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Everything and More 290

Posted by timothy
from the goes-to-eleven dept.
Chris Cowell-Shah writes "If David Foster Wallace can't explain infinity to us, nobody can. At least, that's what I told myself while anxiously waiting for his Everything and More: A Compact History of Infinity. The book promised to be an intellectual history of the mathematical concept of infinity, with heavy doses of history, math, and philosophy. And while it proves heavy going at times, I'm pleased to say that it delivers admirably on this promise." Read on for Cowell-Shah's lengthy review of Everything and More.
Everything and More: A Compact History of Infinity
author David Foster Wallace
pages 319
publisher W. W. Norton & Company
rating 8
reviewer Chris Cowell-Shah
ISBN 0393003388
summary A mathematical and intellectual history of the concept of infinity.

Wallace may be best known for his footnotes. Virtually everything he has written from his strange but mesmerizing novel Infinite Jest to his hilarious essay about cruise ships (the title work in A Supposedly Fun Thing I'll Never Do Again) to his oddly gripping treatise on the philosophy of dictionaries ("Tense Present" in the April 2001 issue of Harper's)has been liberally sprinkled with footnotes. And what footnotes! Many go on wild tangents. Some contain sub- or sub-sub-footnotes. Others are the length of novellas and could legitimately be reprinted separately from the main work. My point is that Wallace is, at heart, a scholar. He's interested in details. Combine this with an impressive background in math and logic (though he modestly claims a "medium-strong amateur interest in math and formal systems"), and he would seem to make the perfect tour guide for infinity, a concept that seems simple enough on the surface but which we generally suspect is far more complex than we realize.

The Book's Audience and Aims

DFW (an overabundance of abbreviations is one of his most prominent literary tics, and I'll follow his lead) calls Everything and More (henceforth EAM) "a piece of pop technical writing" for "readers who do not have pro-grade technical backgrounds." But the fact of the matter is that to truly follow and understand all (or even most) of his points, one needs to know a lot of math. I'm probably typical of the average reader of EAM: I went through the standard two-year calculus cycle in high school and college, and though most of it made sense at the time, these days I generally double-check my long division. While I've had a fair amount of tertiary-level logic and formal systems coursework while studying computer science and philosophy, even those subjects have grown fuzzy with time. But I am interested in this stuff, and I have the patience and analytical practice to wade through almost any argument or proof, so I would guess that my experience with EAM is pretty close to that of most Slashdot readers.

I should note that this work is really an extended essay rather than a book. Granted, it's a 300-page essay, but that's the term DFW insists on and it seems appropriate given the lack of chapters. The only structure is provided by relatively unhelpful section headers like "4b," and the work sometimes seems to lack convenient breaking points where the reader can pause to catch a breath. This is not a criticism, but the style of the essay does demand that the reader do his best to stay aware of where he is in the overall story of infinity and to be prepared for occasional gaps in the narrative thread. Read this like a math proof with lots of reviewing and re-reading and comparing of earlier and later claims and you should do all right. It's also worth pointing out that the word "history" in the essay's subtitle is important. DFW's goal is mainly to chronicle the ways in which early and not-so-early mathematicians approached the concept of infinity, rather than to explain what infinity is useful for or to give us new ways of thinking about the term. It will probably never have the same mass appeal that more colorful but less difficult books like James Gleick's Chaos or Douglas Hofstadter's Gödel, Escher, Bach have enjoyed, but this is not necessarily a bad thing. DFW has a narrower and more technical aim, and he generally hits his target.

What EAM Covers

It's probably better to think of the essay as a series of loosely related arguments and observations rather than a single mathematical story. With this in mind, let's go through some of the essay's sections. DFW opens by discussing what it means to engage in abstract thinking, then investigates the Principle of Induction (a crucial element in the development of infinity) and explains Euclid's proof that there is no largest prime. He (re-)introduces us to a number of high school math concepts, including such things as reductio ad absurdum proofs and the difference between modus ponens and modus tollens. This refresher is very helpful; I consider the book's opening section to be worth the price of admission all by itself.

Once we've got these preliminary concepts under our belt, DFW starts in with ancient Greek philosophers and mathematicians and begins constructing a vast pyramid of mathematical ideas that will eventually support Georg Cantor's notion of infinity at its tip. This nineteenth century German mathematician is the central figure in the book (to the extent that there is one), and DFW makes it clear early on that we're ultimately moving toward his ideas and his vision of infinity. A quick tour through the Greeks covers Pythagoras, Zeno's paradoxes, Aristotle's demolition thereof, and Plato's theory of forms. It's at this point that we are introduced to fascinating questions of mathematical epistemology and ontology, questions that were first mulled over by the Greeks but that remain largely unsettled even today. For example, what do we have to know in order to really know and understand a mathematical concept? And do numbers exist external to people (the Platonist view), or are they purely human constructs (the Intuitionist stance)?

DFW skips ahead to the seventeenth century, where he showcases Galileo's ideas in Two New Sciences and leads us through some of Newton's and Leibniz's independent contributions to the development of calculus. A wonderful discussion of the archetype of the insane mathematician follows (he makes the unsurprising claim that very few world-class mathematicians were terribly well-adjusted). He then chronicles the intellectual shift from math being thought of as empirical (grounded in actual things) to abstract (based on intangibles and relations between them). He does a good job of explaining how this abstraction works surprisingly well when applied to real problems (especially in engineering and physics). It's at this point (in section five of seven) that the mathematical heavy lifting begins. DFW delves deeper into calculus and the notion of limits, and significantly more mental energy is required if the reader wishes to follow carefully. Fortunately, close scrutiny isn't strictly required; even skimming this portion and picking up the thread again in section six yields good results. Now winding down, DFW introduces us to Fourier series and steps through Cantor's delightful diagonalization/denumeration proofs of the mind-warping claims that there are the same number of whole numbers as integers as rationals, and that the cardinality of the reals is larger than the cardinality of any of these other sets. A short excursis into set theory (like most of the rest of the book, it's thrown at us semi-haphazardly rather than being systematically presented), a longish explanation of Cantor's Continuum Hypothesis (a claim about the relations between the various "sizes" of infinity), and we're done. Exhausted and probably more than a little confused, but done.

