hints to odd-numbered exercises.

There are two features of this textbook that I believe are helpful to students and that set this book apart from others at its level: the detailed way in which proofs are analyzed, and the inclusion of a chapter on how to write mathematics well. In most cases, before a proof is presented the authors offer a "proof strategy": a discussion pointing out what needs to be proved and how one might go about proving it. Also, many proofs are followed by "proof analyses" in which some of the interesting or unusual points of the proof are commented on. I believe that students would find these discussions very helpful. In particular, these discussions offer students concrete pointers from which they would learn how to cope with abstract mathematical proofs.

The chapter on writing mathematics (Chapter 0) is unique. While some mathematics textbooks encourage good writing and might devote a few paragraphs to the subject, the present volume offers a brief manual on mathematical writing. The authors begin by explaining why writing is important in mathematics and follow that by offering detailed instructions that would help students in improving their writing. From specific advice like, "Never start a sentence with a symbol" to explanations of "common words and phrases that are peculiar to mathematics," there is a wealth of material on writing from which students can learn. I believe that, by its very existence, this chapter on writing would have a positive influence on students writing.

This book can be used either as a textbook for a course such as the one described above or as a reference that students can consult on certain topics.

Fawzi M. Yaqub
Emeritus Professor of Mathematics
SUNY College at Fredonia ... Read more

Book Description

Starting with the concept that mathematical logic is not a collection of vaguely related results, buta method of attacking some of the most interesting problems which face the mathematician, the author sets the tonefor this classic introduction. The basic concepts are presented in an unusually clear and accessible fashion, keepingin mind the original purpose of mathematical logic to build the foundations of this vast edifice of knowledge in away that helps and intrigues the working mathematician as much as the philosophically minded student of logic.This book has served as a rite of passage to many mature and accomplished researchers. ... Read more

Reviews (1)

Rock-solid introduction to Mathematical Logic
Since my first contact with mathematical logic, I've always seen it as a kind of brainwashing, forcing one's mind to work based on several little pieces of thought. Nevertheless, it can be described as "a necessary evil", because the mindless use of mathematical logic throughout mathematics is very treacherous, as it can be seen in the problems regarding the axiom of choice, the Banach-Tarski paradox in measure theory, the issues about the undecidability of certain assumptions in set theory, and the very limitations of mathematical logic.

Usually, of course, most work in mathematics doesn't require a deep knowledge of rigorous mathematical logic, but it's always a good thing to a serious mathematician to have some acquaintance with it, even if it's just to avoid boobytraps. Then, it's hard to find a better choice than Shoenfield's book. After a long absence from the book market, A K Peters made the wise decision of reprint this masterpiece. Although most of its contents are fairly standard for a book on mathematical logic (unlike the equally marvellous out-of-print book of Yu. I. Manin, which has a more philosophical slant and concerns itself with issues such as quantum logic, literature, etc.), it provides proofs for many propositions that in most of the literature are only stated. It has, of course, some extras not generally found in other books, as for example issues concerning constructibility of sets.

But the most important characteristic of this book is its clarity and precision. It doesn't waste time in unnecessary stuff, and shows why we need mathamatical logic at all. Although it lacks some topics (for example, it doesn't discuss other axiomatic set theories besides Zermelo-Fraenkel. This is not so nice, because it lacks the distinction between classes and sets, one of the tenets of the Goedel- -Bernays-von Neumann set theory, although it is conceptually easier than this last one. But maybe it's a pedagogical choice, because the set theory we all intuitively know is more or less based in Zermelo-Fraenkel), its main concern is pedagogy, so this limitation has a sound reason: this book exposes mainly the logic present in the math most mathematicians and alike scientists (mathematical physicists, etc.) use. Its solidity and razor-sharp precision is great to instruct these people to be more careful with the math they use.

Besides that, some of the missing topics can be complemented by Mendelsson's "Introduction to Mathematical Logic", which is a bit more "merciful" book, which, by the other side, welcomes the thoroughness of Shoenfield. ... Read more

Book Description

This introductory text covers the key areas of computer science, including recursive function theory, formal languages, and automata. It assumes a minimal background in formal mathematics. The book is divided into five parts: Computability, Grammars and Automata, Logic, Complexity, and Unsolvability.

