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on January 28, 2006
I greatly enjoyed this book, but it's not for everyone. To appreciate it, there are two requirements: (1) You must enjoy, or at least tolerate, Wallace's quirky writing style, with its mixture of the conversationsal and the erudite, its frequent footnotes, abbreviations, and discursions. (2) You must have a certain level of mathematical sophistication. This is not one of those popularizations of math for those who never got past high school algebra. It could perhaps be described as a history of calculus, analysis, and set theory, and specifically of their attempts to come to grips with the infinite and the infinitessimal and make them mathematically valid. The focus is on theory and rigor--which maybe makes it sound dry, but it's not, if you like that sort of thing. It's not that Wallace himself gives lots of rigorous arguments, but that he talks a lot ABOUT the search for mathematical rigor.

Reading this book is a little like sitting in on a class taught by an inspiring yet quirky professor (and, indeed, Wallace makes frequent reference to an inspiring, quirky teacher of his own). Such a class would have a prerequisite--I'm not quite sure what, maybe at least a semester or two of calculus. Probably, the more you know about calculus and related subjects, the more you'll get out of this book.
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VINE VOICEon September 5, 2006
A reviewer, below, cites Zeno's Paradox as a metaphor for his experience reading E&M. This is precisely how I felt. Wallace's endlessly annoying flourishes aside, the first half of E&M is very interesting and accessible. Parts of the second half are, too, but those parts are islands in a sea of mounting incomprehensibility. Too much is left unexplained ("you'll have to trust me"-type asides, generally in "IYI" footnotes, abound) and too much else is expressed, formally, in arcane mathematical notation. Granted, Wallace's subject-matter is highly abstruse, and anything remotely approaching mathematical rigor would be both impossible in a book of E&M's size and impenetrable to a popular readership. Faced with these obstacles, Wallace makes a go of writing both to lay readers and specialists. Specialist reviewers on this page criticize Wallace for making numerous errors. As a lay reader, with a deep interest in mathematics (the practice, theory, history, and foundations of), I was (and remain) very eager to read a sophisticated (incipiently rigorous) treatment of the topic of infinity. The first half of E&M delivered a moderately sophisticated (mathematically unrigorous) treatment of the metaphysics of infinity. The second half attempted something like the treatment I had hoped for, but it was pitched entirely too high. The only hope a reader realistically has of navigating the second half of E&M is if (s)he brings a high level of mathematical sophistication to it. Such a reader, however, would almost certainly gravitate towards a genuinely rigorous treatment of the subject (like Dauben's biography of Cantor, oft-cited by Wallace). I'd like to think that I -- an interested and motivated lay reader -- am part of the target audience of E&M. As such, though I credit Wallace for his efforts, I cannot applaud his results.
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on September 15, 2008
Since DFW has committed suicide, we will not see an edition revised by him. In re-reading the reviews, it appears that style means a lot. I personally found the book witty. It was a little slow sometimes because of the convolutions he introduced in style, but mostly I kept plowing (and chuckling) through. The librarian who sent back the book did a disservice to some readers. Not everyone likes to learn in the same way. With that kind of attitude, many years ago I would have had Rudin's books removed as too concise to be useful. Of course, there are many mathematicians who love those books for just that reason, and I would have done them a disservice.

I am a physicist with a math minor. To me, the best part of this book was his explanation of why mathematicians insist on the epsilon-deltas of mathematical rigor. No one ever did that before. If I could have read this in high school, I probably would have finished my math major as well as my physics major. Instead, the whole epsilon-delta thing seemed ad-hoc and inexplicable in purpose. I could never accept the need for rigor demanded in advanced analysis.(a drunken prof and Rudin's book didn't help either) DFW showed how a crisis in dealing with the infinite and with infinitesmals led to the development of the what we call the foundations of analysis. Just excellent.

I envied him his high school math teacher, who seems responsible for much of the really good parts of this book. No, DFW wasn't a mathematician and he (in spite of what some reviewers seem to think) knew it. He made clear that he wouldn't be able to do justice to Godel. But incompleteness is moderate difficult. DFW didn't know much about Fourier series, but did know they were important enough to mention.
For some students, that's the way to get them interested, just mention something and let them go dig (so much easier now with the internet).

