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Everything and More: A Compact History of Infinity
One of the outstanding voices of his generation, David Foster Wallace has won a large and devoted following for the intellectual ambition and bravura style of his fiction and essays. Now he brings his considerable talents to the history of one of math's most enduring puzzles: the seemingly paradoxical nature of infinity.
Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology.
Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and high-profile recognition to the bizarre and fascinating world of higher mathematics.
Is infinity a valid mathematical property or a meaningless abstraction? The nineteenth-century mathematical genius Georg Cantor's answer to this question not only surprised him but also shook the very foundations upon which math had been built. Cantor's counterintuitive discovery of a progression of larger and larger infinities created controversy in his time and may have hastened his mental breakdown, but it also helped lead to the development of set theory, analytic philosophy, and even computer technology.
Smart, challenging, and thoroughly rewarding, Wallace's tour de force brings immediate and high-profile recognition to the bizarre and fascinating world of higher mathematics.
319 pages, Hardcover
First published January 1, 2003
653 people are currently reading
9324 people want to read
About the author
David Foster Wallace
131 books13.3k followersDavid Foster Wallace was an acclaimed American writer known for his fiction, nonfiction, and critical essays that explored the complexities of consciousness, irony, and the human condition. Widely regarded as one of the most innovative literary voices of his generation, Wallace is perhaps best known for his 1996 novel Infinite Jest, which was listed by Time magazine as one of the 100 best English-language novels published between 1923 and 2005. His unfinished final novel, The Pale King, was published posthumously in 2011 and was a finalist for the Pulitzer Prize.
Born in Ithaca, New York, Wallace was raised in Illinois, where he excelled as both a student and a junior tennis player—a sport he later wrote about with sharp insight and humor. He earned degrees in English and philosophy from Amherst College, then completed an MFA in creative writing at the University of Arizona. His early academic work in logic and philosophy informed much of his writing, particularly in his blending of analytical depth with emotional complexity.
Wallace’s first novel, The Broom of the System (1987), established his reputation as a fresh literary talent. Over the next two decades, he published widely in prestigious journals and magazines, producing short stories, essays, and book reviews that earned him critical acclaim. His work was characterized by linguistic virtuosity, inventive structure, and a deep concern for moral and existential questions. In addition to fiction, he tackled topics ranging from tennis and state fairs to cruise ships, politics, and the ethics of food consumption.
Beyond his literary achievements, Wallace had a significant academic career, teaching literature and writing at Emerson College, Illinois State University, and Pomona College. He was known for his intense engagement with students and commitment to teaching.
Wallace struggled with depression and addiction for much of his adult life, and he was hospitalized multiple times. He died by suicide in 2008 at the age of 46. In the years since his death, his influence has continued to grow, inspiring scholars, conferences, and a dedicated readership. However, his legacy is complicated by posthumous revelations of abusive behavior, particularly during his relationship with writer Mary Karr, which has led to ongoing debate within literary and academic communities.
His distinctive voice—by turns cerebral, comic, and compassionate—remains a defining force in contemporary literature. Wallace once described fiction as a way of making readers feel "less alone inside," and it is that emotional resonance, alongside his formal daring, that continues to define his place in American letters.
Born in Ithaca, New York, Wallace was raised in Illinois, where he excelled as both a student and a junior tennis player—a sport he later wrote about with sharp insight and humor. He earned degrees in English and philosophy from Amherst College, then completed an MFA in creative writing at the University of Arizona. His early academic work in logic and philosophy informed much of his writing, particularly in his blending of analytical depth with emotional complexity.
Wallace’s first novel, The Broom of the System (1987), established his reputation as a fresh literary talent. Over the next two decades, he published widely in prestigious journals and magazines, producing short stories, essays, and book reviews that earned him critical acclaim. His work was characterized by linguistic virtuosity, inventive structure, and a deep concern for moral and existential questions. In addition to fiction, he tackled topics ranging from tennis and state fairs to cruise ships, politics, and the ethics of food consumption.
Beyond his literary achievements, Wallace had a significant academic career, teaching literature and writing at Emerson College, Illinois State University, and Pomona College. He was known for his intense engagement with students and commitment to teaching.
Wallace struggled with depression and addiction for much of his adult life, and he was hospitalized multiple times. He died by suicide in 2008 at the age of 46. In the years since his death, his influence has continued to grow, inspiring scholars, conferences, and a dedicated readership. However, his legacy is complicated by posthumous revelations of abusive behavior, particularly during his relationship with writer Mary Karr, which has led to ongoing debate within literary and academic communities.
His distinctive voice—by turns cerebral, comic, and compassionate—remains a defining force in contemporary literature. Wallace once described fiction as a way of making readers feel "less alone inside," and it is that emotional resonance, alongside his formal daring, that continues to define his place in American letters.
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June 10, 2023
Thinking Impossibly
It was the Greeks who discovered that numbers, and therefore mathematics, had only the most tenuous connection with the world in which we live. Numbers constitute a separate order of existence. The number 5 for example has no connection with the five apples that might be sitting on my kitchen table, or with the age of my youngest relative. The number 5 is something all on its own. It is constructed out of other numbers, which are made up of other numbers that may in turn be constructed using the number 5. Mathematics, in other words, is a completely self-contained and isolated world we make up.
No one was aware of this quite separate world before the Greeks stumbled across it. They were, rightly, in awe of its implications. The otherworldliness of mathematics suggested an unnaturalness, indeed a supernaturalness, that demanded religious veneration. Mathematics seemed to literally reveal things that were unknowable in any other way. Numbers must be divine, they thought. Numbers were perfect. What we experienced outside of mathematics were imperfect approximations or distorted reflection of numbers. Within this religion of numbers, only two heresies were recognised: zero and infinity. These were demons which had no place in either the divine or the divine ‘word’ of mathematics.