Key Features
* Computability theory is introduced in a manner that makes maximum use of previous programming experience, including a "universal" program that takes up less than a page
* The number of exercises included has more than tripled
* Automata theory, computational logic, and complexity theory are presented in a flexible manner, and can be covered in a variety of different arrangements ... Read more

Reviews (5)

Pure mathematical view of Computability and Complexity
This is not a common book on Computability and Complexity as Hopcroft-Ullman, Sipser or Papadimitrou. You won't find here too many words describing topics: you'll find the power and elegance of a superlative mathematical approach from one the best authors of the century in the field. Conversely, you'll find here a detailed and elegant treatment of the whole history of computational models that starts at the Primitive Recursive Functions, something you won't find in the other books above mentioned.
A special note goes to the chapter on Blum's complexity, which is about the only good place where I found it and from where I studied for my course on Complexity I.
For this reason the book requires quite more attention than others, but it really worths all the time one can spend reading it. Truly understanding Computability and Complexity as Professor Davis teaches them with this book is in my opinion a definitely high achievement, bringing the sensation that you grasp it totally, with no space for ambiguity or weakness.

Beautiful overview
The authors of this book define theoretical computer science as the mathematical study of models of computation, and they do an excellent job of detailing the major results in the theory of computation as related to mathematical logic. Mathematicians, programmers, and philosophers will find the book an effective one in which to learn computability theory, and it serves well as a textbook for courses in the subject.

After a brief review of elementary mathematics and mathematical logic in chapter 1, the authors move right into the consideration of computable functions in chapter 2. They choose a particular abstract programming language in which to study the computability theory, which is built from variables, and programs that can be built from lists of instructions. Examples of programs are given, which have a Fortran flavor, with examples of computing partial functions. Unfortunately, a plethora of GOTO statements appear in the programs, and throughout the rest of the book, which is surprising given the publishing date. The use of these GOTO statements in the book is a major annoyance.

Then in chapter 3, the authors discuss primitive recursive functions, beginning with a treatment of composition, followed by the all-important concept of recursion. The class (PRC) of primitive recursive functions is introduced, and shown to be computable. The primitive recursive predicates are introduced, followed by a proof that the existential and universal quantifiers over an element of a PRC class are also PRC. This is followed by a discussion of minimalization and Godel numbers.

The next chapter is very interesting, wherein the famous halting problem is discussed and related to Church's thesis. The authors stress, most importantly, that an algorithm cannot be defined outside of the choice of a language, and therefore Church's thesis cannot be proved as a theorem. The authors also introduce recursively enumerable sets and show, via diagonalization, that non-recursively enumerable sets exist. They give an interesting example of a function that is computable but not primitive recursive.

The next chapter extends the results to strings of symbols instead of just numbers, and the authors introduce programming languages for doing string computations. One of these is the famous Post-Turing language, which they use to discuss the halting problem, with a variant used in the next chapter on Turing machines. The authors discuss the famous halting problem for Turing machines in this chapter. This is followed in chapter 7 by a discussion of productions and simulation of nondeterministic Turing machines. A very lucid treatment of Post's correspondence problem is given.

Things get somewhat more complicated in chapter 8, where the authors attempt to classify unsolvable problems. It contains one of the best discussions I have seen in the literature on oracles, and the authors give a very clear treatment of arithmetic hierarchies.

The second part of the book reads more like a book on compilers, as the authors delve into the area of grammars and automata. Regular languages, deterministic and non-deterministic finite automata are discussed, and Kleene's theorem, which states that regular languages and finite automata define the same languages, is proven. The context-free languages, so familiar from the study of compilers, are discussed also, along with a proof that a context-free grammar can be reduced to a Chomsky normal form grammar. Pushdown automata, needed for accepting context-free languages, are treated in detail. The authors give a good explanation here as to the additional facilities needed for a finite automaton to decide if a word belongs to a "bracket" language. Chomsky hierarchies are also discussed, and the authors motivate nicely the need for a linear bounded automaton to accept context sensitive languages.

Part three of the book is an overview of mathematical logic, and begins with a treatment of the propositional calculus. The satisfiability problem is discussed for this system, along with how to reduce formulas to normal form. The important compactness theorem is given a very detailed proof. Predicate calculus is then discussed, and Herbrand's theorem, which effectively reduces logical inference in predicate calculus to a problem of satisfiability of universal sentences, is proven. This theorem is fascinating and has important applications to automated theorem proving, as it ties together semantic and syntactical properties of a formal system. The Godel incompleteness theorem and the unsolvability of the satisfiability problem in predicate logic is proven.