Remember the subtitle -- a compact history of infinity. So it is more history oriented than a mathematical tome. I had recently read Lillian R. Lieber's Infinity (which I see has been reprinted) and it has her sparse, but excellent development of the concepts. It doesn't have much historical detail though. So everything and more was a pleasure.
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on January 25, 2004
Hi, I'm a set theorist. This book is ambitious. For many pages and sections, I really wanted to give it a lot of stars just for effort. There are some good approaches to some hard material. But the errors got to be just too heinous (I'm not at all referring to oversimplifying for the sake of exposition; of course that's necessary. In fact, I reckon the *level* of rigor in this book is just about ideal). If you want to skip right to a cringe-and-sputter bad part, check out his interpretations of the axioms of set theory starting p. 286. Trust me: Bad. And unlike DFW, I'm not gonna tell you to "trust me" unless I know I know what I'm talking about.
He knows a lot of math for a creative writing prof, but he often doesn't know what he does and doesn't know. There was a lot of history and philosophy in the book that I didn't know about, and so I didn't find many errors in those kinds of sections. I probably learned something about that stuff, but unfortunately having seen so much mathematical incompetence I have to distrust DFW as a non-fiction writer.
DFW writes with a dangerous tone. Not a compliment in this case. The tone is: "This is a lot of difficult (but gorgeous) material, but *I've* got it all figured out. So you just trust me to guide you through it (and even when I'm telling you stuff that appears unjustified and kooky, you know it is correct and worth reading because I'm so well-educated and clever)." It's pompous and it's fun and it's fine if you're right. If you take that tone and you're wrong, you suck. Sorry, DFW.
Other reviewers hate the footnotes and other style/organizational whatnot. I agree with a *little* of that. Mostly I thought his willingness to entertain tangents and interpolations and sidebars an appropriate way of handling the material.
DFW refers a lot to items he learned in "college math" and "sophomore math" and so on. The book acknowledges that these math-items may not be familiar to you, but implies they would be if you took math in college and remembered it. That's often probably not quite true; DFW went to Harvard and appears to have had a college math experience atypical even amongst the rather well educated in America. Does he not know this, or did he make a Command Decision that this book is only for people for whom college = Ivy League?
Can you do a 2nd ed. sometime, DFW? It is story that ought to be told well, and I think you have a great draft here.
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Have you thought about infinity recently? If so, it was possibly bound up in religious ideas, in some of which it is integral ("Where will YOU spend eternity?" says one local billboard). Religious infinities have lapped over into mathematical ideas in surprising ways, and if you hanker to do some serious reading about mathematical infinities and their history, you should consider _Everything and More: A Compact History of Infinity_ (Norton) by David Foster Wallace. Wallace is a novelist, author of the huge and well regarded _Infinite Jest_. He isn't a mathematician, except by avocation, but his enthusiasm for his subject is apparent on every page. It's a good thing that this is so; this is definitely not a superficial look at the subject, and Wallace calls upon some high-powered math that you may not even have done in college. The result is a penetrating book from a serious amateur on some of the most important ideas from nineteenth and twentieth century mathematics.
Wallace starts his good-humored and sympathetic tone from the beginning: His "Small But Necessary Foreword" begins, "Unfortunately, this is a Foreword you have to read." There are plenty of footnotes, but half of them are marked "IYI": "If You're Interested," as are many of the paragraphs in the main text (along with "Semi-IYI"). In a history composed of increasing mathematical rigor, Wallace jokes and uses slang. Much of the history has to do with trying to solve the paradoxes of Zeno, like the one about how you can ever get to the other side of the street when you first have to go halfway, then half of the rest of the way, then half of that, and so on. The paradoxes were curious, but when supremely useful calculus came along, the infinitesimals used had never been rigorously defined. It was not until Georg Cantor showed how to deal with infinities as real mathematical entities that calculus had a mathematical foundation. He showed that infinities could be compared, and some infinities were larger than others. The proofs of these ideas (about one of which a mathematician said, "I see it, but I don't believe it!") gave calculus roots, but also gave rise to questions that eventually shook all of mathematics to its foundations.