EAM as a Mathematical History

There are two ways to judge EAM: as a work of mathematical history, and as a piece of English prose. I consider it adequately successful when viewed in the first light, but exemplary when viewed in the second. The math side of the book is probably best assessed by presenting a scattershot collection of my impressions, so let's start with those.

DFW is, in the main, aware of which portions will pose particular trouble for most readers. The prose is peppered with phrases like "Now you can probably feel a headache starting" or "Here's one of those places where it's simply impossible to tell whether what's just been said will make sense to a general reader," which are usually accompanied by extra explanations or illustrations to clarify the point just made. As an amateur mathematician, he may in fact be better at empathizing with his readers' difficulties than many professors are. It's hard to imagine the following passage (with its awestruck tone) appearing in a math textbook or college calculus lecture:

"Let's pause to consider the vertiginous levels of abstraction involved here. If the human CPU cannot apprehend or even really conceive of infinity, it is now apparently being asked to countenance an infinity of infinities, an infinite number of individual members of which are themselves not finitely expressible, all in an interval [0-1] so finite- and innocent-looking we use it in little kids' classrooms. All of which is just resoundingly weird."

As an example of how he leads readers around conceptual landmines, DFW is especially careful to steer us away from thinking that infinity is just a really large number. He invites us instead to consider it and its cousins to be entirely different sorts of objects than finite numbers, with very different properties. This segues into a first-rate explanation of how infinity-related paradoxes (including Zeno's famous arrow paradoxes) often go away, or more properly, cannot be meaningfully stated, once we stop treating infinity as a normal number or (for certain paradoxes) once we are clear on the difference between zero and nothing (or "not applicable"). These are nonobvious points that I had never considered, but which make perfect sense once carefully laid out and illustrated. Resolving these paradoxes turns out to be a crucial propelling force in the history of infinity: "By this point you've almost certainly discerned the Story of Infinity's overall dynamic, whereby certain paradoxes give rise to conceptual advances that can handle those original paradoxes but in turn give rise to new paradoxes, which then generate further conceptual advances, and so on."

Even if you're relatively uninterested in the concept of infinity, DFW's broad and extraordinarily literate survey of concepts like abstractness, limits, and induction make the book worthwhile. He does an especially good job of explaining the nature of abstraction and why abstract thinking is so difficult. The essay is replete with facts not directly relevant to infinity but still interesting to the scientifically inclined. For example, it turns out that 5 x 10^-44 seconds is generally acknowledged to be the smallest interval in which the normal concept of continuous time applies. And Bremermann's Limit (2.56 x 20^92) is the theoretical limit of the number of bits of information that could have been processed by the most powerful computer that could exist on earth (a computer with the mass of the earth that has existed as long as the earth). Problems involving more data than this (such can be found in statistical physics) are considered transcomputable, or not computable in any meaningful sense. These geeky trivia won't improve your life in any way, but it does stave off some of the inevitable monotony of pure math writing.

DFW has lots to say about mathematical pedagogy, including this harsh indictment:

"Rarely do math classes ever tell us whether a certain formula is truly significant, or why, or where it came from, or what was at stake.... And, of course, rarely do students think to ask the formulas alone take so much work to 'understand' (i.e., to be able to solve problems correctly with), we often aren't aware that we don't understand them at all. That we end up not even knowing that we don't know is the really insidious part of most math classes."

Perhaps this concern for how math is taught leads him to focus his efforts strictly on core concepts rather than on the biographical gossip so often found in popular science writing. There are some fun notes about Cantor's personal life, but he's the only one who gets an extended biographical exegesis. This appears to be a conscious and reasoned decision on his part rather than an oversight ("Again, most of this personal stuff we're skipping") and I think it is a wise strategic move in that it keeps the reader's attention focused and undistracted.

As expected, this work does indeed swim in a sea of footnotes. DFW fans would be disappointed in anything less, but I have to confess to lightly skimming most of the footnotes after the first third of the essay. The most difficult or technical notes are marked "IYI" (for "If You're Interested"), but even the non-IYIspasm notes are full of some pretty thorny math; I found that they often proved more confusing than helpful. But readers more familiar with the subject matter might appreciate the additional historical context and suggestions for further exploration provided in the footnotes.

Overall, EAM is more successful at explaining the small problems, paradoxes, and steps in the creation of infinity than it is at stringing them all together into a coherent, easily followed, transparently structured whole. As an example of how well DFW deals with the small-scale issues, consider the following mind-boggling concept. It is of course impossible to fully wrap your mind around this sort of thing, but in the text that follows this quotation he does a sterling job of steering us toward comprehension:

"The Number Line is obviously infinitely long and comprises an infinity of points. Even so, there are just as many points in the interval 0-1 as there are on the whole Number Line. In fact, there are as many points in the interval .00000000001-.00000000002 as there are on the whole N. L. It also turns out that there are as many points in the above micro-interval (or one one-quadrillionth its size, if you like) as there are on a 2D plane, even if that plane is infinitely larger in any 3D shape, or in all of infinite 3D space itself."