In part 4, issues in computational complexity are addressed, the measure of complexity given in terms of the Blum axioms. This is a very abstract way of introducing complexity theory, as it introduces measures of complexity that more general than time and space complexity. The fascinating gap theorem, comparing program performance on two computing machines via complexity measures, is proven. This is followed by a detailed discussion of the speedup theorem, which essentially states that there is a wildly complicated recursive function such that for any program computing this function, there exists another program computing the function that works a lot faster for almost every input. The polynomial-time computability is discussed along with the famous P vs NP problem, with the discussion given in terms of Turing machines. Examples of NP-complete problems are given.

The last part of the book covers semantics, with operational and denotational semantics defined and compared. The emphasis in this part is on programming languages and constructions that one would actually find in practice, and so the preceding chapters on computable functions must be extended. The concept of an approximate ordering is introduced to allow for the instantaneous of a computation at some point before its completion. The denotational semantics of recursion equations and infinitary data structures are discussed, with the latter put it in to deal with the sophisticated systems that are constructed here. The discussion here is very involved, but the authors do a fair job of explaining the need for these types of data structures. The same is done for operational semantics, and the authors finally show that the computable numerical functions are actually partially computable. They then show the existence of computable irrational numbers.

My favorite book on the theory of computation
I first learned computability from this book and I loved every minute of it. It has lots of material and is superbly written. In fact, I think the chapters on logic are the most painless way to learn that subject. There are many other books around on this subject, but this is the ultimate!

CS Theory at it's best
I haven't found a better book on the Theoretical foundations of Computer Science. However since this IS theory the text can be a bit cryptic. Still, I'd recomend this book to any PhD Candidate or full Professor. Even a lowly Master's student like myself could use it.

This is a wonderful text about the theory of computation.
It taught me how to think about the theory of computation. The exercises added to the second edition are a big improvement over the first editon. ... Read more

Book Description

In one of the finest treatments for upper undergraduate and graduate level students, Professor Suppes presents axiomatic set theory: the basic paradoxes and history of set theory, and advanced topics such as relations and functions, equipollence, finite sets and cardinal numbers, rational and real numbers and more. Exercises. References. Indexes.
... Read more

Reviews (4)

Still interesting...and still important.
One does not hear about set theory too much these days, no doubt due to the de-emphasis of foundational discussions in mathematics. Foundational questions of course were the focus of much attention in mathematics in the early twentieth century, this taking place because of the many paradoxes in set theory and due to the influence of the philosophers. Set theory, the theory of types, and mathematical logic are still very important though in computer science and in artificial intelligence, due to the needs in these fields for knowledge representation, computational models of intelligence, and automated reasoning. This book could serve to introduce these topics or as an historical reference to the issues as they were hotly debated in the last century.

The first chapter gives an informal introduction to the notion of a set, first-order predicate logic (notions of bound and free variables and quantification), and the Zermelo-Fraenkel axioms of set theory. The author describes the difficulties in the "axiom of abstraction" in the writings of Frege as pointed out by Bertrand Russell. It is pointed out that the axiom of abstraction is in fact an infinite collection of axioms, thus motivating the concept of an "axiom schema". The axiom schema that is used explicitly in the book is the "axiom schema of separation" due to Ernst Zermelo, which he formulated in order to make precise the notion of a statement as being "definite". More of the set-theoretic paradoxes are discussed, along with their classification due to F.P. Ramsey into "linguistic" and "semantical" ones.

The advantage of an older book on set theory is that more of the underlying details are explained, instead of just being formally developed. The author gives a thorough discussion of the concepts throughout the book, beginning with an organized development in chapter 2. He begins immediately with discussing the distinction between the object language and metalanguage, and the symbols to be used in the object language: constants, variables, logical connectives, quantifiers, and grouping symbols. These symbols are used to construct formulas, a subclass of which, the primitive formulas, are defined recursively, and which all formulas in the object language can be expressed in terms of. Throughout the book though the author uses additional notation that allows formulas not to be written in terms of primitive formulas. This is done to make the text more readable, but he requires that the added notion satisfy the criterion of eliminability and non-creativity. The notion of a set is defined formally, and then the axiom of extensionality, which gives a criterion for two sets being equal, and the axiom schema schema of separation. The pairing axiom, which gives the existence of a non-empty set; the sum axiom, which gives the existence of the union of a family of sets; the power set axiom, which gives the notion of the set of all subsets of a set; and the axiom of regularity, which prohibits infinite descending sequences of sets, are all discussed in detail.