The theological prejudice of the Greeks was tempered a bit in late antiquity. As mathematics inched its way from geometry to algebra, zero was recognised as a useful addition to mathematical doctrine - much like free will later became essential in strict Calvinism to motivate virtue. Zero seemed real enough since it was possible to point to an empty basket of fruit as a purported proof of its existence. But even today, there is debate about whether zero is a number or merely a digit which is useful in mathematical expression - something like a decimal point for example.
Infinity, however, is a different matter altogether. Although infinity is an essential concept in modern mathematics, there is no way to throw shade about what it is. Infinity can’t be pointed to nor represented except by symbols for something that is entirely beyond anyone’s experience. As Wallace’s title so concisely says, infinity is more than everything there is - more than the number of gluons, muons, bosons, and all other elementary particles in the entire universe, for example.
And the distance of infinity from any reality we know only increases when we recognise that there are many ‘orders’ of infinity - infinities that are more or less than other infinities. These higher orders of infinity weren’t discovered until the 19th century. And we appear still to have resisted the implications of these discoveries in the same way that the Pythagoreans did by keeping the indeterminacy of the infinitely long decimal expression of π, the relation between the circumference and the diameter of a circle, as a cultic secret which might undermine faith in mathematics. Infinity for them meant ‘mess.’
And infinity today, although less of a mess, is still very messy indeed about what it implies. Wallace quotes the great German mathematician, David Hilbert, approvingly: “The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Infinity is an abstraction, the ultimate mathematical abstraction. But an abstraction of what? No one has ever seen an infinitely full basket of anything in order to make such an abstraction.
No, infinity is an abstraction from a system of numbers, which themselves are supposedly abstractions. It is at this point that the ultimate revelation of mathematics takes place: numbers are indeed abstractions but abstractions of each other not of some experience of baskets of various items. Numbers produce each other; they have no existence except in their relationship with each other. 2 + 2 = 4 is not an inductive generalisation of market experience of baskets and their contents; it is an entirely intellectual proposition/discovery/definition. Which of these you choose to describe infinity is a reflection of one’s already established metaphysical position. It fits with and confirms them all.
Here’s the thing: infinity shows that the world created by mathematics has only an obscure and unreliable connection to our experience. This applies not just to the infinite in all its manifestations but also to the number 5 and its colleagues and associates. Like infinity, no one has ever experienced the number 5, or the way it interacts with other numbers to produce itself or yet further numbers. If you doubt this, just try to provide a precise statement of, say, the square root of 5. Numbers don’t cut the world at its joints. Sometimes they don’t even know their own joints. As Wallace summarises the situation: “... mathematical truths are certain and universal precisely because they have nothing to do with the world.”
And the revelations generated by infinity are not limited to mathematics; they extend to that more general realm of which mathematics is a part: language. Wallace nails this too: “... the abstract math that’s banished superstition and ignorance and unreason and birthed the modern world is also the abstract math that is shot through with unreason and paradox and conundrum and has, as it were, been trying to tie its shoes on the run ever since the beginning of its status as a real language.” Mathematics is the most precise language we have. Yet, ultimately it doesn’t know what it’s talking about, except itself.
None of this means that mathematics, or language in general, isn’t immensely useful. Of course it is; but for rather complex and often mysterious reasons. The revelation of infinity is simply that mathematics is not reality. Nor is any other language. Like all language, mathematics can be beautiful, and compelling, and inspirational. But it is never the way the world is. Confusion about this simple fact is something that human beings seem to have a great deal of trouble with. Language especially political language, easily reverts to religion (and vice-versa). Wallace’s little book is appropriate therapy for reducing this confusion.
And by the way, Neal Stephenson’s introduction alone is worth the price of admission.
It was the Greeks who discovered that numbers, and therefore mathematics, had only the most tenuous connection with the world in which we live. Numbers constitute a separate order of existence. The number 5 for example has no connection with the five apples that might be sitting on my kitchen table, or with the age of my youngest relative. The number 5 is something all on its own. It is constructed out of other numbers, which are made up of other numbers that may in turn be constructed using the number 5. Mathematics, in other words, is a completely self-contained and isolated world we make up.
No one was aware of this quite separate world before the Greeks stumbled across it. They were, rightly, in awe of its implications. The otherworldliness of mathematics suggested an unnaturalness, indeed a supernaturalness, that demanded religious veneration. Mathematics seemed to literally reveal things that were unknowable in any other way. Numbers must be divine, they thought. Numbers were perfect. What we experienced outside of mathematics were imperfect approximations or distorted reflection of numbers. Within this religion of numbers, only two heresies were recognised: zero and infinity. These were demons which had no place in either the divine or the divine ‘word’ of mathematics.
The theological prejudice of the Greeks was tempered a bit in late antiquity. As mathematics inched its way from geometry to algebra, zero was recognised as a useful addition to mathematical doctrine - much like free will later became essential in strict Calvinism to motivate virtue. Zero seemed real enough since it was possible to point to an empty basket of fruit as a purported proof of its existence. But even today, there is debate about whether zero is a number or merely a digit which is useful in mathematical expression - something like a decimal point for example.
Infinity, however, is a different matter altogether. Although infinity is an essential concept in modern mathematics, there is no way to throw shade about what it is. Infinity can’t be pointed to nor represented except by symbols for something that is entirely beyond anyone’s experience. As Wallace’s title so concisely says, infinity is more than everything there is - more than the number of gluons, muons, bosons, and all other elementary particles in the entire universe, for example.
