Reviews (2)

The standard work, but there are better
This is the "standard" book on the subject. It is referenced everywhere. It has a lot in it. I have not read it cover to cover, just used it for reference, but if you are new to the subject I think Kolmogorov and Fomin looks beeter, and Shilov's books look better too.

Reviews (2)

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

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

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

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

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

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

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

Editorial Review

Until about 1975, logarithms were every scientist's best friend. They were the basis of the slide rule that was the totemic wand of the trade, listed in huge books consulted in every library. Then hand-held calculators arrived, and within a few years slide rules were museum pieces.

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

Editorial Review

At the very beginning of his book on i, the square root of minus one, Paul Nahin warns his readers: "An Imaginary Tale has a very strong historical component to it, but that does not mean it is a mathematical lightweight. But don't read too much into that either. It is *not* a scholarly tome meant to be read only by some mythical, elite group.... Large chunks of this book can, in fact, be read and understood by a high school senior who has paid attention to his or her teachers in the standard fare of pre-college courses. Still, it will be most accessible to the million or so who each year complete a college course in freshman calculus.... But when I need to do an integral, let me assure you I have not fallen to my knees in dumbstruck horror. And neither should you."

Reviews (6)

Avoid this book, droogs
I've taken a first semester Discrete Math course, and am currently taking the second semester of it. I bought this book on a whim, hoping it might supplement my text, or at least clarify a few points. It fails at both of those things. Here's why:

1) Abundant errors: I read the first 15 pages and found at least one *serious* typo per page (i.e. a typo that could impede learning). Plus, the grammar ranges from illegal to ambiguous. Thankfully, I was familiar with all of the material that I was reading -- were I not, severe confusion and discouragement would have been the result.

2) Poor examples: They're too abstract or too simple -- and there aren't even very many of them. Oftentimes, he contradicts what he's trying to illustrate due to a small oversight or typo. It's truly bad.

3) Gratuitous brevity (yes I know that may sound paradoxical): The author uses compound sentences in his definitions; sometimes going as far as to define two or three concepts IN THE SAME SENTENCE! It's infuriating.

4) Chapter Zero: This deserves its own rant section. Chapter zero contains nearly all of the material from the first four chapters of my current textbook: Logic, Set theory, Induction, Relations, etc. Somehow the author crams all of it into about 24 pages (plus 4 or 5 pages of exercises). He fails at clarity or lucidity. It's an ambomination -- it reads like lecture notes (you know, the ones only the professor looks at).

OK -- I WANTED to like this book. It's kind of cute, I'll admit it. And the price is sweet. But friends, you get what you pay for. Even after I came across the first 5 or so serious typos I was willing to forgive. Eventually, the sheer amount of contradictory examples and ambiguous sentences riled me up so much that I considered tearing the book in half. Really. I doubt I'll ever open the thing again.

Excellent Text
As with any Dover text, it is important to remember that this text is designed to teach the material, not to coddle the reader.This text provides broad and deep coverage of the various topics that fall under discrete mathematics (set theory, boolean logic, graph theory, etc.) with clarity and simplicity.This book is not designed to help you pass a test, but is instead designed to help you grasp and understand the topic, which it does very well.Easily the best book I own on this topic (I often joke that the author covered my first semester course on discrete math in the first chapter!).

Too succinct for the discrete novice
I learned much more from the Schaum's Outline (ISBN: 0070380457 -Schaum'sOutline of Theory and Problems of Discrete Mathematics (Schaum's OutlineSeries) by Seymour Lipschutz, Marc Lipson (Contributor), Seymour Lipschultz).

That book overcomes the two shortcomings of this one: for aself-proclaimed introductory work on discrete mathematics, this textcontains too few worked out in-chapter examples, and too many omitted stepsin the reasoning.On this latter point, there were many times my readingbrought me to the phrase "It follows from the definition that..."or "obviously..." when, for me, it didn't follow, or it wasn'tobvious. Contrary to another reviewer's assessment, I found quite a lot oftypos, but none too serious.To its credit, the book does contain a lot ofend-of-chapter problems with solutions, and it is inexpensive.