Chapter 3 treats relations and functions, so important not only in mathematics but in computer science, especially in the theory of relational databases. Then in chapter 4, the author begins a study of cardinality and the cardinal numbers, proving that the finite cardinal numbers have the properties of the natural numbers, as one would expect. The author is careful to point out the need for the axiom of cardinal numbers in this study. Chapter 5 then goes into the theory of ordinal numbers, wherein it is emphasized that no special axioms are needed for the development of this theory. The author is also careful to note the special problems that arise in defining the arithmetic of natural numbers, such as defining addition recursively without using set theory. But including the apparatus of set theory does allow the replacement of the recursive definition by a proper definition. The axiom of infinity is brought in to permit the construction of arithmetical operations as certain sets. The theory of denumerable sets is then discussed, followed by one of the most fascinating concepts in all of mathematics: the theory of transfinite and infinite cardinals.

The author then shows that set theory can allow the construction of the real numbers, which takes place after the construction of the rational numbers. The famous "Dedekind cut" is discussed, along with the method of Cantor, which defines real numbers as equivalence classes of Cauchy sequences of rational numbers. The author uses the Cantor approach in the rest of the book. He also proves the famous Cantor theorem on the non-denumerability of the real numbers, and gives a brief discussion of the Continuum Hypothesis.

Chapter 8 then gives an overview of the fascinating topics of transfinite induction and ordinal number theory. Recursion theory makes its appearance again in the transfinite recursion for ordinal numbers, using the axiom schema of replacement. The non-commutativity of ordinal addition and multiplication is brought out, and the falsity of Fermat's Last Theorem and Goldbach's Hypothesis in ordinal number theory is shown. The author then shows to what extent cardinal number theory can be done without using special axioms by defining cardinal numbers as initial ordinals. The axiom of choice however is needed to show that every set has a cardinal number. The author then restates the Zermelo-Fraenkel axioms in their final form at the end of the chapter.

The final chapter gives an overview of the most controversial topic in all of set theory, if not in all of mathematics: the axiom of choice. The author shows that the use of this axiom allows one to prove that an infinite set has a denumerable subset, and he shows the equivalence of the axiom of choice with the numeration theorem, the well-ordering theorem, Zorn's lemma, and the law of trichotomy. The counterintuitive Banach-Tarski paradox is discussed, and the author shows the existence of axioms which imply the axiom of choice.

Decent intro book.
This is a basic introduction to axiomatic set theory. You dont need much experience with informal set theory or formal logic to begin it. The book is rigorous and follows a definition - theorem - proof format, broken with clear exposition and historical notes. Enough formal mathematical logic is introduced only to express the axioms (that is, formal proof systems are not used or discussed). He uses the ZF (Zermelo/Fraenkel) system and gives footnote comparisons to the NBG (von Neumann/Bernays/Godel) system. In chapter 4 he introduces a special axiom (outside of ZF) to simplify the development of cardinal arithmetic, and this involves the addition of a new primitive notion. All theorems relying on this special axiom are clearly marked. While this admirably allows Suppes to avoid employing the axiom of choice or developing a much more complicated strictly ZF-based construction of the cardinals, it does make the book unacceptable for more advanced readers. In chapter 6 he gives a detailed construction of the rational numbers and the real numbers (using Cauchy sequences). He uses only the axiom of separation until the axiom of replacement becomes necessary. He does a good job explaining why each axiom is needed and how it arose historically. The book is comparable to Monk's _Introduction to Set Theory_, though a little easier and less advanced.

An Excellent Text for Self-Study
This book presents a rigorous, axiomatic development of classic set theory, introducing the axioms as needed and founding nearly all results upon theorems derived earlier in the book (or on the axioms themselves). It is genuinely gratifying to see the development proceed in such a regimented fashion, from basic sets to natural numbers to reals, and then on to transfinite induction and the axiom of choice. There are numerous exercises; no answers are provided, but the intelligent reader who proceeds carefully should not find this a hindrance. It is however, not a modern book; readers who want to understand current ideasin set theory (inaccessible, supercompact cardinals, etc.) should look elsewhere.