On a similar theme, DFW gives a brilliantly simple and utterly convincing explanation of the cortex-withering claim that "the number of points in the closed-interval [0,1] is ultimately equal to the infinity of points on the whole Real Line stretching infinitely in both directions." But (and this is my biggest criticism) this essay really has to be read twice (or more) to get anywhere near full comprehension of the material. In this respect, it's a lot like an extended math proof or a very long philosophy paper. Repeated exposure makes it easier to follow the narrative flow and string the arguments and proofs together into a consistent thread of thought rather than isolated, self-contained concepts.

EAM as a Literary Work

As mentioned above, where EAM really shines is not as a math history, but rather as an example of pure writing. DFW's prose is clear, precise, witty, and creative. His literary idiosyncrasies may be an acquired taste, but once the reader gets used to the aesthetic feel of the essay it becomes hard not to consider it a stylistic tour de force. In many ways this doesn't feel like a math book at all. This is perhaps not surprising given that the author is, after all, mainly a novelist. He loves to make up words, use obscure words, or use common words in strange new ways. Your appreciation for this style will vary depending on your tolerance for neologisms like homodontic (meaning "having only a single type of tooth") or epistoschizoid (meaning, well, your guess is as good as mine), or unusual punctuation (Does he really need parentheses nested inside of other parentheses? As it turns out, yes.). But you also get exposed to real (and entertaining) words like clonic (involving muscle spasms -- nothing to do with clones), cephalalgia (headache), and peruke (the goofy hats worn by Dutch burghers in seventeenth century portraits). Sometimes it doesn't quite work (What does "We are now once again sort of out over our skis, chronologically speaking" mean? Anyone?), but the overall effect is a refreshing and fun change of pace from standard math or science writing.

DFW uses shorthand to an almost pathological degree. This takes some getting used to, but ultimately it makes his text wonderfully compact (OK, his sentences can be almost unparsably long, but he packs a ton of content into each one) and produces virtually no loss of comprehension. The text is sprinkled with abbreviations like "w/r/t" for "with respect to" and useful sentence fragments like "Meaning it doesn't seem logically impossible or anything," and "Goes on forever." This sort of shorthand is pervasive, but really is more of a help than a hindrance. They may not be everyone's cup of tea, but informal parenthetical phrases such as "they're reversed from the axes in the motion-type graphs you're apt to have had in school (long story; good reasons)" are usually very helpful and inject a nicely colloquial tone into a topic that is traditionally treated in the most formal (and dullest) of styles. Descriptions like this are what keep you going when the math gets tough:

"[T]he whole enterprise becom[es] such a towering baklava of abstractions and abstractions of abstractions that you pretty much have to pretend that everything you're manipulating is an actual, tangible thing or else you get so abstracted that you can't even sharpen your pencil, much less do any math."

Everything and More: A Compact History of Infinity is more or less what its title promises. I found it well worth the (not insignificant) effort to plow through, and I recommend it to anyone interested in mathematical and/or intellectual history, or to anyone curious about how difficult mathematical concepts can be discussed in a lively and engaging way. While most readers won't be able to follow all of the subtleties of his arguments with just one pass through the text, a single pass can still be well worthwhile. Those looking for an introduction to David Foster Wallace would be better served by one of his less difficult books (I especially recommend A Supposedly Fun Thing I'll Never Do Again), but for fans of his more technical, scholarly essays, this book is a welcome arrival.


Chris Cowell-Shah is a consultant with Accenture Technology Labs, the R&D branch of Accenture. His website is cowell-shah.com. You can purchase Everything and More: A Compact History of Infinity from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.

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Everything and More

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  • I read it (Score:5, Funny)

    by Anonymous Coward on Friday March 19, 2004 @05:01PM (#8615323)
    I found it a bit short. I expected infinity to be longer.
    • Actually (Score:3, Funny)

      by The Queen (56621)
      I think you're supposed to flip the book over and begin again at page 1... kinda Moebius Strip style.
  • by graveyhead (210996) <fletch AT fletchtronics DOT net> on Friday March 19, 2004 @05:02PM (#8615346)
    Zero: The Biography of a Dangerous Idea [amazon.com]

    Sounds similar in concept, though from the review, it seems to me like the Zero book is a lighter read.
    • by graveyhead (210996) <fletch AT fletchtronics DOT net> on Friday March 19, 2004 @05:05PM (#8615393)
      Whoops :)
      the Zero book is a lighter read.

      No pun intended.
    • Brings up an interesting (to me, at least) point: which came first, the idea of zero, or of infinity?

      I don't think this is as simple as a monkey-case of "I have ALL the food" versus "I'm starving," but more of a rigorously defined "This is mathematical zero" and "This is mathematical infinity." I'd be interested in hearing from a (certified?) Mathematical Historian about when/where/under-what-circumstances each of these ideas evolved.

      • The greeks did not have the mathematical concept of 0. They thought of infinity, but didn't like it and excluded it from their mathematics.
      • Might I suggest "Alpha and Omega" by the same guy that wrote Zero (as referenced above), als Zero would be worth a read
      • by saforrest (184929) on Friday March 19, 2004 @06:19PM (#8616315) Homepage Journal
        Brings up an interesting (to me, at least) point: which came first, the idea of zero, or of infinity?

        Well, I'm no certified mathematical historian, but I don't know if you'll find one on Slashdot.

        The standard claim is that zero was invented in India around the 7th century, as wikipedia says [wikipedia.org]. There is some controversy over this, largely because other cultures had previously invented various forms of placeholders to indicate 'nothing' or 'no value', but I don't think there's any proof that these placeholders had been elevated to the class of an actual number.

