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Everything and More
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 HistoryThere 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.
I found it a bit short. I expected infinity to be longer.
Zero: The Biography of a Dangerous Idea
Sounds similar in concept, though from the review, it seems to me like the Zero book is a lighter read.
std::disclaimer<std::legalese> sig=new std::disclaimer; sig->dump(); delete sig;
less difficult books like James Gleick's Chaos or Douglas Hofstadter's Godel, Escher, Bach
If GEB is less difficult, count me out!
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. :)
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.
Does 0.99999999 (repeating forever) equal 1?
Rhymes that keep their secrets will unfold behind the clouds.There upon the rainbow is the answer to a neverending story
Or four lines longer than the review. Or five lines longer than the review. Or six...
webpage
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
After 3 days without programming, life becomes meaningless
- The Tao of Programming
I think you're supposed to flip the book over and begin again at page 1... kinda Moebius Strip style.
The House Between - Original Sci-Fi Series
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 ;)
I don't read your sig, why do you read mine?
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 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.
Protect your liberties. Donate to the ACLU
Infinitely unreadable, like most everything from David Foster Wallace.
A History of Pi. I found it very interesting.
People who disagree with you are not automatically evil, greedy, or stupid.
I, for one, grew out of being mystified by infinity shortly after I graduated middle school and began to learn about truly mystifying things like "women" and "alcohol's effect on the human body". Why can I drink on Friday without a hangover, but when it comes Monday morning my head is being pounded by sledgehammers?
Discuss.
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 modern mathematics and then physics without fictitious concepts like infinity and time would force a shift in perspective that would have far reaching implications.
Am I off my rocker, or does this make sense to anyone in the know?
Please mod this post only if you think others should/n't read this. I have enough ego^H^H^Hkarma. Thanks!
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.
1. No sig. 2. ???? 3. Profit!!!
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 all of DFW's main technical points. Despite the fact that the main topic is logic and math, DFW's unique style of writing shines through. If you enjoy Everything and More, try reading Infinite Jest-it is the Gravity's Rainbow of our generation.
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.
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.
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.
Unlimited growth == Cancer.
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 sadly if it is infinite it becomes miserably excessive. Life can be like a good book, you don't necessarily want it to end, but if it never ends it will become terrible.
(Sponsored by cheeseSource for President 2012)
"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....
The party of stupid and the party of evil get together and do something both stupid and evil, then call it bipartisan.
A History of Pi" really satiated my appetite.
e: The story of a number really expanded my mind.
An imaginary tale really grabbed my imagination.
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?"
A pizza of radius z and thickness a has a volume of pi z z a
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 good opinion of the book, but I'm taking a break and re-reading Cryptonomicon (which, oddly shares some of the same concepts towards the beginning when we're getting introduced to Woe-to-hice and his early education).
Nice review.
Not A Sig
is a test. To see if my posts are actually getting through.
SSLKJM SSDOL! MKLWPYQ, LKJYXMK. TAFN.
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 relative to the change in position for the hands of a clock is real. But to say that "time" exists in a physical sense because we can say those things is a stretch.
Think about it this way: If the physical reality of time isn't real and is nothing more than a useful mathematical abstraction (as is infinity by the way), then how would time travel be possible? Wouldn't traveling through some sort of space be predicated upon finding that space first?!
Where time is concerned, we say to ourselves "well it must exist, because there has been a change in the time on the clock". But the clock only changes because we made it that way. The clock doesn't actually measure anything after all, it's just a contraption who's parts move around in a predetermined way. AFAIK, all of physics is based around such clocks. But, again, these clocks don't measure anything physical. Yet, the verity of time as a physical property is assumed.
You think time physically exists? Give me a thought experiment that would appear to prove its existence. Keep in mind that I say that all anyone proves by pointing to a clock of any kind, is that there's some matter and, "Oh look! It moves around". At this point in time, I can not conceive of a proof for the existence of time which somehow doesn't rely upon an external, irrelevant event.
Honestly, I'm not trolling here. Review my posting history and you'll see I'm not a troll. I am pointing out the arbitrariness of human perception though.
Please mod this post only if you think others should/n't read this. I have enough ego^H^H^Hkarma. Thanks!
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.
Most writers regard truth as their most valuable possession, and therefore are most economical in its use - Mark Twain
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 decimal expansions that approach infinity, but which are infinite. is a number, not just a succession of slightly better approximations. These means, of course, that there are numbers out there which are effectively uncomputable.
There are really four classes of numbers, each of which contains the previous one.
Numbers with a finite decimal expansion in base 10. Examples are 0.125, 0.12957235, 456, and 5/2.
Rational numbers. Numbers which can be written as a/b for a and b integers. This includes the previous numbers, as well as numbers like 1/3, which have infinite decimal expansions in base 10 (but not in base 3!).
Algebraic numbers.