Reviews (4)

Excellent Introduction to Combinatorics
A COURSE IN COMBINATORICS covers a great breadth of topics under the label of "combinatorics," including graph theory, enumeration, and some algebra. The book is comprehensive; the instructor can pick-and-choose appropriate material from the huge array provided without detriment to understanding.

Each chapter is written in a friendly, accessible manner: plenty of interesting and instructive examples follow the clear definitions and preliminaries. To give the reader an idea of the topics presented in the book, a list of chapters follows:

1. Graphs
2. Trees
3. Colorings of graphs and Ramsey's theorem
4. Turan's theorem and extremal graphs
5. Systems of distinct representatives
6. Dilworth's theorem and extremal set theory
7. Flows in networks
8. De Bruijn sequences
9. Two (0,1,*) problems: addressing for graphs and a hash-coding scheme
10. The principle of inclusion-exclusion; inversion formulae
11. Permanents
12. The Van der Waerden conjecture
13. Elementary counting; Stirling numbers
14. Recursions and generating functions
15. Partitions
16. (0,1)-Matrices
17. Latin Squares
18. Hadamard matrices, Reed-Muller codes
19. Designs
20. Codes and designs
21. Strongly regular graphs and partial geometries
22. Orthogonal Latin squares
23. Projetive and combinatorial geometries
24. Gaussian numbers and q-analogues
25. Lattices and Mobius inversion
26. Combinatorial designs and projective geometries
27. Difference sets and automorphisms
28. Difference sets and the group ring
29. Codes and symmetric designs
30. Association schemes
31. (More) algebraic techniques in graph theory
32. Graph connectivity
33. Planarity and coloring
34. Whitney duality
35. Embeddings of graphs on surfaces
36. Electrical networks and squared squares
37. Polya theory of counting
38. Baranyai's theorem

The problems in the book are generally very rich and well-written, with helpful hints from the appendix that provide motivation but do not spoil. However, the relative difficulty of the problems is not readily made appparent, so over- or underthinking of problems often occurs with misjudgments.

For the interested high-school student to the beginning graduate, this book is ideal for the study of combinatorics. Truly a nice read that connects many areas of mathematics and combines them into a thing of true beauty.

A nice tour of combinatorics
The first word that comes to my mind when I think of this text is "encyclopedic". It contains around 40 chapters, hitting most of the high points of combinatorics that a graduate student should see. The exposition is generally good with nice examples. The one thing that I fault it for is the number of statements that the authors claim are "obvious". In a way, this is good, because it makes you pay attention and understand the material, but sometimes the statement isn't obvious until you've thought about it for an hour and written out a lengthy proof. At that point, it does become completely obvious and you can't believe that you ever thought it wasn't, so I can understand why van Lint and Wilson fell into the trap so often. (In fact, I've heard that Wilson even stumbles over some of those points in lectures.) This is a great book to have on your shelf if you need somewhere to look up combinatorial ideas.

A gentle introduction to combinatorics
This book was the text for a graduate-level course I took.The presentation is very laid-back, much like the lecturing style of one of the authors (Wilson), and so it was quite readable (unlike many other mathbooks which you have to stop every few pages and pick apart everythingbefore it sinks in).

Combinatorics is a relatively recent development inmathematics, one which is generally easy to explain, but with manydifficult open questions.Van Lint and Wilson do an excellent jobexplaining, but there are a few places where the reader needs to know somebackground to place the particular problem in the appropriate mathematicalcontext.Understandably, if the authors were to include all themathematical machinery needed, the book would be huge!Instead, they havechosen to describe as many facets of the field as possible, and thereforehave written a broad, well-balanced book which approaches the topic in anon-threatening way.

My one criticism, then, is that there is a lack ofdepth in several areas of the book, with further discussion of advancedtopics or open problems.But even so, I can appreciate the omission forthe sake of accessibility.

Reviews (24)

A pourry of combinatorics
I want to start saying that this is a book designed for Engineers, not for Mathematicians. It focuses on the tecniques, not on the arguments. This is not a book about combinatorics, it is a wide raging introduction (it lacks on definitions, and his proofs are a lot far away from mathematical ones). The Enegineers can use this book as a good reference. The Mathematicians can improve their lateral thinking, for them (well: us) it is book about problem-solving strategies.
I will never use this book as a textbook for a graduate/undergraduate course, it can be helpful if used with another book about combinatorics: when you study a combinatorial object, you can read from this book the techniques it involves.
The exercises are extremely exciting, when I read this book I spent a lot of time about its exercises (proportion read:solve = 1:3), and they led me to interesting results.