EXERCISES
I skim this book one day while looking for some reference books at my local bookstore.The clarity of the book makes it a good book for complementing set theory courses.The examples are given in a consice manner without obstructing the learning material to the readers. However, the lack of answers to the given problems makes the book unfitting to first time readers who may want to learn the subject just for plain curiosity; none of the steps are hinted for solving any problem. It is unfair having to buy this book because readers buying this book are expected to be experts in the field. Rather, it should also consider general science readers who have interest for the subject and want to learn the material. I think having full answers to problems allows all readers to have a good understanding of the subject at hand since it clarifies the bridge of ideas that mathematicians are trying to let the world see. Furthermore, it motivates readers to critically think challenging problems once enough practice has been establish through the ones with answers. However, because there is a lack of communication and guidance to achieve a full census with the subject, this book has a mild sour taste. So in hopes that the author of this book improves the book for the benefit of his readers, "Please provide answers next time." ... Read more

Book Description

This book presents a systematic, unified treatment of fixed points as they occur in Godels incompleteness proofs, recursion theory, combinatory logic, semantics, and metamathematics.Packed with instructive problems and solutions, the book offers an excellent introduction to the subject and highlights recent research. ... Read more

Book Description

This book categorizes, identifies and explains the various techniques that are used repeatedly in all proofs and explains how to read proofs that arise in mathematical literature by understanding which techniques are used and how they are applied. ... Read more

Reviews (7)

Do not buy this book!
I had one of my math courses at Macalester College use this book in addition to the main textbook. It is one of the most boring readings in math ever. If you want a beginning book that would teach you how to do problem-oriented math, you had better consider George Polya's How to Solve It or consult a math teacher near you;-) In order to further underscore my poit, I will just say that it was not only I but the whole class that disgruntled against the book and we hated it so much that the professor does not use it anymore.

Big Improvement in Second Edition
Contrary to the review by the person from Louisiana I feel the second edition is better than the first. The typesetting is greatly improved, and there are a few new tools for your toolbag in the second edition.

As to the criticism that the second edition only has solutions for the odd numbered problems, the reviewer failed to mention that there are twice as many problems in the new edition and that all the problems from the first edition were carried into the second (along with their solutions). I found it more satisfying working through the second edition knowing that the problems were correctly solved - not because the answer matches the back of the book - but because the arguments are compelling and demonstrably correct.

I heartily recommend this book to anyone who feels mystified at the process of writing proofs.

Great Introduction to proofs
When I order this book it was not at all clear if I would like it, because of it low grades contain and examples. I was very surprised when I received it, because the book is written very clearly. The author make a great job when he explains the technics of proof. The exercises and the examples are definitely too easy for an undergraduate or graduate but you must take a look at it just for the explanation that the author gives for the technics of proof. The examples and exercises are mostly for high school students. The price of the book make it very affordable. It worth the price. I would recommend this book to anyone who want to introduce himself into the basic of proofs.

YOUR FIRST BOOK IN MATHEMATICS
YOU REALLY NEED THIS BOOK IF YOU ARE TRYING TO STUDY ABSTRACT MATHEMATICS BY YOURSELF OR WITH A TEACHER. LIKE ELECTRONIC ENGINEERING THAT LIKE TO DO MATHEMATICS THIS BOOK REALLY HELP ME TO UNDERSTAND HOW TO READ PROOFS AND MADE MY OWNS.

no comparison to 1st edition
while the author's approach to proofs is refreshingly logical, he takes away from the wholeness of his work by including ONLY odd-numbered answers. his first edition had all of the answers to all of the questions. most students of mathematics are self-taught and highly motivated. with the exclusion of even- numbered answers, the author has saved a few pages and a bit of work.

i feel fortunate to have a copy of the first addition which i purchased at a used book store for a buck. you can't beat that! :-) ... Read more

Book Description

Eli Maor examines the role of infinity in mathematics and geometry and its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement. "Maor explores the idea of infinity in mathematics and in art and argues that this is the point of contact between the two, best exemplified by the work of the Dutch artist M. C. Escher, six of whose works are shown here in beautiful color plates."--Los Angeles Times "[Eli Maor's] enthusiasm for the topic carries the reader through a rich panorama."--Choice "Fascinating and enjoyable.... places the ideas of infinity in a cultural context and shows how they have been espoused and molded by mathematics."--Science ... Read more

Reviews (7)

To the limits of infinity
Even as children we have a vague concept of infinity, thinking of it as the largest number; remember the familiar exchange of "I dare you!" "I double-dare!" "I dare you to infinity!" "I dare you to infinity plus one!" or some such thing. Even then, we realize to some extent that infinity is not truly the largest number because there is always something bigger.