And the distance of infinity from any reality we know only increases when we recognise that there are many ‘orders’ of infinity - infinities that are more or less than other infinities. These higher orders of infinity weren’t discovered until the 19th century. And we appear still to have resisted the implications of these discoveries in the same way that the Pythagoreans did by keeping the indeterminacy of the infinitely long decimal expression of π, the relation between the circumference and the diameter of a circle, as a cultic secret which might undermine faith in mathematics. Infinity for them meant ‘mess.’
And infinity today, although less of a mess, is still very messy indeed about what it implies. Wallace quotes the great German mathematician, David Hilbert, approvingly: “The infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Infinity is an abstraction, the ultimate mathematical abstraction. But an abstraction of what? No one has ever seen an infinitely full basket of anything in order to make such an abstraction.
No, infinity is an abstraction from a system of numbers, which themselves are supposedly abstractions. It is at this point that the ultimate revelation of mathematics takes place: numbers are indeed abstractions but abstractions of each other not of some experience of baskets of various items. Numbers produce each other; they have no existence except in their relationship with each other. 2 + 2 = 4 is not an inductive generalisation of market experience of baskets and their contents; it is an entirely intellectual proposition/discovery/definition. Which of these you choose to describe infinity is a reflection of one’s already established metaphysical position. It fits with and confirms them all.
Here’s the thing: infinity shows that the world created by mathematics has only an obscure and unreliable connection to our experience. This applies not just to the infinite in all its manifestations but also to the number 5 and its colleagues and associates. Like infinity, no one has ever experienced the number 5, or the way it interacts with other numbers to produce itself or yet further numbers. If you doubt this, just try to provide a precise statement of, say, the square root of 5. Numbers don’t cut the world at its joints. Sometimes they don’t even know their own joints. As Wallace summarises the situation: “... mathematical truths are certain and universal precisely because they have nothing to do with the world.”
And the revelations generated by infinity are not limited to mathematics; they extend to that more general realm of which mathematics is a part: language. Wallace nails this too: “... the abstract math that’s banished superstition and ignorance and unreason and birthed the modern world is also the abstract math that is shot through with unreason and paradox and conundrum and has, as it were, been trying to tie its shoes on the run ever since the beginning of its status as a real language.” Mathematics is the most precise language we have. Yet, ultimately it doesn’t know what it’s talking about, except itself.
None of this means that mathematics, or language in general, isn’t immensely useful. Of course it is; but for rather complex and often mysterious reasons. The revelation of infinity is simply that mathematics is not reality. Nor is any other language. Like all language, mathematics can be beautiful, and compelling, and inspirational. But it is never the way the world is. Confusion about this simple fact is something that human beings seem to have a great deal of trouble with. Language especially political language, easily reverts to religion (and vice-versa). Wallace’s little book is appropriate therapy for reducing this confusion.
And by the way, Neal Stephenson’s introduction alone is worth the price of admission.
September 15, 2012
David Foster Wallace was a great writer of fiction. He was not a great writer of popular math exposition, as this book shows.
The main reason I read this book, besides just curiosity about one of the lesser-read Wallace books, was my interest in figuring out a certain infamous scene in Wallace's wonderful novel Infinite Jest. In that scene, one character (Michael Pemulis) dictates to another a description of a mathematical method, based on the Mean Value Theorem, that he says will simplify the calculations involved in playing a certain complicated wargame. But Pemulis' proposed method does not actually make any mathematical sense. (He states the Mean Value Theorem correctly, but there is no useful way to apply it to the problem he wants to solve.) Ever since reading that scene, I've wondered if this was a mistake on Wallace's part or a deliberate choice intended to cast doubt on Pemulis' mathematical ability. Since Everything and More deals with some of the same sort of math that appeared in that scene (elementary calculus), it seemed like a good place to look for answers about Wallace's own grasp of that material.
Unfortunately, it was. This book is full of errors. A lot of them are just terminological solecisms that general readers won't notice or care about, but there are also some mathematical arguments in the book that are seriously flawed -- some of them much worse, in fact, than Pemulis' argument. (Some of them are wrong in an utterly weird, "only a stoned undergrad at 3 AM could think like this" way, which makes me wonder how on earth they got found their way into the book -- extreme time pressure, maybe?) I'm now forced to conclude that the Mean Value Theorem thing in IJ is not a sly bit of characterization, but simple authorial incompetence.
Everything and More is also very poorly written and organized. There's very little of the usual Wallace charm and cleverness, and a lot of aimless rambling, needless distinctions and clarifications-that-don't-really-clarify. Anyone who reads this book without no knowledge of the relevant math will come out of the experience with the impression that it is incredibly thorny and complicated and that Wallace has done his heroic best to shape it into some popularly presentable form. As it happens, most of the math is actually quite simple, and most of the appearance of complexity here is an artifact of Wallace's style -- the result of inconsequential (or incorrect!) nitpicking and a dizzying, needlessly scattered order of presentation.
It makes me sad to think that there are people out there whose first impression of Wallace will come from this book.
The main reason I read this book, besides just curiosity about one of the lesser-read Wallace books, was my interest in figuring out a certain infamous scene in Wallace's wonderful novel Infinite Jest. In that scene, one character (Michael Pemulis) dictates to another a description of a mathematical method, based on the Mean Value Theorem, that he says will simplify the calculations involved in playing a certain complicated wargame. But Pemulis' proposed method does not actually make any mathematical sense. (He states the Mean Value Theorem correctly, but there is no useful way to apply it to the problem he wants to solve.) Ever since reading that scene, I've wondered if this was a mistake on Wallace's part or a deliberate choice intended to cast doubt on Pemulis' mathematical ability. Since Everything and More deals with some of the same sort of math that appeared in that scene (elementary calculus), it seemed like a good place to look for answers about Wallace's own grasp of that material.