Wallace doesn't get much into Gödel and his eventual Incompleteness Theorem, and it is just as well. There is enough excitement here, at least exciting for anyone who finds paradoxes a charming way to make the neurons spin. Admittedly, he has included equations and propositions full of Greek letters and advanced functions that only math buffs will absorb. He says of Weierstrass's demonstration of continuity, for instance: "There's a reason this all looks so hideously abstract: it _is_ hideously abstract." The substance is tough going, but there is enough style here, in jokes, curious illustrations, and piquant asides, to make this a fascinating show of intellectual prowess.
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on April 19, 2004
Inspired by praise for David Foster Wallace's "Everything and More" in publications including The Onion and Wired, I bought it hoping to revive in myself and instill in my kids an enduring excitement about mathematics.
Wallace begins with a series of anecdotes that promised to fill the bill, leavened with plain talk and a bracing occasional bit of scatology. But the book's reliance on advanced notation -- much of it impenetrable even to this reader, despite four years of college math (up to differential equations!) -- soon kills the narrative flow.
Wallace's parenthetical asides and copious footnotes sometimes provide illumination, but the book's scattershot structure belies the dust jacket's promise of "a literary masterpiece."
Even Wallace himself acknowledges the book's shortcomings, apologizing at several points for convoluted sentences, bewildering explanations and jumbled storytelling. A good editor could have helped him cut those knots, isolating the advanced math or otherwise rendering it intelligible, allowing him to deliver what author James Gleick hails in his promotional blurb as "exquisitely (and hilariously) original science writing." (Did Gleick and the other reviewers survive the entire book? Or did they just get the funny parts?)
Reading "Everything and More" was like being trapped in a literary version of Zeno's Paradox: Finishing half the book, then struggling to complete half of what remained, then half of that ... I finally just gave up, disillusioned.
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VINE VOICEon November 17, 2003
I think the first thing to be said of this book (or booklet, as Wallace recurrently refers to it) is that it's rather a lark to read. This will surprise no reader familiar with Wallace's literary and critical works. But, unlike his previous works, this one deals with extremely (towards the end) technical mathematics which the author is obliged to gloss over.-Quite a contrast to, say, Infinite Jest.
I was, by turns, frustrated with this lack of rigour, and appreciative of it. I can't put it better than Wallace does in a footnote on pp.220-221, "Rhetoricwise, let's concede one more time that if we were after technical rigor rather than general appreciation, all these sort of connections would be fully traced out/discussed, though of course then this whole booklet would be much longer and harder and the readerly-background-and-patience bar set a great deal higher. So, it's all a continuous series of tradeoffs." - Informed readers take note of his use of the term "continuous series" here!
Thus, Wallace does the best that I think any writer could in walking the tightrope between over-the-top technical mare's nests which only a few members of the faculty at Mathematics departments (and a few autodidacts) could grasp, and what he derides as the "Pop" accounts of such things as the development of Set Theory.-So, nobody, including Wallace, and myself, is going to be completely satisfied. While not a complete technical purist, I do wish he'd chosen to be more technical in some parts and less so in others. As a former student who has always wished his "formal" training in Mathematics went further that first year college Calculus (though I later worked my way through more advanced textbooks on my own), I was genuinely interested in the technical illuminations this book might provide. On the other hand, as an appreciator of fine writing, I know the two do not go hand in glove.
So, in the end, I should say that this book is as good a "tradeoff" as you're going to find. I was pleased to see that Wallace's wit and style haven't suffered from the subject matter. He rather resembles, in this respect, another writer who is more often quoted herein than any other for, as Wallace terms it alliteratively, his "pellucid prose": to wit, Bertrand Russell, a mathematician of first order, whose renegade life and pixie wit served him well throughout his (as Wallace puts it, wryly, in the penultimate footnote of the "booklet") long, distinguished life. Let's hope Wallace's life and output are equally as long and energetic.
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on December 25, 2003
As a logic graduate student who really loved Infinite Jest, once I heard this book existed, I had to read it. In particular, I wanted to see if DFW had gotten anything out of his one year of philosophy study at Harvard. It seems like he got something out of that year certainly, but not real philosophical/mathematical rigor. I definitely recommend reading this book if you're a fan of his fiction and haven't done any math beyond calculus, but the more math you've done and the less you like his fiction, the less you'll enjoy this book.