        The notion of infinity is rather older than this, going back to the Greeks. One early mention of the concept was by Aristotle [wikipedia.org] in Physics:

        "... it is always possible to think of a larger number: for the number of times a magnitude can be bisected is infinite. Hence the infinite is potential, never actual; the number of parts that can be taken always surpasses any assigned number." [Physics 207b8]

        It does make some sense that the notion of infinity precedes zero, simply because it's easy to think of bigger and bigger numbers, wonder if they ever stop, and realize they cannot. This is an intuitive argument, though, and its plausibility may depend heavily on the historical development of these ideas.
        • You are implicitly equating two different mathematical concepts which are often times referred to using the same word "infinite":

          1. unbounded
          2. infinite

          Something that is unbounded can "keep going" forever, but infinite is something different. It is the idea that you can reason about a completed unbounded thing. Yes, the notion of a completed unbounded "thing" is problematic (almost self contradicting) and much has been written on the subject especially with regards to foundations of mathematics.

          I wo
          • You are implicitly equating two different mathematical concepts which are often times referred to using the same word "infinite":

            I agree there is a distinction there, but it's not clear to me that a lot of other people make it, or that the original poster was referring only to the second. It doesn't help that they mean the same thing in two different languages, too.

            I would say that the notion of an infinite object, distinct from the idea of 'arbitrarily large', is probably due to Cantor.

            Obviously there
    • I've read that! And it was a light read, but very interesting and highly entertaining as well. It's difficult to imagine not having zero as a number, or the idea of it being controversial, but that's how it actually was.
    • by dmeranda (120061) on Friday March 19, 2004 @05:36PM (#8615815) Homepage
      One of the best non-mathematical books I've read on the modern theory of Infinity is

      "The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity"

      And it's still the best book which also contains a lot of very interesting biographical treatments of Cantor, the father of the modern theories.

      Of course nothing replaces actually reading the original (English-translated) works of say the great Georg Cantor or my favorite, Bertrand Russel. If you have the mathematical fortitude I highly recommend those, there is so much detail in those, not just mathematical but philosophical as well. Dover publishers is a great source to find these important original translated works of lots of mathemeticians, and they are surprisinly cheap too.
      • by saforrest (184929) on Friday March 19, 2004 @06:06PM (#8616171) Homepage Journal
        One of the best non-mathematical books I've read on the modern theory of Infinity is

        "The Mystery of the Aleph: Mathematics, the Kabbalah, and the Search for Infinity"


        I read this book. (Actually, I bought it from Aczel himself, when I saw him speak.) The title certainly sounds meaty, and I figured the author was enough of a mathematician that he couldn't be outright lying when he connected Cantor's work with Jewish mysticism.

        The book was, unfortunately, way too light and fluffy. And he seems to put wayyy too much emphasis on the mystic implications of what is really just simple notation. Sure, the cardinality of the natural numbers is denoted by aleph, but does that really have anything to do with the mystic aleph, except as a coincidence?

        There was also a bit too much of the "mad Icarus" imagery in the book with regard to Cantor. Mathematicians are often not the most stable people around, but the insane ones aren't all cutting-edge theorists driven to madness by the profundity of their ideas, which are too great for a fragile human vessel; some of them are just plain nuts. With Cantor it's kind of hard to say, though he fits the bill more than most.

        Of course nothing replaces actually reading the original (English-translated) works of say the great Georg Cantor or my favorite, Bertrand Russel.

        Sorry if I'm being a bit unfair here, but the fact that you mispelled Russell's name makes be a bit suspicious about whether you're really read Principia. Adding to this suspicion is the fact that I don't know anyone personally who's actually read Cantor, simply because the set-theoretic language and notation has changed so much since his time. Apologies if my suspicion is unjustified.
        • Yes, you are being unfair. The spelling mistake is called a typo. And you may not know me personally, but I guess you now do know someone who has read Cantor's own words...me! I'm glad you met Aczel, I've never had the pleasure.

          And yes, I have read the Principia (English translation), several times. That's pretty much a mandatory read for anybody serious about pure mathematics. And I've also read original (translated) works from Cantor, Euclid, Barendregt, Godel (sorry for the missing accent, I know


          • Yes, you are being unfair. The spelling mistake is called a typo. And you may not know me personally, but I guess you now do know someone who has read Cantor's own words...me! I'm glad you met Aczel,


            Sorry. I guess I shouldn't leap to conclusions like that.
          • "And yes, I have read the Principia (English translation), several times."

            Hmm, I thought it was written in English, Mr. Russell being a British guy, taught at the university of New York, another English-language institution.

            Then again, I guess he was kind of bright, maybe he picked another language to doodle in. I heard that it's easier to discuss quantum mechanics in Hopi than English. Maybe Mr. Russell was ahead of his time.

            Unless your talking of another Principia . . . written by some other, less fa
    • by SmackCrackandPot (641205) on Friday March 19, 2004 @05:47PM (#8615941)
      A History of Pi" [amazon.com] really satiated my appetite.

      e: The story of a number [amazon.com] really expanded my mind.

      An imaginary tale [amazon.com] really grabbed my imagination.
    • by ortholattice (175065) on Friday March 19, 2004 @06:24PM (#8616368)
      Readers (or rather web surfers) might also enjoy the Metamath [metamath.org] web site. Try the "Metamath Proof Explorer" which "constructs mathematics from scratch" quite literally, up through Cantor's infinity and beyond.
    • by Vagary (21383)
      Seriously, don't waste your time. Like most popular mathematics books, everything it has to say beyond some basic history (which you've probably already heard much of if you're a geek) is trivially obvious. It's written in such a light style that I found it patronising and questioned whether the author had any knowledge of higher-level math whatsoever.