Unfortunately, it was. This book is full of errors. A lot of them are just terminological solecisms that general readers won't notice or care about, but there are also some mathematical arguments in the book that are seriously flawed -- some of them much worse, in fact, than Pemulis' argument. (Some of them are wrong in an utterly weird, "only a stoned undergrad at 3 AM could think like this" way, which makes me wonder how on earth they got found their way into the book -- extreme time pressure, maybe?) I'm now forced to conclude that the Mean Value Theorem thing in IJ is not a sly bit of characterization, but simple authorial incompetence.
Everything and More is also very poorly written and organized. There's very little of the usual Wallace charm and cleverness, and a lot of aimless rambling, needless distinctions and clarifications-that-don't-really-clarify. Anyone who reads this book without no knowledge of the relevant math will come out of the experience with the impression that it is incredibly thorny and complicated and that Wallace has done his heroic best to shape it into some popularly presentable form. As it happens, most of the math is actually quite simple, and most of the appearance of complexity here is an artifact of Wallace's style -- the result of inconsequential (or incorrect!) nitpicking and a dizzying, needlessly scattered order of presentation.
It makes me sad to think that there are people out there whose first impression of Wallace will come from this book.
August 27, 2020
L'infinito... cos'è l'infinito? Esiste davvero l'infinito?
L'universo, forse, è infinito o comunque è talmente vasto che per noi potrebbe già essere considerato infinito o talmente infinitamente infinito...
DFW in questo libro cerca di raccontarci come si potrebbe confermare l'infinitesimalità dei numeri e degli insiemi, in matematica. Attraverso i secoli, dagli albori degli studi matematici, da Aristotele a Pitagora a Galileo a Newton, fino ad arrivare a Weierstrass, Dedekind, ma soprattutto George Cantor ed in ultima analisi a Godel e Russell, DFW scandaglia gli studi fatti da questi straordinari matematici, sull'annoso problema dei numeri infiniti o meglio transfiniti, principalmente attraverso le teorie degli insiemi.
Il libro che ho appena letto, non è semplice, anzi direi di averlo trovato abbastanza ostico, soprattutto per quei concetti puramente da matematica ed analisi universitaria, che io non ho frequentato, ma è talmente affascinante ed appassionante da avermelo fatto leggere e rileggere diverse volte, alla fine non penso di aver capito tutto, ma va bene lo stesso, il concetto di infinito è imprescindibile per chiunque abbia voglia di capire la vita, l'universo e come girano le cose al mondo, come cita bene DFW nel libro: "I numeri sono libere creazioni della mente umana; essi servono come mezzo per cogliere con maggiore facilità e acutezza la differenza tra le cose". E quale mezzo migliore dell'infinito ci potrebbe essere?
Un mondo che oggi ruota in un nuovo tipo di Vuoto, tutto formale. La matematica continua ad alzarsi dal letto.
DFW continua a confermarsi un autore straordinario, qui, differentemente dai suoi romanzi, si mette nei panni del divulgatore scientifico, a suo modo, cioè con note su note, che sono un mondo a parte (chi ha letto un qualsiasi suo libro, capisce che cosa intendo) e...
L'universo, forse, è infinito o comunque è talmente vasto che per noi potrebbe già essere considerato infinito o talmente infinitamente infinito...
DFW in questo libro cerca di raccontarci come si potrebbe confermare l'infinitesimalità dei numeri e degli insiemi, in matematica. Attraverso i secoli, dagli albori degli studi matematici, da Aristotele a Pitagora a Galileo a Newton, fino ad arrivare a Weierstrass, Dedekind, ma soprattutto George Cantor ed in ultima analisi a Godel e Russell, DFW scandaglia gli studi fatti da questi straordinari matematici, sull'annoso problema dei numeri infiniti o meglio transfiniti, principalmente attraverso le teorie degli insiemi.
Il libro che ho appena letto, non è semplice, anzi direi di averlo trovato abbastanza ostico, soprattutto per quei concetti puramente da matematica ed analisi universitaria, che io non ho frequentato, ma è talmente affascinante ed appassionante da avermelo fatto leggere e rileggere diverse volte, alla fine non penso di aver capito tutto, ma va bene lo stesso, il concetto di infinito è imprescindibile per chiunque abbia voglia di capire la vita, l'universo e come girano le cose al mondo, come cita bene DFW nel libro: "I numeri sono libere creazioni della mente umana; essi servono come mezzo per cogliere con maggiore facilità e acutezza la differenza tra le cose". E quale mezzo migliore dell'infinito ci potrebbe essere?
Un mondo che oggi ruota in un nuovo tipo di Vuoto, tutto formale. La matematica continua ad alzarsi dal letto.
DFW continua a confermarsi un autore straordinario, qui, differentemente dai suoi romanzi, si mette nei panni del divulgatore scientifico, a suo modo, cioè con note su note, che sono un mondo a parte (chi ha letto un qualsiasi suo libro, capisce che cosa intendo) e...
January 1, 2021
I never intended to read this one, given that it's almost universally panned. But I found a hardback copy for sale for a literal dollar, and sometimes a bargain of that magnitude can feel strangely like divine providence...
The subject is perhaps DFW's most incongruous since Signifying Rappers, but you have to give him credit for the commitment: this is a thoroughly researched book, and the effort needed to put this together must have been immense, especially for a non-expert. But the problem here is that the author's writing style is totally at odds with the subject matter. DFW as a nonfiction writer is at his most compelling when exploring hidden facets of mundane things. He uses frequent digressions and footnotes to create complexity, and he uses complex language to frame simple concepts. This creates a sort of fervent energy, which (since there is no risk of losing the reader on the subject matter) allows him to generate interest on the strength of his voice alone.