Steep learning curve, the definitive prerequisite for TAOCP.
Why I got this book:
It's a great feeling to know how computers work, when I decided that I want to make a career and a life out of computers, as its truly a passion for me, I delved deeper, discovering the true beauty in the Science part of Computer Science, so I decided to get Donald Knuth' "The Art of Computer Programming" - to describe that seminal, huge work, it's like biting more than you can chew while trying to drink from a fire hose, moreover, the technical and mathematical prerequisites for the work are sometimes too demanding, they require a huge amount of experience with discrete mathematics, although I had some lectures and read some books, none came close "Concrete Mathematics", it covers, from ground up (though with a dangerously steep learning curve) a lot of discrete mathematics topics, it is by far the most extensive work I've read about Sums and really teaches the algorithmic problem solving thinking skill the authors preach so much about, with small amusing comments written by actual students of this course, a comfortable format, and very good writing skills, you can feel these guys are great professors who enjoy this material and are passionate about teaching it.

Recommended, though some better, less steep, introductionary text books are probably out there.

Enjoy.

Only one problem with this textbook
Basically, I like this textbook. The material is interesting, the way the authors presented the material is inspiring, and they provided a lot of jokes to make even studying for exams not that boring. But there is one big problem which made me decided to rate this book only 3 stars instead of 5 stars: the authors like to use non-standard notations. For example: m\n means "m>0 and n=mk for some integer k". One of the worst thing in scientific world is writing things others cannot read, and the authors did this by introducing many strange notations. These things makes the good work sometimes almost unreadable. This is not computer systems in which we use "cp" for the copy command and "cd" for change directory command.

Reviews (3)

Very helpful
Combinatorics is a bit of an oddity. Although a few principles (like pigeonholing) apply in many cases, every combinatorial problem has unique features. Attacking a new situation is almost like starting all over again, unless you can recognize an old problem in your new one.

This book gives a number brief case studies. Its 18 chapters (not counting intro and closing) span a variety of interesting topics. Cameron doesn't write down to the reader - it takes serious thought and some mathematical background to get full value from the reading. The examples are nowhere near as concrete as you'd expect in a popularized version. Still, the author avoids opaque references to specialist terms, and keeps the text approachable.

I have personal reason to like this book more than it's high quality warrants. I was thumbing through it in a store, and skimmed a page that described Kirkman's schoolgirls (a two-level problem in selecting subsets). Quite abruptly, I realized that those charming young ladies exactly represented a problem I had in connecting the parts of a multiprocessor. One or two references later, I had a practical way out of a potentially ugly quandry. This material is not just fun for its own intellectual challenge, it has application to real engineering, too.

Good introduction to combinatorics!
Every discipline has key introductory texts that motivate the subject, whet the appetite for more, and guide a novice to see the forest despite the trees.'Combinatorics' by Peter Cameron is one such gem!Combinatorics has a reputation for being a collection of disparate clever ad hoc arguments.The author has carefully presented binding principles such as double counting, the pigeon-hole principle, generating functions, enumeration via group actions, sets ofdistinct representatives,...see the book for more, that makes coherent combinatorics as a discourse, despite its diverse application.

Reviews (5)

so-so
This book was my first contact with measure theory. I read this for self study -- more or less as a leisure book. The material is aimed at undergrads, and probably doesn't assume much past plain ol' college calculus. However, the more you know, the easier it will be to read. Any experience with analysis, and proofs will be helpful. And, in chapter 7, "Function Spaces", a linear algebra course will come in very handy.

Anyway, it's not a hard book to read, but it is very dry. Because the book is so short, there is not much room for anything other than a list of definitions, lemmas and theorems. There isn't really much insight. All the way through chapter 7 I was basically plodding along, simply because I wanted to finish the book. However, I'm glad that I did, because I found chapter 8 really fascinating. I think this chapter (Fourier Series in L^2) really ties the book together because you get to see measure theory and lebesgue integration working in harmony with linear algebra. I never really liked linear algebra that much until I read this chapter.