      Everything & More sounds much more like the way math books should be.
  • by Best ID Ever! (712255) on Friday March 19, 2004 @05:05PM (#8615400)
    less difficult books like James Gleick's Chaos or Douglas Hofstadter's Godel, Escher, Bach

    If GEB is less difficult, count me out!
    • by Vagary (21383)
      Seriously, if you've ever taken a computational theory course, you have to admit that nothing in GEB is profound (sure, the ideas were profound originally, but Hofstadter is just reporting them). And, in fact, Hofstadter fills the book with vacuous connections to art and little games which I can only surmise are in there to show you how clever he is. Maybe all that junk is necessary to make it interesting to the uninterested layman, but personally I find the concepts interesting enough on their own.

      The pop
  • by AGTiny (104967) on Friday March 19, 2004 @05:11PM (#8615501)
    I don't know if I even want to think about a book that is more difficult than GEB! Egads! I still haven't made it all the way through... although I do think it's a great book, just over my head in most places. :)
    • Heh -- not-quite-great minds think alike! (see my above post)
    • I found Everything And More a good bit easier than GEB; maybe it's just shorter, but I think it's also a bit less ambitious in what it tries to cover. I'd definitely recommend EAM to someone who was interested enough to try to read GEB.
    • I actually enjoyed reading GEB - it's one of those books that once you get into it, it's very difficult to put down until you've finished it. It also stands up to multiple readings - there are layers upon layers of information in it.

      One book that I'd love to read but has utterly defeated me is "On Growth and Form" by D'Arcy Thompson. The ideas in it are fascinating, but trying to read even a chapter is a monumental battle to simply get past the language! Pop Science it ain't...
  • by nebaz (453974)
    Does 0.99999999 (repeating forever) equal 1?
    • I would think that 0.99... would be approaching the value of one. Sorta think of it in the context of limits, where a function can sometimes approach but never reach a number. (as if you couldnt tell, my math reasoning is less than stellar)
      • I would think that 0.99... would be approaching the value of one. Sorta think of it in the context of limits, where a function can sometimes approach but never reach a number. (as if you couldnt tell, my math reasoning is less than stellar)

        You're right, in the sense that 0.9... with any finite number of nines after it would approach 1 as a limit, as the number of nines goes to infinity.

        But with real numbers, in theoretical terms, you're allowed to have numbers with infinite decimal expansions. Not just
    • Re:So.... (Score:2, Informative)

      by oracleofbargth (16602)
      Does 0.99999999 (repeating forever) equal 1?

      Yes, it does. My calc1 prof showed us the proof.
      I can't remember the whole proof (it's been many many years since i took calc1), but here's the gist of the idea:

      start with 0.9, you add 0.1 to it to get 1.
      then look at 0.99, you add 0.01 to it to get 1.
      now look at 0.999, you add 0.001 to it to get 1.
      repeating infinitely, you would eventually need an infinite number of zeros before the 1 to be able to add it to the repeating 9.

      That infinitessimally small number
      • Or...
        1/9 = .1111111111111111...
        2/9 = .2222222222222222...
        3/9 = .3333333333333333...

        ...

        9/9 = .9999999999999999... = 1
      • repeating infinitely, you would eventually need an infinite number of zeros before the 1 to be able to add it to the repeating 9. That infinitessimally small number actually turns out to be equal to zero, thus, 0.9(repeating) is equal to 1.

        all you've done is transform the question of whether or not .999... = 1 to whether or not .00001 = 0! here's a stronger argument:

        let r = .999...
        then 10r = 9.999...
        and 10r - r = 9,
        so 9r = 9,
        and r = 1.

        voila! you can use the same argument to find the rational form of a

        • Cool! I'd forgotten about this handy trick.

          As an arbitrary example of using it on another repeating irrational number (useful in that it there are fewer 9s involved):

          What's 0.151515... ?

          100r = 15.151515...
          100r - r = 15
          99r = 15
          r = 15 / 99 = 5 / 33

          Spifftastic!
    • Re:So.... (Score:5, Funny)

      by SigmoidCurve (188795) on Friday March 19, 2004 @05:37PM (#8615830) Homepage Journal
      Q: Does 0.99999999 (repeating forever) equal 1?

      A: Yes, for sufficiently small values of 1.
    • Just ask Dr. Math [mathforum.com].
    • Re:So.... (Score:3, Informative)

      by ornil (33732)
      1/9=0.111111(1)
      1=1/9*9=0.99999999(9)
    • Here's a better one. We all know that 1/6=0.16666... So let's take the sum of six of these...

      0.166...6
      0.166...6
      0.166...6
      0.166...6
      0.166...6
      + 0.166...6
      ------------
      0.999...6
      So 6/6 equals point 9 repeating, with a six on the end.
      • Re:Better one... (Score:3, Interesting)

        This discussion of infinity reminds me of this "proof" that pi==2:

        - Begin with two points A and B that lie on opposite ends of a semi-circle with diameter 2. Let us call the length of the curve between A and B pi.

        - Take a point equidistant to A and B, C, that lies on two smaller semi-circles, AC and CB. Note that the length of the curves AC + CB is still pi, and the height of the curve above the straight line AB is less than the height of the AB circle above the line AB.

        - Repeat this. As you do this an i
    • The argument goes kind of like this:

      All real numbers are the limit of two sets of numbers, the set of rational numbers for which the solution x to an expression E is definitely lesser than the set members, and the set of rational numbers for which x is definitely greater. This pair of "least upper" and "greatest lower" bounds is identical to the real number "solution" to E.