While this approach works well when writing about something as quotidian as a concert or sporting event, it is misapplied to the already very complex subjects tackled in Everything And More. DFW tries earnestly to explain each concept, but he constantly gets in his own way, offering branching digressions where they are not needed, and serving only to obscure the core narrative or point, making the whole thing more difficult to understand, and less enjoyable. What he needed to do is rein himself in; simplify; focus on clarity and the logical organisation of his ideas.
The subject is perhaps DFW's most incongruous since Signifying Rappers, but you have to give him credit for the commitment: this is a thoroughly researched book, and the effort needed to put this together must have been immense, especially for a non-expert. But the problem here is that the author's writing style is totally at odds with the subject matter. DFW as a nonfiction writer is at his most compelling when exploring hidden facets of mundane things. He uses frequent digressions and footnotes to create complexity, and he uses complex language to frame simple concepts. This creates a sort of fervent energy, which (since there is no risk of losing the reader on the subject matter) allows him to generate interest on the strength of his voice alone.
While this approach works well when writing about something as quotidian as a concert or sporting event, it is misapplied to the already very complex subjects tackled in Everything And More. DFW tries earnestly to explain each concept, but he constantly gets in his own way, offering branching digressions where they are not needed, and serving only to obscure the core narrative or point, making the whole thing more difficult to understand, and less enjoyable. What he needed to do is rein himself in; simplify; focus on clarity and the logical organisation of his ideas.
April 13, 2011
HERE IS WHY THIS BOOK IS AWESOME:
This book addresses three related enthusiasms: for mathematics itself, for math history (the lives of the mathematicians & the historical chain of deduction that gave us the math of today) and for DFW's high school math teacher (who sounds totally amazing). A book about any one of these might be more straightforward but DFW conflates the three in a breezy, entertaining mess. The operating concept is the history of infinity as a topic that has driven mathematicians nuts. The designated hero of the story is Gregory Cantor, but you hardly even see him until the last chapter. The rest is foreshadowing & background material & lots and lots (lots!) of math.
Lovers of DFW's prose couldn't ever find a purer source of it. I was constantly laughing at footnotes, loving the intertwine of math and history, enjoyed all the ways he bent the conventions of mathematical writing to the weird shape of his brain. If you like DFW but have been putting this one off, this really is not the one to put off. His stated goal is to make a bunch of boring math more interesting and to walk you through the hard parts. I am probably his ideal reader: an interested and smart yet lazy & unconcerned person who hasn't thought about infinity lately.
My measure of a good book is how much it makes me think, and this book gets five starts for reminding me that Math is a planet and not just a multi-tool. And for succeeding in highlighting that the paradoxical nature of infinity is hiding right behind all the math-tricks I learned in high school, had anyone ever pointed them out to me or had I ever bothered to look. (I recall the opposite: we were encouraged not to go there.) The nature of the infinitely large and the infinitely small has felt, at least for a few days, like a metaphor for all sorts of other failures of logic and rationality. Likewise, the concept of a discrete set vs. a continuum is ably and interestingly highlighted here. The many ways in which it seems all of geometry and all of arithmetic are non-identical conjoined twins, even though that distinction divides math history into two warring camps, is suitably made deep. My appetite for understanding is bolstered. Infinity is fun!
I have to admit my main discouragement in following the math presented here is that I can't seem to summon the sense of dread and confusion that comes from, for instance, asserting that 9.999... repeating forever is equal to 10. Maybe because I didn't have DFW's high school math teacher, I find I'm blasé about infinity in a way that I gather would appall most of the mathematicians who have grappled with the concepts. In a sense, I just don't care. And so many different ways of talking about the problems of infinity and discontinuity, from Zeno up to Cantor, as presented by DFW, really do feel like a long series of restatements of the obvious: that infinity is a paradox math can't ever straighten out, but if you don't worry too hard it's actually present everywhere and quite handy.
A sad truth this book drives home, once again, is that high school Math is too often taught -- was taught to me, even in "Honors" math courses -- as Computation: come, kids, and learn about these nifty, cryptic, useful symbolic systems we found over here on this bookshelf! Do some drills, get some practice using them to solve certain kinds of problems, and just maybe (via the dreaded Word Problems) develop some intuition about which of these solutions might apply to which of your upcoming future questions.
Wheras Math, as understood by mathematicians (such as DFW's amazing-sounding HS math teacher) is more like another planet -- an actual landscape, a real thing that exists and can be perceived, initially by our intuitions (i.e. that two grapes and two oranges are similar in the sense that there are two of them, and therefore "twoness" exists and can be known, as can the nesses of other integers) and then later by deducing from just those truths plus our intuitions, just as astrophysicists can know the likely orbits of habitable planets in far-off galaxies. There is this incredible detail to the mathematical landscape, and the people who discovered it were real explorers. This version of Math relates to mere Computation in about the same way that the study of physics relates to auto shop. But efforts to base grade school mathematical education in intuition of mathematical truths instead of computation drills (see: New Math) are constantly met with deep suspicion by all the parents and administrators who themselves only learned Computation and don't get the difference. So DFW lucked out there.
(One thesis of Neal Stephenson's introduction is that this was a direct result of DFW growing up in a midwest college town, overpopulated with humble degreed braniacs who did things like teach high school math. Whereas I -- in defense of my own quite likable Honors Math teacher -- grew up in Silicon Valley, a society fairly fixated on Computation for Computation's sake.)
Which in the end means that, to me, Cantor's diagonal proof about the rational number set & subsequent branding of the real number set as a higher order of infinity seems like much ado about nothing, just another rephrasing of the fact that the latter is continuous and the former is not, which means that the former is composed of numbers and the latter of spaces containing numbers, which really doesn't seem so "hard" to me, but i'm totally willing to accept that I'm just missing something. Perhaps this is the inevitable result of an education in Computation of math instead of Comprehension of it: I'm too quick to discount the divine & take the rest for granted.