Unfortunately, chapter 9 was a let down and I actually quit reading a few pages before the end of the book. I had already got what I needed out of it. It's a good intro to measure theory if you just want to see what it is, and not really go into detail with it. A lot of the lemmas, propositions and corollaries are left as exercises. I tried to do a handful of problems from every chapter, especially the ones that fill in the text, and had little or no difficulty with any of them.

I would have given this 3 stars if not for chapter 8. For the price, I would recommend it, especially if you love calculus, but never liked linear algebra, because it will hopefully tie them together for you. Now I can't get enough linear algebra! I know, it's sick ;) Then, with your newfound love of linear algebra, read Hubbard and Hubbard's _Vector_Calculus,_Linear_Algebra,_and_Differential_Forms:_A Unified_Approach_, which is currently blowing my mind :)

Good review of Lebesque integration
This is more about Lebesgue integration, covered in the first 112 of 154 pages, than about Fourier Series.But do not be fooled by the low price; this book makes for very good review of the Lebesgue theory, well organized and concise.Or with moderate mathematical maturity, the reader could learn the theory here.

Gentle Intro to Measure Theory
I am using this book with two advanced undergraduates, after working through a basic introduction to analysis text the previous semester.Each section is about 3 or 4 pages long, giving the main results and most proofs, although some are left as exercises.Each chapter ends with a nice set of exercises, most of which are accessible to students who feel comfortable with epsilon/delta proofs.The book is very short and concise, 145 pages plus an appendix summarizing basic analysis results on sets, countability, functions, and sequences.

Chapter 1 reviews the Riemann integral and some of its drawbacks. Chapter 2 introduces the idea of outer measure and measurable sets, all on the unit interval.The next two chapters discuss properties of measurable sets and measurable functions.Chapters 5 and 6 then cover the Lebesgue integral and convergence theorems.The last three chapers introduce L2 spaces, Fourier series, and proofs of convergence.

Reviews (6)

Getting started in math analysis
This book by Shilov covers the fundamentals in beginning analysis(both real and complex). It has in common with Walter Rudin's book (entitled 'Real and Complex Analysis') that it covers both real functions (integration theory and more), as well as Cauchy's theorems for analytic functions. Shilov's book is at an undergraduate level, and it can easily be used for self-study. The Dover edition is affordable. Rudin's book is for the beginning graduate level, and it is widely used in math departments around the world. Both books have stood the test of time.
Comparison of Shilov with Rudin: Rudin's 'Real and Complex' has become an institution, and I have to admit I have loved it since I was a student myself, but conventional wisdom will have it that Shilov is a lot gentler on students, and much easier to get started with: It stresses motivation a bit more, the exercises are easier (some of Rudin's exercises are notorious, but I find the challenge charming--not all of my students do though!), and finally Shilov gets to touch upon a few applications; fashionable these days. But that part easily gets dated. I will expect that beginning students will enjoy Shilov's book.
Personally, I find that with perseverance, students who keep at it with Rudin's book, will end up with a lot stronger foundation. They are more likely to have proofs in their blood. I guess Shilov can always serve as a leisurely supplementary reading to Rudin.
There will never be another book like Rudin's 'Real and Complex', just like there will never be another van Gogh. But the fact that we love van Gogh doesn't prevent us from enjoying other paintings.

Possibly too simple
As Shilov write in the introduction "I have tried to accomodate the interests of larger population of those concerned with mathematics" and at that he seems to do. However, the book does require some mathematical background as he appears to omit defining a few things. I believe the book would be ideal for those who want a handy reference, or an easier book when struggling with an analysis course.

However, for the more mathematically inclined readers, the problems are often too easy, and many things are proved that could be better left as exercises. For a more difficult Analysis book, I would reccomend Rudin.

A wonderful text -- Highly recommended!
I purchased this book as a reference book for my first analysis course.It is very well written, and easy to follow.Dr. Shilov has a very nice way of organizing this text:He puts all the definitions at the beginning of the chapter and the subsequent sections are results of those definitions.It makes for a very quick reference.His presentation of the included proofs is also very nice.There were several occasions I found myself thumbing through it for a second perspecitve.