      For example, suppose the number in question is the square root of two. The sets of numbers that you wish to consider are all q in Q su
  • Makes you wonder if there really is an infinity... Mankind used to believe that the universe was infinite (physical infinity), but we're getting more and more proof that it might actually be finite.

    Today we believe that there is a mathematical infinity. Maybe in a few generations, a genius will discover that there is no such thing either...

    Maths can be scary sometimes
    • The lack of a physical manifestation of something with a property of infinite magnitude (such as the dimensions of the universe) would not render the concept useless. Two completely different things.
      • by Phisbut (761268)
        What I meant is, in mathematics as well as in physics, because we believe something is true today does not mean it really is. I gave the phisical example of the finitude of the universe, but there are also examples of when mankind believed something to be true mathematically but then was proved otherwise.

        At one time, there were only integer numbers from 1 to infinity. Then came zero. Back then, people thought that nothing could be smaller than 1, but then zero arrived. So we had a lot of integer numbe

    • I'm no mathematician, but I've often wondered what math/physics today would be like if physicists refused to use concepts like infinity and even time in equations. After all, AFAIK, we haven't proven either of them exists and there is very little science to suggest that they do. You should be able to use physical concepts like time and infinity in equations until you at least have a solid scientific basis for positing their existence.

      I'm guessing that stripping away those constants and redeveloping moder
      • I'm no mathematician, but I've often wondered what math/physics today would be like if physicists refused to use concepts like infinity and even time in equations. After all, AFAIK, we haven't proven either of them exists and there is very little science to suggest that they do. You should be able to use physical concepts like time and infinity in equations until you at least have a solid scientific basis for positing their existence.

        IMHO Your whole premise is wrong. Lack of understanding something is

        • I have to protest the idea that I'm saying that time doesn't exist because I'm simply ignorant about it. I'm saying that no one, AFAIK, has bothered to look into whether it has any physical reality whatsoever. Yes, it's a useful abstraction. My problem with this abstraction though is that people continually treat like it's something real in the physical sense. My hand is real. This desk is real. Position within space is real. Change of position in space is real. Change of position in space measured

      • einstein didn't think so...

        People like us, who believe in physics,
        know that the distinction between past,
        present, and future is only a stubbornly
        persistent illusion. (Albert Einstein)

        'HEAT IS THE FOURTH DIMENSION'

        regards,
        john [earthlink.net].

    • Sometimes I find it easier to comprehend the universe as being infinite.

      If it's finite, then what's at the border? Is there a wall you can't pass through? Does it phase out? Can you jump across the border and be wiped from existence, violating ideas about conservation of energy?

      Maybe the universe is expanding at the speed of light though... and we're prevented from knowing what happens at this border because we can't travel faster than the speed of light. Sort of a mechanism to keep the universe from
  • Rudy Ruckers "White Light" a fictional account of the concept of infinity. A lot of it reads like an LSD trip but it got me thinking about things in a whole new way. I don't think its quite as technical as the one reviewed above though ;)
    • Rudy Ruckers "White Light" a fictional account of the concept of infinity.

      Rudy Rucker also has a non-fiction book about infinity that I liked more than White Light: "Infinity and the Mind, The Science and Philosophy of the Infinite"
  • by sdedeo (683762) on Friday March 19, 2004 @05:19PM (#8615630) Homepage Journal
    I am a huge David Foster Wallace fan, and think Infinite Jest is the greatest book of the 1990s.

    However, the reports on Everything and More have not been good. The reviewers who have demonstrated some understanding of the mathematics involved (not particular heavy, but somewhat obscure), have come down pretty hard on DFW for his errors. Here [lrb.co.uk] is a representative review (from the LRB), which covers DFW's book and a slew of other "books on infinity" at once:

    "As for Wallace's book, the less said, the better. It's a sloppy production, including neither an index nor a table of contents, and after a while his breezy style grates. No one who is unfamiliar with the ideas behind his dense, user-unfriendly mathematical expositions could work their way through them to gain any insight into what he is talking about. Worse, anyone who is already familiar with these ideas will see that his expositions are often riddled with mistakes. The sections on set theory, in particular, are a disaster."

    (You might put this down to academic anxiety, since the reviewer, A. W. Moore, is a professional philosopher with an anthology on "infinity" to his name as well.)

    It is strange, since DFW did spend part of his youth (not the alcohol and drug-addicted part) in a philosophy and logic Ph.D. program. I'm not sure if I'll read it; on the bright side, he has a new collection of short stories coming out in June [amazon.com].

    • Yes, based on the /. review alone, I am a bit worried about the acuracy of this book. For example:

      [DFW is] especially careful to steer us away from thinking that infinity is just a really large number


      In fact, the various infinite cardinal and ordinal numbers can be thought of as numbers, in a way that is not difficult to make mathematically precise. From the review, it seems like DFW focuses on how myserious and abstract infinity must be, rather than on the mathematical details of how various infin
      • Yes, it was given an *extremely* negative review in the January 16th issue of Science. Basically, the reviewer demonstrated that Wallace has no real understanding of the subject by pointing out glaring errors such as confusing Cantor's continuum problem and Cantor's continuum hypothesis -- two entirely different things. But I'm sure many Wallace fans in English departments and the like will buy the book and think the reason why it doesn't make any sense is that it soooo deep.
    • I take some issue with the mathematics of the cited reviewer as well though. He says:

      Not only is this optimism controversial and expressed without adequate justification, it is also inconsistent with the equally controversial and equally unjustified pessimism expressed on the final page of the main text, where we are told that a problem with which Cantor wrestled throughout his life - whether any set is intermediate in size between the set of positive integers and the set of sets of positive integers - is
      • It's more properly a ground for saying that we can choose different versions of set theory which are extensions of ZF or ZFC and further, have something to say about the Contiuum Hypothesis.