Maor gives a brief history of the concept of infinity and how it fits into the worlds of art and science. This is a generally good book although there are a couple of errors (such as when he mixes up the concepts of whole numbers and integers). Maor is good at illustrating just how big infinity is without getting either overly technical or metaphysical (a problem with the last book I read on infinity, whose title I forget). Maor also shows how there are different sizes of infinity; it is often hard to conceive that there are more irrational numbers between 0 and .00001 then there are rational numbers along the whole number line.

With the exception of the couple of minor errors mentioned above, this is a good book. Infinity is a difficult concept to grasp, but with this book, you can do just that.

The finest generally accessible math book I have seen.
I have read other books by Eli Maor. After "June 8, 2004", I had doubts about this one, but I wanted to clarify some Cantorian issues. Once I started this one, I could not put it down. It also answered my questions.

Most, if not all of the material should be accessible to a motivated high school senior. It presents the history of infinity in a manner as fascinating as a mystery or adventure story (a true one, better than fiction); it reminds me of "Terrible Lizards" in that sense. Interspersed with the historical narrative, but easily separable, it contains good solid mathematics in a clear and concise fashion. Only the section on Bertrand Russell's paradoxes failed to satisfy.

The Infinite in Nature
Maor titles his book "a cultural study," but the cultural work domainates the second half of the book. The first half--which is more interesting than the second half--is a truly amazing analysis of just what the infinite is. Maor goes into detailed discussion of the nature of infinity in prime numbers, irrationals, rationals, and so on. The patterns, surprises, and mysteries of number fields are discussed with perfect clarity. Other issues involving infinity are mapped with equal precision and clarity for the beginner. The second half of the book involves studying the infinite in Escher's art, in geometric systems before and after Euclid, and in art, theology, science, singularities, and etc. Overall, for those interested in the mecahnics of nature, this book is not to be passed up!!! But be cautioned, this book is for beginners, for those only interested in grasping basic concepts of mathematics, not intense formulas that lead to singularities, for example. I am a graduate student in philosophy, so it served my purposes to the maximum level.

Very interesting
If you have an appreciation for math, you'll love this book. If not, you'll get an appreciation of math. The writer tells interesting stories about various fields of mathematics, geometry and even astronomy. You will learn stories of famous people who contributed to the advance of the science and the ultimate quest to reveal what lays beyond the infinite. Highly recommended

A masterpiece of scholarship!
Maor is a great scholar! He's a professional mathematician with a deep knowldege of history of mathematics and astronomy and also a great writer. In addition, he has a deep love for music and culture. The book will give you a great sense of the diversity of mathematics. I strongly recomends all the four books by Maor! ... Read more

Book Description

Recursive methods offer a powerful approach in dynamic macroeconomics. This book contains both an introduction to recursive tools, including standard applications such as asset pricing, and advanced material, including analyses of reputational mechanisms and contract design. The tools are presented with enough technical sophistication to get the reader started working on practical problems. When numerical simulations are called for, the book provides suggestions for how to proceed, as well as references for further reading.

The applications cover many substantive issues in macroeconomics, such as equilibrium asset prices, market incompleteness, wealth distribution, fiscal-monetary theories of inflation, government debt, optimal labor and capital taxation, time consistency and credible government policies, optimal social insurance, economic growth, and labor market dynamics. ... Read more

Reviews (7)

get it free
This is a great book. But you can download the second addition free on Sargent's website, so I wouldn't recommend buying it.

Perfect book
This is a perfect book for three reasons; i) it is perfect for those who wish to learn modern macroeconomics. The book develops necessary knowledge and tools to be applied to dynamic economics, ii) Sargent is one of most prominent and leading macroeconomists of the world, and he should be Nobel prize winner in Economics, iii) the book is published by MIT.