As far as the actual material presented, Dr. Shilov starts off with funtions of one real variable, then rather quickly generalizes to complex variables and N dimensional functions, so you'll quickly see metric theory and some topology.He does keep in mind this is intended for undergrads and first year grads though.

Reviews (7)

Should appeal to both mathematicians and poets
Maor has written a book for both mathematicians and poets. Since he is a mathematician himself there is, to be sure, plenty of math in Maor's book. But the book should also appeal to the aesthetic side of many readers (me included) by exploring human perspectives of infinity, such as how we try to relate to the concept at a personal level, and how different people have tried to capture the notion in art and prose.
The book is arranged in four parts, dealing with the mathematical concept of infinity (how it shows up in algebra, etc.), geometrical infinity, aesthetic infinity (both art and poetry) and cosmological infinity.
The section on mathematical infinity has the typical assortment of historical examples, beginning with examples like the runner's paradox made famous by Zeno. There are also examples of infinite series that converge, including examples of how ancient mathematicians invented infinite series for transcendental numbers like pi. There's a plethora of little tidbits found throughout this section in little mini chapters that are short essays, only a few pages long, that give surprisingly succinct, tantalizing, and often delicious examples of mathematical infinity. Reading this book I was struck by what good reading it makes for any student preparing to take a class in calculus.
Some of the author's most interesting material is the author's discussions about infinite series. I particularly enjoyed hisexamples how the associative property doesn't hold for infinite series (a non-intuitive fact that often comes as a surprise to many new students). Ordinarily, if you have a string of numbers that are connected by addition (x1+x2+x3+..+xn) for example, you can rearrange their order and get the same result. One of the strange things about infinity, though, is that rearranging the terms in an infinite series can result in the limit of the series changing from one number to another.
Of course no discussion about infinity would be completewithout mentioning Cantor, which Maor does with particularclarity for first-time readers. Indeed, this is one of the things Ilike about Maor best - he's written a book that is fun to read, even if you already know most of the stuff. It's engaging and entertaining, and full of "ahh" and "ohhh" even when you find yourself reading about something you studied many years ago. At the same time this is a good introductory text for anyone (I'm thinking youngsters in high school) who wants to start exploring some of these mathematical concepts, and need a friendly introductory text. If you can manage first-year algebra you have the tools you need to follow what Maor is talking about, though be advised that he doesn't shirk when producing equations, though most of the math is relegated to the appendices.
The section on geometric infinity is punctuated by nice illustrations and those geometrical shapes that you may have heard about - the ones with things like finite volume but infinite surface area. This was one of those rare occasions where I found myself wishing Maor had gone a little further. Instead of simply showing how such objects exist in mathematics, he really should have explained the apparent "paradox" (it's not hard). Instead, he makes the example more of a "paradox" than it really is by mixing metaphors in talking about "painting" the surface. Of course mathematicians have one idea about painting a surface (mathematical paint has no thickness), but the beginning reader is likely to be mostly confused - too bad, since Maor clearly has the skill to explain the trick.
Maor's exploration of the infinite is (almost) infinite. He has a wonderful section on tiling, and some brilliant plates representing some of the best mathematical art that attempts to depict the nfinite. The section on cosmology and the infinite is a nice summary of the history of astronomy and how astronomers and cosmologists have vacillated over the years between a cosmos that is infinite, then finite and bounded. I thoroughly enjoyed reading this book. It is well written and both easy and fun to read. My only complaints are rather minor. Several times Maor treats infinity as a "big number" (it's not a number at all, and he makes that clear, but his terminology on this score isn't asconsistent as it should be). And, he refers to mathematics as a science. Well, I suppose he's entitled to his opinion on that one,
though I imagine it will continue to be debated. Count me as one of those who puts mathematics in the "tools" category, separate from science.
The fact these inconsequential gripes are all I can find to complain about tells you what a really fine book this is. If you love mathematics, this book really needs to be in your library.

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.