He's done an excellent job of "popularizing" the philosophy and mathematics of the infinite (if his readership can really be considered popular), but with the emphasis much more on the history than the philosophy or mathematics. Even with all the background I have, I learned a lot about the history, particularly with the early analysts, and what prompted Cantor to do his stuff. But this book could have used an editor with a mathematical background. While he states the Extreme Value Theorem pretty much correctly, every time he uses it, it is applied both unnecessarily and completely incorrectly. Slightly more egregious is his misstatement of the Continuum Hypothesis and confusion between cardinal exponentiation and successorship. (This is perfectly understandable in someone just learning the subject, as DFW obviously was, but in writing a book on it, he should at least state this central problem clearly.) Also, where his idiosyncratic footnotes and acronyms were great in his fiction, they seem a bit tacky in a more mathematical work - especially where they clash with perfectly standard notation that is at least as clear and just as concise. I often got the feeling that I was reading a draft, rather than a finished book.
But whenever he switched back to a more philosophical stance from a mathematical one, he was much more on top of things. And if you like his style, then this book's for you. But after you've read it, I'd also recommend reading a reputable book on the subject to get some of the mathematical facts straight. If you understood them that is, rather than just reading it as a work of art, which seems fairly appropriate.
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on November 17, 2003
A fascinating, irreverently funny, and accurate (as far as I can tell) tour of humankind's dealings with infinity, with more (and welcome) attention to the history of analysis and its transfinite travails than most other books nominally aimed at the general reader. Tough chewing as a first course (see below for more digestible introductions), but a wonderful second course for the whetted appetite. I'd give it 5.29 stars if an editor had made it read less like a (very good) third draft of an exuberant series of lectures, full of self-directed notes on how to better arrange the material, clunky phrases (e.g., "a nice opening-type quotation"), and idiosyncratic abbreviations. Keep a dictionary handy, unless gems like "apodictic," "horripilatively," and "deracinate" are already part of your word hoard.
Okay, 4.84 stars if you have something beyond basic calculus under your belt (foundations, point-set topology, functional analysis, etc.), 4.41 stars if you can think back (not necessarily remembering much) on introductory calculus without discomfort, 4 stars if you're not undebauched by basic function and set-theoretic notation at the level of a serious high school algebra course, and 3.61 stars otherwise if you're interested in the topic and are willing to skip the gnarly parts.
My candidate for the best general-audience introduction to the mathematics of infinity (both content and style) is Robert and Ellen Kaplan's marvelous "The Art of the Infinite: The Pleasures of Mathematics." See also Leo Zippin's "Uses of Infinity," Rudy Rucker's "Infinity and the Mind," and Eli Maor's "To Infinity and Beyond" for other excellent introductions. Shaughan Lavine's "Understanding the Infinite" and Joseph Dauben's "Georg Cantor: His Mathematics and Philosophy of the Infinite" are, like Wallace's book, more suited to professionals and experienced amateurs (as is, I believe, Edward Huntington's "The Continuum and Other Types of Serial Order," which I'm still waiting for).
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on December 14, 2012
I wish my calculus teachers had covered the material the way Wallace does. Science and math textbooks authoritatively present the material as if it had descended from heaven and is so obvious that no one wants to argue about it anymore. But that's just bluff. As recently as 200 years ago the real number system was an ungrounded mess, and today mathematics is still reeling from Godel's Incompleteness Theorem. Worse, math textbooks leave out the mathematicians, the humans.

Wallace gives you a sense of the mindbending toil and real genius necessary to dream up mathematical theories and proofs. His story starts with the Greeks (they didnt believe infinity and irrational numbers existed), moves through the wild success of the Calculus to calculus's masochistic foundational crisis (how can we simultaneously have n = 0 and n ≠ 0?) and culminates in Cantor's invention of set theory. It's a story of great scope and allows Wallace to indulge in his metaphysical interests and explore different expository techniques.

A lot of reviewers think the book is a failure and find fault with Wallace's grasp of the subject and with his baroque style, its curlicues and flourishes. But I dig his style; and I think he does a super job explaining how the different concepts work, as well as the historical and metaphysical background. Still, if the reader has had some college math courses it would help her make it through the book without eliding great chunks of the text.
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