        To say 'for ever undecidable' is strange. What is the 'for ever' for? It makes it sound like the problem has an answer, but one which will forever be outside our capability to calculate. On the contrary, Goedel and Cohen have resolved the issue. So DFW's phrase is -- at least a little -- misleading.
  • Infinitely unreadable, like most everything from David Foster Wallace.
  • A History of Pi [amazon.com]. I found it very interesting.
  • The excerpt arguing for "There are as many numbers [0,1] as [0,0.1]" just illustrates a common misconception of people - that we can somehow count infinity. So often people associate infinity with a number... "I have infinity billion dollars" or "The biggest number is infinity plus one." Infinity is a concept, one that has inexorably become tied with numbers, to the detriment of both.

    I, for one, grew out of being mystified by infinity shortly after I graduated middle school and began to learn about trul

    • Why can I drink on Friday without a hangover, but when it comes Monday morning my head is being pounded by sledgehammers?

      Easy, you probably just drank enough that you were still drunk on Saturday and Sunday. My own record in that regards is 3 days. The hangover lasted for a good portion of the following week too.
    • Why can I drink on Friday without a hangover, but when it comes Monday morning my head is being pounded by sledgehammers?

      Maybe because you've been drinking for over 2 days, non stop?
      Then again, its friday night. So as drunk as you probably are right now, I can see how this can be a bit of a mystery at the moment ;-)
    • In fact, you can COUNT. That's the whole point of Cantor's argument. You just can't exhaust it. You might think I am pulling your leg, but counting, in a mathematical sense, means establishing a one-to-one map between it, and the integers.
    1. There are a larger number of irrational numbers than rational numbers (Cantors diagonal argument).
    2. Between any two points on the number line there lies a rational number.

    Discuss.

  • ....there [boston.com] is a complete interview with him.

    May I also ask :

    Combine this with an impressive background in math and logic (though he modestly claims a "medium-strong amateur interest in math and formal systems")

    What exactly this "impressive background" is as I was unable to find any information except for his litterature classes? I nice link to a complete bio would be appreciated, thank you.

    • He has an BA from Amherst in lit and an MFA from University of Arizona. He has also been given one of the Macarthur Genius awards. I think his interest in math comes from the fact that his father is/was a professor of mathmatics at the University of Illinois. Currently he is teaching at Pomona College..Basically, he is a god in post-modern literature-read Infinite Jest if you want proof.
      • Thank you, but unless I'm wrong all of those are in the field of litterature/fiction. I certainly would read at least one of his books, but I honestly don't think his past is the description of an "impressive mathematical background". He may be a genius of thought & theory, but that does not automatically qualify you for maths' god. Interesting person tho...
  • I am a huge DFW fan so you might consider my opinion to be slightly biased... Everything and more was a fantastic read even though I had to spend about half to the time going back to certain points in the text to understand his explainations and abreviations. Basically, it is a 300+ page essay on why Cantor's work on set theory was so important in the grand scheme of mathmatics and the way we percieve math. I only took the requisite 2 years of calculus in high school/college and I was able to understand
  • I wish someone else would have written it. David Foster Wallace's 10 footnotes per page style is very tedious to read. Maybe I'm just not scholarly enough or something.
  • Obfuscation (Score:3, Interesting)

    by phliar (87116) on Friday March 19, 2004 @05:37PM (#8615834) Homepage
    Bah! I thought it used so-so writing style combined with the overuse of the worst of the opacity of math notation. We use notation because we don't know any better way to communicate, not because we think talking 'leet jargon is kewl. So instead of helpful chapters and sections with names we get crap like "3b." And he cross-references to things like "as we talked about in 2c." Try shuffling throught the book trying to find that damn "2c". I wanted to throw the thing across the room when I was reading it. I wanted so much to like it... I was hoping it would be a decent popular account of infinite sets (and related concepts) so my friends could see how cool this shit is, instead of thinking (as they do now) that I'm speaking in tongues.

    This book will just convince people who read it that this stuff is obscure jargon-ridden crap that only lunatics are involved in... stay out, because you're too stupid to understand all this.

    • It should be noted that this is common practice in the mathematical community... that is a heavy use of formalisms. I would disagree that mathematicians do it to appear "leet". If you have a better idea for communicating mathematical ideas, please fill us in. You might just invent the COBOL of mathematics!

      errr, uhhh... wait a second... COBOL wasn't a good thing, that is, natural language sucks at communicating a restricted type of mathematics: algorithms. You can bet it would suck at communicating math
      • You've misunderstood me: in a former life I was one so I know math doesn't use jargon to appear 'leet. My objection is that this hack does see to be using it that way -- there's absolutely no reason to write a plain-language book with crap like "2c." instead of a chapter name. Ditto garbage like IYI, which isn't even standard math jargon so I kept tripping over it.
        • Oh, sorry. I haven't read the book, and it just bugs me when people think that mathematical objects are best described using verbose english prose, e.g. COBOL.
    • In case you weren't, you should know that all higher-level Math is jargon-ridden crap that only lunatics are involved in. Math is like a religion with all these incantations and rituals. Part of the problem might be that the content of Mathematics has expanded so quickly in the last hundred years that the communication techniques are not keeping up. Or maybe it just has something to do with maintaining a high barrier to entry for the field of Professional Mathematicians.

      Either way, it is a travesty that it
      • And the fact that someone with an undergrad degree in Math is not knowledgable enough to read your average journal paper is symptomatic of a serious problem.