HERE IS WHY THIS BOOK SUCKS:
I had a big objection to Infinite Jest based on one mathematical footnote DFW gave which convinced me his grasp of mathematics was not all he thought it was. I must look up and revisit that, because this book really thoroougly convinces me that he knew way more about Math than I ever will. Reading it, I have not just been entertained by a whole bunch of chaotic, burbling DFW-prose; I have also come to believe that I learned something.
However, there are quite a few Real Mathematicians who would dispute that. This book was not well-reviewed by mathematicians, in two senses First, it seems not enough of them were asked to review the manuscript for errors before publication. Second, upon publication, many of them found the math to be full of holes. Here is probably the most charitable review in this vein; Here is one that really slides the knife in. I will not get into them. Suffice it to say that I have two warring concepts of DFW: one is True Genius, the other is Bullshitter In Genius Clothing. Reading the book, I was lured back to the Genius side. But I felt a necessity to check the facts, and when I did -- just like with Infinite Jest -- the odor of Bullshit again became detectable.
Mathematicians, of course, are just the sort of fun-free jerks who would be anal enough to poke holes in a lyrical work of math fantasy that the rest of us are trying to enjoy. How you feel about that is a really important question. Please take a moment to ponder it; it is pertinent across the entire Popular Science section of your local bookstore.
The math-reviewers don't hesitate to label DFW a "fiction writer" although his best work IMHO is journalism. But yes, he writes to entertain. This book is entertaining. And Popular Science, taken as a genre -- with Popular Math, its more recent sub-genre -- strives to entertain. That's how it gets Popular. Publishers put these books out to sell them, and the idea of DFW writing a treatise on the history of infinity had to sound good in the boardroom. He wrote something -- apparently something a bit more erudite and symbol-encrusted than they were hoping -- but they printed it anyway. It seemed entertaining enough. Print it! Sell it!
And that's fine for fiction, but this book purports to relay mathematical and historical fact. In such a book, facts should be checked and then double-checked -- that is, if the book is really striving to educate. It would not have been hard AT ALL. But, if you believe science is a decorative art and history is "true stories", it's not much of a stretch to consider Mathematics a flexible world of witchcraft akin to that found in Harry Potter books.
Can you tell how much that offends me? It really does.
This book, brilliant as it is, comes across as a first draft, despite at least one mention of a previous, even more chaotic draft, and despite what undoubtedly must have been a fair amount of research. Then again, he's faulted by some for not researching better; for not having read more of the available research on Cantor, for instance -- recall that Gregory Cantor is the purported star of this book, and DFW screws up certain facts about his life. Meanwhile, an extremely mathy-looking organizational scheme is invented on the fly for the sole purpose of making the book seem more organized than it is.
In a word: sloppy. DFW was a writer who's so talented at rhetoric, forming excellent sentences and entertaining voices, and also with a certain talent for bedazzling us with concepts from math, philosophy and tennis, that he could just ramble on about anything he thought was really interesting and sell the first draft to a major publisher. He was absolutely brilliant at sounding brilliant. But I keep on catching him trying to sound erudite without checking his facts, and it keeps eroding my faith in him.
This book addresses three related enthusiasms: for mathematics itself, for math history (the lives of the mathematicians & the historical chain of deduction that gave us the math of today) and for DFW's high school math teacher (who sounds totally amazing). A book about any one of these might be more straightforward but DFW conflates the three in a breezy, entertaining mess. The operating concept is the history of infinity as a topic that has driven mathematicians nuts. The designated hero of the story is Gregory Cantor, but you hardly even see him until the last chapter. The rest is foreshadowing & background material & lots and lots (lots!) of math.
Lovers of DFW's prose couldn't ever find a purer source of it. I was constantly laughing at footnotes, loving the intertwine of math and history, enjoyed all the ways he bent the conventions of mathematical writing to the weird shape of his brain. If you like DFW but have been putting this one off, this really is not the one to put off. His stated goal is to make a bunch of boring math more interesting and to walk you through the hard parts. I am probably his ideal reader: an interested and smart yet lazy & unconcerned person who hasn't thought about infinity lately.
My measure of a good book is how much it makes me think, and this book gets five starts for reminding me that Math is a planet and not just a multi-tool. And for succeeding in highlighting that the paradoxical nature of infinity is hiding right behind all the math-tricks I learned in high school, had anyone ever pointed them out to me or had I ever bothered to look. (I recall the opposite: we were encouraged not to go there.) The nature of the infinitely large and the infinitely small has felt, at least for a few days, like a metaphor for all sorts of other failures of logic and rationality. Likewise, the concept of a discrete set vs. a continuum is ably and interestingly highlighted here. The many ways in which it seems all of geometry and all of arithmetic are non-identical conjoined twins, even though that distinction divides math history into two warring camps, is suitably made deep. My appetite for understanding is bolstered. Infinity is fun!
I have to admit my main discouragement in following the math presented here is that I can't seem to summon the sense of dread and confusion that comes from, for instance, asserting that 9.999... repeating forever is equal to 10. Maybe because I didn't have DFW's high school math teacher, I find I'm blasé about infinity in a way that I gather would appall most of the mathematicians who have grappled with the concepts. In a sense, I just don't care. And so many different ways of talking about the problems of infinity and discontinuity, from Zeno up to Cantor, as presented by DFW, really do feel like a long series of restatements of the obvious: that infinity is a paradox math can't ever straighten out, but if you don't worry too hard it's actually present everywhere and quite handy.
A sad truth this book drives home, once again, is that high school Math is too often taught -- was taught to me, even in "Honors" math courses -- as Computation: come, kids, and learn about these nifty, cryptic, useful symbolic systems we found over here on this bookshelf! Do some drills, get some practice using them to solve certain kinds of problems, and just maybe (via the dreaded Word Problems) develop some intuition about which of these solutions might apply to which of your upcoming future questions.