Technically, these are numbers which appear as the root of a polynomial equation with rational coefficients, like x2 - 2=0. This means, basically, that these are the numbers you can get by starting with some rational numbers and dividing, multiplying, adding, subtracting, taking powers, and taking square roots. The algebraic numbers include all the previous numbers, plus irrational numbers like the square root of 2.
Real numbers. This is the so-called 'number line' as you probably learned it.
They include all algebraic numbers, plus so-called 'transcendal' numbers, which basically fill in the gaps in the number line left by the algebraic numbers. Since they're not algebraic, you can't produce them from simpler numbers by a finite series of steps involving algebraic operations. Transcendental numbers have to be gotten at using infinity in some way. Examples of transcendental numbers are and e. Here's a simple-to-understand way of producing e via an infinite-running algorithm:
e
n
for i from 1 to infinity do
n
e
end do;
Chop that off wherever you want and you'll get an approximation; after 5 steps it's already at 2.716666667.
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!
Can I just say that Infinite Jest is like the best fucking novel ever?
I have nothing else to contribute.
jack's bicycle is music to my ears
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.
Physics relies intrinsically on the notion of time - take that away and there's not much left. You might take away space too, while you're at it ;-)
... is that a watch that you're carrying? if so (and time is fictious), then yes, you're off your rocker :-)
You're mistaken in requiring a proof of time. It's like requiring a proof of the existence of the Universe - you can't do it. It's an empirical reality. "Time" is actually meaningless in physics, too - if you want to be really picky all that has a meaning is "time interval" (between events) and that in a specific reference frame.
and about it being fictious
Sorry, I just thought they would make an interesting comment - I won't link to any commercial sites in the future.
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 finish it at some point, but it's not easy stuff to digest for someone even with a couple calculus classes under their belt. I was taking a calculus review last fall, and when I saw the book come out, I was pretty excited to check it out. Honestly, at this point I'm sort of burned out on math. Not just math, but Math.
No matter what you think of Infinite Jest being too long, or whether you believe that DFW is fashionably being worshipped as the post-modern god of contemporary literature, I think that what he's doing is incredibly important. That is, amongst all the tech and science illiterate people involved in the liberal arts, he introduced science and math into the aesthetic realm, in a way that most people probably never imagined. In Infinite Jest, he included some intriguing passages that require at least a little bit of understanding of calculus in a way that was entertaining and enlightening. Math underlies so much of the modern world, and most people don't have any real conscious realization of it.
Is it well written? I think it's written well enough. I wouldn't have the slightest clue how to begin writing such a book. It wasn't as accessible as Hawking's Brief History of Time, but it has certainly been enlightening to me, in the same way that James Gleick's book on chaos theory was enlightening to me. For people who are not otherwise familiar with DFW, definitely check out A Supposedly Fun Thing I'll Never Do again and Brief Interviews with Hideous Men. Wallace's incredibly precise awareness and (almost scientific) attention to linguistic detail is seriously unparalleled. At least in my reading experience.
I'm not sure why you would think that time is any less real than space. By your own argument, how can you prove that space has any physical reality? To many quantum gravity theorists, neither time nor space are fundamental -- the only fundamental things are certain sorts of events. Everything else, including both time and space, are just convenient representations for the human mind to perceive the patterns that occur among those events.
I'm not saying that your argument is wrong -- I'm just saying that you haven't taken it far enough, and once you take it to its logical conclusion, you quickly come to realise that time is just as real (or unreal) as everything else that we typically consider to be real.
|>oug
Its basically equivalent to the infinate geometric series .5 +.25 +.125 + .0625 ... etc. Try it in your calculator and you'll see it amounts to .9999
.5 in for the first term and .5 for the constant (each term is half the preceding term) and you get .5/(1-.5), which is .5/.5 = 1.
Now the equation for a geometric series
(fir)/(1-(constant)) where the absolute value of the constant is less than one.
So just plug
...
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.
|>oug
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.
-DNA
Your hand doesn't "exist" in the same way as "change of position in space" exists. Think about what it would mean to doubt the former, then think of what it would mean to doubt the latter.
If "we" were able to "prove" time is "real", it would be "real" in a different sense than your hand, the color blue, and (perhaps) spatial relations.
What does it mean for "reality" to be "real"? Are we talking about time travel or talking about talking about time travel? If "time" didn't exist at least as a concept, "time travel" wouldn't be impossible, it would be meaningless. As to "traveling through some sort of space" being "predicated upon finding that space first" -- I don't understand what you're trying to say. In order to "move through space" in the conventional sense of the phrase, I certainly don't have to understand what space "is", any more than the Earth has to understand the law of gravitation in order to orbit the sun. Are you saying otherwise?
If understanding "understanding" necessarily proceeded understanding, we could never understand. I take Socrates' position in this endless argument, "that is, that we shall be better, braver, and more active men if we believe it right to look for what we don't know than if we believe there is no point in looking because what we don't know we can never discover." (Plato, Meno).
No, as a matter of fact "we" don't ususally say that, unless "we're" doing philosophy.