        My original message may have been unclear: I was a mathematician in a former life, and the area (set theory) is near and dear to my heart. If there is a lot of jargon and notation in a math proof it's because we haven't found a better way to talk about those ideas in an unambiguous manner. Just like music: it may look arcane and 'leet, but it's the b

  • concept of... (Score:2, Insightful)

    by cheeseSource (605209)
    I used to think of infiniity as the end all (without end, of course). If only something could be infinite - being immortal for instance. Wouldn't that be the ideal; to be able to watch and learn and absorb even a fraction of the spectrum of time. But somewhere it occured to me that an infinite life and inifinity itself is just as meaningless as a finite one. There is no beating it: either life is too short - no matter how long life could be if it isn't infinite then it might as well not have happend. And sa
    • If you were immortal... and able to observe and learn everything there is to be observed and learned... you'd still never know everything there is to know.

      The brain is not infinite in size... eventually you'll reach the limit to how much information the brain can store, and some stuff will be overwritten. I wonder what happens when a brain is "full."
  • "We are now once again sort of out over our skis, chronologically speaking..."

    I think we're getting a bit dangerously ahead of ourselves, here....
  • by Andy_R (114137) on Friday March 19, 2004 @05:51PM (#8615990) Homepage Journal
    on BBC radio 4 a few months back. A nice moment was when the mathematician recalled trying to explain infinity to a very young child:

    Child: "what's the biggest number there is?"
    Mathematician: "what do you think it is"
    Child: "um, 380?"
    Mathematician: "but if you add one to that, don't you get 381"?
    Child: "Wow!"
    (pause, in which the mathematician assumes the child has grasped the idea that you can ALWAYS add 1 and get a bigger number)
    Child" "I was really close, wasn't I?"

    • That exact concept (minus the child's misunderstanding) was in the children's book "The Phantom Tollbooth", which was a really cool kids book that actually got me thinking about alot of cool ideas, both scientifically and philosophically.

      Basically, there's a town that mines 'numbers' from the ground. They say they mine every number. The kid asks what's the biggest number they get, and they show him a huge number 8. he then says, "no, i mean, what's the longest number" and they show him a number 25 that

      • ObSimpsons:
        It also put global conflicts into perspective, as there was a letter city that did things with letters, and the number people and the letter people hated each other. A peacemaker, trying to better their world, made them each come to a common agreement on something. They wound up agreeing that they disagreed! not so different from the real world.

        "I don't agree to that."
        "Neither do I!"

        --Stephen

  • (What does "We are now once again sort of out over our skis, chronologically speaking" mean? Anyone?)

    So, I think it's kind of a complex way of saying "we're getting ahead of ourselves, here." I don't imagine that he's implying we're about to do a temporal face-plant, just that we've gone wandering forwards towards the end before we've really explored the middle.

    Otherwise, I'm about 85% of the way thru (given that I've just started section 7), and find it a good read (if sloggy to get thru), and share your

  • In Keeping With (IKW) David Foster Worth's (DFW's) literary preponderance of using acronyms (LPOUA), I'll be brief (IBB).

    SSLKJM SSDOL! MKLWPYQ, LKJYXMK. TAFN.

  • by DukeyToo (681226) on Friday March 19, 2004 @06:41PM (#8616527) Homepage
    I propose we add infinity to the supported values for an integer in SQL. I find the whole NULL thing somewhat unbalanced.

    SELECT * FROM Articles WHERE Len(ReviewText) = INFINITY

    RESULT
    ------
    NULL

    In case you read this far, I don't really have a point, but it is Friday afternoon, so I have an excuse.
  • Infinity easily explained in one phrase:

    The Cream of Wheat box. ....

    Ok, let me explain. On the old Cream of Wheat box, there's a picture of a man. In that picture, that man is holding a Cream of Wheat box, which of course has a picture of a man on it, holding a box of delicious Cream of Wheat. Rinse. Lather. Repeat. Infinite recursion at it's finest!
  • by xmutex (191032)
    Can I just say that Infinite Jest is like the best fucking novel ever?

    I have nothing else to contribute.
  • I don't think it's his best book ever. And I certainly haven't read or studied enough math to say that his book is either really good or really bad.

    I do think that some of the introductory stuff that he wrote about basic math (like Principle of Induction, how counting is taught in elementary school in Platonic fashion, etc) was really informative as well as fun to read. I haven't gotten past 60% of the way through the book, because a lot of the stuff is confusing and going over my head. I'll probably
  • by nessus42 (230320)
    Another book that folks interested in the topic of infinity might want to read is the book Infinity and the Mind by Rudy Rucker. He's a great popular math/science writer, a math professor, and a seminal science fiction author. This book is easy to read, yet is accurate and informative enough to have been used as a textbook in MIT's Infinity and Paradox course. The book is a bit heavy on Rucker's Buddhist philosophy, but it is easy to ignore that stuff if you don't go for it and stick to the math.

    |>ou
  • Just a minor (and not particularly relevant, shaving along the edge of off-topic) point: a peruke is not a goofy hat -- it's a goofy wig. Comes from the Dutch word "pruik", apparently.
  • Very well written review. I probably won't ever read the book, but you did a nice job reviewing it. Kudos. KUDOS.
  • Just six weeks ago I posted about countable and uncountable infinities [slashdot.org] in my /. journal, as much for future reference as because of anything in particular at the time.

    I guess that future has now arrived.

    Skipping past the fluff, my central point was that understanding the difference between countable and uncountable infinities is often really useful, but that even esteemed mathematicians often miss that point.

    It really doesn't sound from this review that either the author or the reviewer really get that p

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