Wheras Math, as understood by mathematicians (such as DFW's amazing-sounding HS math teacher) is more like another planet -- an actual landscape, a real thing that exists and can be perceived, initially by our intuitions (i.e. that two grapes and two oranges are similar in the sense that there are two of them, and therefore "twoness" exists and can be known, as can the nesses of other integers) and then later by deducing from just those truths plus our intuitions, just as astrophysicists can know the likely orbits of habitable planets in far-off galaxies. There is this incredible detail to the mathematical landscape, and the people who discovered it were real explorers. This version of Math relates to mere Computation in about the same way that the study of physics relates to auto shop. But efforts to base grade school mathematical education in intuition of mathematical truths instead of computation drills (see: New Math) are constantly met with deep suspicion by all the parents and administrators who themselves only learned Computation and don't get the difference. So DFW lucked out there.
(One thesis of Neal Stephenson's introduction is that this was a direct result of DFW growing up in a midwest college town, overpopulated with humble degreed braniacs who did things like teach high school math. Whereas I -- in defense of my own quite likable Honors Math teacher -- grew up in Silicon Valley, a society fairly fixated on Computation for Computation's sake.)
Which in the end means that, to me, Cantor's diagonal proof about the rational number set & subsequent branding of the real number set as a higher order of infinity seems like much ado about nothing, just another rephrasing of the fact that the latter is continuous and the former is not, which means that the former is composed of numbers and the latter of spaces containing numbers, which really doesn't seem so "hard" to me, but i'm totally willing to accept that I'm just missing something. Perhaps this is the inevitable result of an education in Computation of math instead of Comprehension of it: I'm too quick to discount the divine & take the rest for granted.
HERE IS WHY THIS BOOK SUCKS:
I had a big objection to Infinite Jest based on one mathematical footnote DFW gave which convinced me his grasp of mathematics was not all he thought it was. I must look up and revisit that, because this book really thoroougly convinces me that he knew way more about Math than I ever will. Reading it, I have not just been entertained by a whole bunch of chaotic, burbling DFW-prose; I have also come to believe that I learned something.
However, there are quite a few Real Mathematicians who would dispute that. This book was not well-reviewed by mathematicians, in two senses First, it seems not enough of them were asked to review the manuscript for errors before publication. Second, upon publication, many of them found the math to be full of holes. Here is probably the most charitable review in this vein; Here is one that really slides the knife in. I will not get into them. Suffice it to say that I have two warring concepts of DFW: one is True Genius, the other is Bullshitter In Genius Clothing. Reading the book, I was lured back to the Genius side. But I felt a necessity to check the facts, and when I did -- just like with Infinite Jest -- the odor of Bullshit again became detectable.
Mathematicians, of course, are just the sort of fun-free jerks who would be anal enough to poke holes in a lyrical work of math fantasy that the rest of us are trying to enjoy. How you feel about that is a really important question. Please take a moment to ponder it; it is pertinent across the entire Popular Science section of your local bookstore.
The math-reviewers don't hesitate to label DFW a "fiction writer" although his best work IMHO is journalism. But yes, he writes to entertain. This book is entertaining. And Popular Science, taken as a genre -- with Popular Math, its more recent sub-genre -- strives to entertain. That's how it gets Popular. Publishers put these books out to sell them, and the idea of DFW writing a treatise on the history of infinity had to sound good in the boardroom. He wrote something -- apparently something a bit more erudite and symbol-encrusted than they were hoping -- but they printed it anyway. It seemed entertaining enough. Print it! Sell it!
And that's fine for fiction, but this book purports to relay mathematical and historical fact. In such a book, facts should be checked and then double-checked -- that is, if the book is really striving to educate. It would not have been hard AT ALL. But, if you believe science is a decorative art and history is "true stories", it's not much of a stretch to consider Mathematics a flexible world of witchcraft akin to that found in Harry Potter books.
Can you tell how much that offends me? It really does.
This book, brilliant as it is, comes across as a first draft, despite at least one mention of a previous, even more chaotic draft, and despite what undoubtedly must have been a fair amount of research. Then again, he's faulted by some for not researching better; for not having read more of the available research on Cantor, for instance -- recall that Gregory Cantor is the purported star of this book, and DFW screws up certain facts about his life. Meanwhile, an extremely mathy-looking organizational scheme is invented on the fly for the sole purpose of making the book seem more organized than it is.
In a word: sloppy. DFW was a writer who's so talented at rhetoric, forming excellent sentences and entertaining voices, and also with a certain talent for bedazzling us with concepts from math, philosophy and tennis, that he could just ramble on about anything he thought was really interesting and sell the first draft to a major publisher. He was absolutely brilliant at sounding brilliant. But I keep on catching him trying to sound erudite without checking his facts, and it keeps eroding my faith in him.
February 14, 2024
[T]he infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to. Can thought about things be so much different from things? Can thinking processes be so unlike the actual process of things? In short, can thought be so far removed from reality?
- DFW, Everything and More
I've probably only got a few DFW books left. I like math, but I've been saving this one for some random time when I really wanted both DFW and something new. This book delivered. I own the whole closed set of Great Discovery books. They are great. Vollmann, DFW, and others writing about and explaining various scientific concepts, mostly focused around famous (Einstein, Godel, Darwin, etc) and less famous (unless you are moderately into science/math).