Not for beginners
When I recently left my job as cryer in a grim, north-eastern town, I was made the head of recursive macroeconomic theory at a major international bank. I could have done with a simpler introduction than this, to be honest, as my knowledge of RMT was limited. But now I hold my own in meetings simply by spouting a few long words from this book (mainly "macroeconomic" and "recursive" - theory doesn't seem to impress as much) and delegating to underlings.

Not the first book in Dynamic Macro, but excellent afterward
The first time i read the book, i'm sure this should not be the first text book for Dynamic Macroeconomics everyone should read. It's better to read somewhere else as an introduction to the idea of dynamic macroeconomics. Romer 'Advanced Macroeconomics' and Stokey, Lucas, Prescott 'Recursive Methods' are more appropriate to start. After gainning some similarity with Dynamic Methods, it would be much better to study models about macroeconomics presented in the book.

This book is the presentations of various models using Dynamic / Recursive Macroeconomics. It makes them easier and time-saving to study many kinds of model in a semester. It's GOOD & HELPFUL IN THIS SENSE. However, it might not be a good book for study in depth. You are better to study from the original papers for the same topics.

I think, this book is similar to Tirole 'Theory of Industrial Organization' in spirit, but different in content. They both show the simplified version of various models in the fields.

If you think you like this style, you would like to have it. But if you don't, it might be better just to skim (from the library) and read the original papers.

Hope this comment would be helpful for you to make a decision :)

a review from a skeptical student, now a convert.
This text is perhaps the most accessible introduction to modern macroeconomics available. What I feel to be the greatest contributions of the text are the problems-- in each chapter, they start from the basics and build upon one another until you are formulating elaborate models that are the basis for much of the current discourse in the literature. The approaches used are so powerful and the questions tackled so varied that you cannot help believing that the recursive method is the future, not only for traditional issues in macroeconomics, but throughout the discipline. Hey, the book stands out so much, I decided to write a review! ... Read more

Reviews (1)

Nice coverage of Wolframs published work
This is a nice collection of wolframs work on cellular automata (which first appeared as a number of papers in various physics journals). It is a nice coverage of cellular automata, but it would have been nice to give more credit to von Neuman for his pioneering work in cellular automata theory.

There is also an annoying habit for all of his work to concentrate on deterministic cellular automata, and the mathematics is constrained to this. Recent work has indicated that the origin of complexity in our universe is from random sources that are preserved.. not that the complexity all came from the initial conditions.

It is especially interesting to note in his book how the different rules of cellular automata play out to create varying degrees of complexity. It takes a very specific rule set indeed to allow for interesting complex behaviors to show up, as evinced by the long search Conway undertook to discover "life".

Hopefully Wolfram will comment on the recent research that indicates that complexity is introduced into our universe through nondeterministic phenomena. He also should have presented Fredkins ideas about reversible computation to more fully flush out the relationship between cellular automata, computability and reversibility. ... Read more

Book Description

A complete treatment of fundamentals and recent advances in complexity theory Complexity theory studies the inherent difficulties of solving algorithmic problems by digital computers. This comprehensive work discusses the major topics in complexity theory, including fundamental topics as well as recent breakthroughs not previously available in book form. Theory of Computational Complexity offers a thorough presentation of the fundamentals of complexity theory, including NP-completeness theory, the polynomial-time hierarchy, relativization, and the application to cryptography. It also examines the theory of nonuniform computational complexity, including the computational models of decision trees and Boolean circuits, and the notion of polynomial-time isomorphism. The theory of probabilistic complexity, which studies complexity issues related to randomized computation as well as interactive proof systems and probabilistically checkable proofs, is also covered. Extraordinary in both its breadth and depth, this volume:
* Provides complete proofs of recent breakthroughs in complexity theory
* Presents results in well-defined form with complete proofs and numerous exercises
* Includes scores of graphs and figures to clarify difficult material
An invaluable resource for researchers as well as an important guide for graduate and advanced undergraduate students, Theory of Computational Complexity is destined to become the standard reference in the field. ... Read more

Reviews (2)

Blah.
I can only concur with my fellow Ann Arborite - dense, badly written, often wrong, the works. There's really no perfect textbook that covers this material, but Papadimitriou beats this one handily - plus, he quotes the Clash.