Reviews (7)

Test the limits !
In my teaching of the basic tools of mathematical analysis; and even going back to my student days, I noticed the hurdle that separates the beautiful definitions from the `messy' examples. Often students tell me that the theory looks so easy, `but how do we construct an example to illustrate the limits of the theory?' -----A counter example?
Part of the difficulty is that the definitions involve quantifiers; and how do you check the quantifier `for all' ? And on top of that, there are the axioms of set theory: the axiom of choice, or one of its equivalent variants.

The lovely little book by Gelbaum-Olmsted was a savior to many of us when we started out in math, and it appeared first in 1961. But I had almost forgotten about it until by accident (while browsing in the bookstore) I stumbled over a new edition of it about a year ago, a lovely Dover reprinted edition. And so affordable !

In all the other books you learn about the wonderful things that are true about convergence, sets on the line or in the plane, modern variants of the so called Fundamental Theorem of Calculus, and in Gelbaum-Olmstead you learn the things that aren't true. And then there are all the lovely Cantor constructions, The Devil's Staircase, space filling curves, and much more; beautiful, but little known constructions going back to Lebesgue, and some to Riemann.

But more importantly the book gives students an edge when they have to do the assigned exercises in your analysis course. Many told me that the book is a 'secret weapon'.

Palle Jorgensen, October 2004.

Great Book -- A "must have"for your bookshelf
The counterexamples here are a wonderful aid to educating intuition about definitions in Real Variables. It may sound strange, but I always thought of this book as entertaining reading: If you glance at the table of contents, you'll may find youself saying, "wait, no, that can't -- well, I guess so, but what does that look like?" In later conversations you may find youself saying: "wait a second, I seem to recall seeing somewhere a continuous nowhere differentiable function," or someting of the sort. Unfortunately, there are not a whole lot of these creatures in the book, but they are worth spending some (enjoyable) time with.

Reviews (2)

The theory of object-oriented typing
Abadi and Cardelli have written a very thorough, formal analysis of the basic theory of object oriented (OO) languages. The first parts of the book present the mathematical tools needed for the discussion. There, they extend formal logic so that it can make statements about classes and subclasses, the kind of statements that must be made in order to determine whether a program, even a whole programming language, make good sense.

The authors introduce a notation I haven't seen elsewhere, having to do with the object instance bound to a method instance. This subtlety describes a number of language constructs, including Java's inner classes. They add further notation for describing languages where object structure can be highly dynamic. Although of theoretical interest, the dynamics do not apply directly to commercial OO languages such as Java, C++, or Ada. Dynamics may also complicate reasoning about the type systems. Other type analyses are simplified by acting on the static program representation. Dynamic analysis will have to invoke heavier mechanisms, like the ones used in traditional formal verification of programs.

I have to admit that I haven't gone through the book's entire content because that discussion doesn't address my current needs. Right now, I'm working with very static systems; this book creates solutions for problems that I don't have. Still, I've gotten some value out of the basic discussion of covariance and contravariance in subclassing, so the book has helped me somewhat.

This book is intended for researchers in computing theory, or possibly for practitioners who develop languages and language tools. It's way beyond the needs of most OO programmers, and is decidedly not for OO beginners. If you need deep, rigorous understanding of OO foundations, beyond what's needed for mainstream languages or applications, then this book may be very helpful.

//wiredweird

Reviews (11)

Nice book for readers with a background in math
I really enjoyed reading this book that describes some background on Euler and his work. It is written in an informal style, so for people with a math background it reads like a novel.

The book is not suitable for people who want to learn more about the person Euler, but do not have a math background, because 75% of the book is about real math (equations). So if you don't enjoy reading equations, do not buy the book.

Summary: as enjoyable as the other Dunham books, although a bit more expensive (but still worth the money).

William Dunham has done it again!
With the publication of this, his third book, Dunham has once more shown himself to be a master himself of mathematical explanation. Unlike his previous two books, The Mathematical Universe and Journey Through Genius, which covered results by a variety of mathematicians, this book focuses on selected results that sprang from the remarkable mind of Leonard Euler, one of the most prolific and important mathematicians of all time. What sets Euler apart is not only the vast quantity of his output (the publication of his collected works, the Opera Omnia, spans six dozen volumes, or over 25,000 pages in all!), but also the breadth and originality of his work. Not only did Euler contribute to a wide array of mathematical fields -- from number theory to complex analysis to geometry -- but in many cases, he was the founder of those fields. For example, Euler invented the field of analytical number theory, and he was the first mathematician to recognize the importance of and to discover the important properties of complex numbers.