Yes, to keep track of time. Not to understand its deeper meaning.
An observation, which is conveniently corollated with other observations which happen to be more difficult to directly quantify. Isn't that all we mean by "measurement"?
No, they move in predictable ways. Based on this property, they can be used to measure other things. They certainly don't "do the measuring" themselves...
Yet the verity of "verity" is not assumed. We have deeper problems! Somebody call the philosophy police!
What do we mean by "time's existence"? What would it mean for time to "not exist"? In short, what are we trying to prove with our "thought experiment" (and i
That's pretty good ... except I don't understand how e really expanded your mind unless you use one of the other forms for e ... primarily
e = sum(1/n!, n, 0, inf)
If I am not mistaken.
Nice puns though, and thanks for posting the links. It seems like the moderators didn't mind either.
Cheers.
Mathematician, n.:
Someone who believes imaginary things appear right before your i's.
I think the most convincing argument for the existence of time is special relativity. Since identical clocks move at different rates in moving objects than in stationary objects, it must be true that time actually exists, and the clock just doesn't tick at a certain rate because it was made so.
"Knowledge makes us accountable." - Che Guevara
We also find out in the interview DFW has a crippling dependence on chaw. Who knew?
The argument goes kind of like this:
... (which is itself a fraction), and q_r be the sum of fractions 1 + 0/10 + 0/100 + 0/1000 ... and you can prove that there are no other q strictly greater than q_l, nor q' strictly lesser than q_r that satisfy the expression x * x = 1. (Go ahead, try to construct a number that is provably in the set but has a finite difference between itself and that previously greatest member q_l)
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 such that q x, where x satisfies x * x = 2. We can drop any q or q' replaced for x and trivially see which set it belongs to (lesser or greater). In each case, we are guaranteed a least member of q', and greatest member of q, even if the sets are infinite (!) and we call them q_l and q_r. (Proof: there are an infinite number of unique rationals between any distinct rationals a and b, but every countably infinite set contains a minimum or maximally valued element) The pair q_l and q_r are the real number sqrt(2).
In the case of 0.99999... = 1.00000 we could say we wish to solve x * x = 1, and let q_l be the sum of fractions 9/10 + 9/100 + 9/1000
The proof is an inductive proof (and induction is an axiom of the set used to define this bracket defintion of real numbers).
I may have a few details/terms confused, as this was 4 years ago that we looked at this.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
Take a look at the The Universal Computer.
It's really a study of the history of Computability. It contains chapters on Hilbert, Godel, Turing, Boole, Frege, and many others.
The author is a logician. He solved one of Hilbert's problems. He studied under Emil Post and Alanzo Church. He also worked at the Institute of Advanced Study with Von Nuemann and Godel.
The book requires a small dose (hardly any) mathematical maturity. It explains concepts like Turing Machines and Cantor's Diagonal Arugument very well.
IMHO, it's the best popular science book on logic.
What do you mean my sig is repetitive? What do you mean my sig is repetitive? What do you mean....
The answer to the question in this thread is unknown. It is the same question as proving 1/infinity = 0.
In EAM, DFW subtly conceded this point. He mentioned that if you accepted infinitesmal (currently, there are many mathematicians who hold that infinitesmal calculus is consistent) then the answer is a no. 0.99999... != 1. They differ by an infinitesmal. If you hold to the traditional pre-Cantorian view, you are likely to say that 0.99999... approaches 1 but stop before assigning equality.
My personal view is that 0.999... and 1 have essentially the same value, but that 0.999... contains different information than 1. For example 0.9999.... does not belong in the set of numbers that begin with a 1.
In other words, there are equally valid mathematical models which have different answers to this question. There is no way to prove either the view that they are or are not equal.
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.
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.
/. is one of the best pieces of evidence that you should ignore all the comments having anything to do with computer science.
The popularity of GEB on
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 is possible, with the technology we have available today, to publish a proof that contains an error. 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.
Finally! A chance to post this link!
Girlfriend Stops Reading David Foster Wallace Breakup Letter At Page 20
They will never know the simple pleasure of a monkey knife fight
Very well written review. I probably won't ever read the book, but you did a nice job reviewing it. Kudos. KUDOS.
#19845
Don't forget the classic One Two Three . . . Infinity : Facts and Speculations of Science by George Gamow (AbeBooks might have it for cheaper.).
Have you read my journal today?
Just six weeks ago I posted about countable and uncountable infinities 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 point either.
-- Our systemic servants do not good masters make.
I started Infinite Jest, then left my copy in a restaurant, and couldn't be bothered to go back and pick it up.
I realised that I had read >150 pages and I still hadn't met a character about whom I could bring myself to care what happened to them. Which is, IMHO, a bad thing.
evil math within Nature's Cubic Creation!
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.
Try _Elements of Set Theory_ by Enderton. The foundations of math are open for anybody to learn. Why settle for some 2nd-hand account?