Terrible
My pity goes out to those unfortunate enough to take a course using this book.
Aside from being nearly inscrutable, this text is also incorrect in many places. Exercises range from trivial to nearly impossible, with no indication of which is which. Proofs are verbose with little indication of the relevant ideas.
Perhaps a useful reference, but do not expect to learn much from this text.

Example errors (in most recent edition at this time):
p.21 Thus, A $\in$ DTIME(c t(n)) (Didn't define A...)
p.47 EXP is closed under $\leq_m^P$ (EXP (def p.21) isn't closed this way... maybe they meant EXP POLY)
p.66 Note that the graph G'... do not satisfy the triangle inequality. (Actually it does satsify the triangle inequality... maybe they meant strict triangle inequality here)

Ad Nauseum. ... Read more

Amazon.com

The Loom of God takes an entertaining, indeed playful, look at numbers and mathematical patterns and the mystical properties that have often been ascribed to them. Clifford Pickover takes you on a romp through numerological history, introducing both its characters (such as the great mathematician Pythagoras) and its concepts: triangle numbers, "perfect" numbers, Fibonnaci numbers, and more. Pickover describes how ancient--and sometimes no-so-ancient--cultures and religions interpreted the significance of various numbers; he examines the geometry of Stonehenge and considers the probability of earth's annihilation by collision with an asteroid. While many authors could chronicle the history of mathematics and its relationship with mysticism and religion, few could do it with the verve and flair that Pickover manages. ... Read more

Reviews (3)

The Loom of God is a rich source for bored programmers
If you've gotten bored of hunting the wumpus, check out The Loom of God. It covers vast mathematical areas, many of which make excellent computer programs. One intriguing concept presented was that of "sociable numbers." That is, numbers A, B, C, D, E (or more) for which the factors of A add up to B, the factors of B add up to C, and so one, until the factors of E add up to A. As you might imagine, the search for sociable numbers requires either VERY powerful computers, or VERY innovative algorithms... none of which are discussed in the book. It does however, provide an excellent introduction this and many other mathematical topics.

Entertaining bored programmers is not, of course, the primary focus of the book, but it alone makes the book worth buying.

Math made fun, non-fiction and fiction in one
The author has done a wonderful job in taking math and making it interesting. By weaving non-fiction and fiction into one coherent story, Pickover has been able to take math and give it a life of its own. Certainly a good addition to any mathematicians library, but an even better addition to everybody's library because everybody can understand it!

"Dazzling tour of number and the numinous." - Publ. Weekly
"Clifford A. Pickover leads readers on a dazzling, lushlyillustrated tour of the intersection of number and thenuminous." - Publisher's Weekly, April 1997 ... Read more

Book Description

This book covers the emerging area of logic computation—the use of advanced mathematical methods to solve complex problems in logic. This logic system may be used for the construction of expert systems, such as automated handwriting analysis, traffic control systems, and data mining. The topic of this book is essentially the starting point for solving these problems, i.e. ways in which a complex problem may be broken down into a number of smaller ones. ... Read more

Reviews (1)

Required Reading for all students of logic computation
A serious book for those interested in algorithms to solve the 'satisfiability' and 'minimum satisfiability' problems.While mostly dealing with the theory behind Truemper's practical algorithms, the book also provides discussions of applications to major combinatorial problem classes. ... Read more

Book Description

Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.
... Read more

Reviews (2)

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

Mapping the Outer Limits of Computation
The book introduces the theory of computability and non-computability to the mathematically-comfortable. The theory of recursive functions provides entry to that theoretical territory at the limits of what is computable and what is solvable. The theory is relevant to important philosophical questions and also in the theory of computing and what is possible (and never possible) by use of computing machines.

The result for philosophy is establishment of absolutely unsolvable problems and undecidable questions, even ones that can be completely and precisely formulated using rigorous logic. The result for computing is problems that are absolutely unsolvable by use of a computer program.

So what problems are theoretically solvable by a computer program? First, the Universal Turing Machine (UTM) is presented along with the famous demonstration that all universal computers are equivalent in the sense that any one of them can be made to simulate any of the others, using a suitable representation.

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

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

Davis maps the boundary between the impossible (the unsolvable) and the merely inhumanly difficult (the computable). With that foundation, one can move on to other work that introduces what has been learned about computational compl