This book in many ways resembles Dunham's Journey Through Genius. As in that book, Dunham has selected 15 or so theorems to present in detail, and he makes an effort to keep the proofs similar in spirit to the original proofs. Although the proofs are complete and the book is full of equations, they are accessible to anyone with a high school level of mathematics education. But in addition to the proofs, Dunham also provides historical context, as well as commentary on how later mathematicians used and improved upon Euler's work. For example, we learn that Euler began to loose the sight in his right eye at the age of 32, and that despite his virtual blindness by the age of 65, he continued his prolific rate of output until his death at age 84.

The book's title is taken from a quote by Laplace, who said, ``Read Euler, read Euler. He is the master of us all.'' Indeed, if you have any interest in mathematics, you will almost certainly find yourself in complete agreement with Laplace's sentiments by the time you finish reading this wonderful book. ...

Reviews (6)

Solid introduction to analysis
I used this book in conjunction with Little Rudin while studying analysis.While Rudin provided deeper insight, it provided almost no motivation or exposition.Rosenlicht and Little Rudin are very good complements to one another.I found Rosenlicht much easier to learn from, but use Rudin as a reference now that I understand the theory.

Solid Introduction to Metric Spaces and Continuous Functions
Introduction to Analysis by Maxwell Rosenlicht is another bargain from Dover Publications. I used this inexpensive mathematics reprint to help fill in gaps in my background before tackling more advanced mathematics. I found the first 150 pages to be challenging, but manageable. I had less success with the last the last 100 pages.

My college work was limited to applied mathematics, but in recent years I have developed some familiarity with metric spaces, topology, and analysis. (I previously reviewed Metric Spaces by Victor Bryant and Introduction to Topology by Bert Mendelson.)

Throughout his text Rosenlicht emphasizes how the same idea or theorem can be formulated in various ways. I found his approach to be quite helpful in clarifying more abstract representations of key ideas.

The first two chapters review set theory and the real number system and should be familiar to many readers. However, Chapter 3 (Metric Spaces) and 4 (Continuous Functions) are critical and will require substantially more effort. My pace slowed dramatically.

For the reader new to metric spaces, Chapter 3 will likely be challenging, although metric space concepts are not really that difficult, just unfamiliar.

Rosenlicht demonstrates how statements concerning the open subsets of a metric space can be translated into statements concerning closed subsets, or alternatively into ones concerning sequences of points and their limits. Rosenlicht closes Chapter 3 with definitions and discussions of Cauchy sequences, completeness, compactness, and connectedness.

Rosenlicht begins Chapter 4 by illustrating that the familiar epsilon-delta definition of continuity of functions can be reformulated using the metric space open ball concept, or by using open subsets in metric spaces. He further explores the interdependence of theorems about continuity, limits, and convergent sequences. Chapter 4 concludes with discussions on continuous functions on a compact metric space and on continuous sequences of functions (analogous to sequences of points).

In chapters 5 (Differentiation) and 6 (Riemann Integration) we discuss the fundamental ideas of calculus using concepts and theorems introduced in the previous chapters. At this point I revisited a favorite calculus book by Salas, Hille, and Etgen. I was pleased to find that a had a much better understanding of topics that had previously been somewhat nebulous. Rosenlicht was indeed helping me.

Nonetheless, I had substantial difficulty with the longer and more complex proofs common in the remaining 100 pages, chapters titled Interchange of Limit Operations, the Method of Successive Approximations, Partial Differentiation, and Multiple Integrals. I again visited other textbooks, but this time looking for help with topics like power series, the fixed point theorem, and the implicit function theorem. Although familiar with partial differentiation and multiple integration, I only skimmed the final chapters. I hope to return to Rosenlicht later after exploring another text on analysis.

I do recommend Introduction to Analysis, especially for students looking for a review of analysis. This Dover reprint is a good buy, even if like me, you find the later chapters to be rather difficult.