This is my first one. And it seemed appropriate for DFW in a couple ways: 1) DFW likes big concepts and was exceptionally good at formal logic, etc, 2) DFW's penchant for footnotes/endnotes/asides seems perfect for a book that is looking to write about a complicated topic in a way someone who isn't a graduate student in Mathematics could approach (with some difficulty) without completely pissing off specialists (from the introduction to later editions written by Neal Stephenson* it appears he wasn't really successful in irritating Mathematicians who take themselves seriously, but to be far, mathematicians seem primed to be irritated with layman and mathematicians alike), 3) his ability to translate high concepts into not just prose for the layman, but do so with interesting prose and a narrative that flows. Anyway, based on this first one that I've read, I'm definitely going to read the rest in 2024 and 2025. 4) Dr. E. Robert Goris of U—— Sr. High School’s AP Math I and II who DFW refers to periodically throughout this book and shows how critical great teachers are to the formation of curiosity. Also, his terms or summaries mentioned by DFW throughout this book are some of the best parts. 5) Some of the words/phrases DFW uses or makes up or both -- for example: epistoschizoid, attic facts, "D.E.s can be thought of either as integral calc on some sort of Class IV hallucinogen", "apodictic hygiene", ""math’s absolute Prince of Darkness"", etc. 6) I love too how often his writing becomes coversational, often with DFW speaking directly to the reader: "It would maybe be good to prepare yourself, emotionally, for having to read the following more than once" .
* BTW, my hardcover version of this book didn't have the Stephenson intro, so if you read this book and own the HC version, I'd also seek out the intro by Neal Stephenson. It was definitely a good move to include. Also, Stephenson's The Baroque Cycle is a fascinating read if you are into the enlightened period when Calculus (Newton and Leibniz) was starting to bend shit up.
- DFW, Everything and More
I've probably only got a few DFW books left. I like math, but I've been saving this one for some random time when I really wanted both DFW and something new. This book delivered. I own the whole closed set of Great Discovery books. They are great. Vollmann, DFW, and others writing about and explaining various scientific concepts, mostly focused around famous (Einstein, Godel, Darwin, etc) and less famous (unless you are moderately into science/math).
This is my first one. And it seemed appropriate for DFW in a couple ways: 1) DFW likes big concepts and was exceptionally good at formal logic, etc, 2) DFW's penchant for footnotes/endnotes/asides seems perfect for a book that is looking to write about a complicated topic in a way someone who isn't a graduate student in Mathematics could approach (with some difficulty) without completely pissing off specialists (from the introduction to later editions written by Neal Stephenson* it appears he wasn't really successful in irritating Mathematicians who take themselves seriously, but to be far, mathematicians seem primed to be irritated with layman and mathematicians alike), 3) his ability to translate high concepts into not just prose for the layman, but do so with interesting prose and a narrative that flows. Anyway, based on this first one that I've read, I'm definitely going to read the rest in 2024 and 2025. 4) Dr. E. Robert Goris of U—— Sr. High School’s AP Math I and II who DFW refers to periodically throughout this book and shows how critical great teachers are to the formation of curiosity. Also, his terms or summaries mentioned by DFW throughout this book are some of the best parts. 5) Some of the words/phrases DFW uses or makes up or both -- for example: epistoschizoid, attic facts, "D.E.s can be thought of either as integral calc on some sort of Class IV hallucinogen", "apodictic hygiene", ""math’s absolute Prince of Darkness"", etc. 6) I love too how often his writing becomes coversational, often with DFW speaking directly to the reader: "It would maybe be good to prepare yourself, emotionally, for having to read the following more than once" .
* BTW, my hardcover version of this book didn't have the Stephenson intro, so if you read this book and own the HC version, I'd also seek out the intro by Neal Stephenson. It was definitely a good move to include. Also, Stephenson's The Baroque Cycle is a fascinating read if you are into the enlightened period when Calculus (Newton and Leibniz) was starting to bend shit up.
August 8, 2023
Nach Unendlicher Spaß nun Viel Lärm um nichts.
David Foster Wallace hat drei Romane geschrieben „Der Besen im System“, „Unendlicher Spaß“ und „Der bleiche König“, der Fragment geblieben ist. Neben den Romanen und auch Kurzgeschichten schrieb er für Zeitungen, hielt Reden, rezensierte. Seine Essays sind mehrheitlich auch auf Deutsch erschienen in Sammlungen, die bunte Namen tragen wie „Am Beispiel des Hummers“ oder „Der Spaß an der Sache“. Hinzukommen auch Sachbücher. Zu ihnen gehört „Die Entdeckung des Unendlichen“:
„Sobald eine Zahl all diesen Bedingungen [der logischen Konsistenz] genügt, kann und muss sie als existent und real in der Mathematik betrachtet werden. Hier erblicke ich [Cantor] den … Grund, warum man die rationalen, irrationalen und die komplexen Zahlen für durchaus existent anzusehen hat wie die endlichen positiven ganzen Zahlen.“
Foster Wallace zeichnet in seinem Buch die Geschichte des Unendlichkeitsbegriffes in der Mathematik nach. Dieser gipfelt in Georg Ferdinand Ludwig Philipp Cantors Bemühen über die rationalen Zahlen hinaus eine Zählbarkeit einzuführen, die sich überabzählbar nennt und die Grundlage für eine völlig neue Form der Mengenlehre und der Zahlen bildet. Wie genau Foster Wallace die Sache nimmt, lässt sich sofort an dem Zitat ablesen. Dort, wo bei Foster Wallace „den … Grund“ steht, heißt es im Original: „Hierin blicke ich den in §. 4 angedeuteten Grund“. Cantor weiß also selbst, dass er keinen Beweis für das gibt, was er behauptet. Der Grund bleibt angedeutet. Foster Wallace kümmert sich um solche Nebensächlichkeiten nicht, schließlich gilt es einen Helden zu verehren:
„Die verführerischen [im Original steht sexy] Mathe-Ausdrücke spielen im Moment keine Rolle. Der Cantor der letzten Zeile ist Professor Georg F.L.P Cantor, geboren 1845, ein naturalisierter Deutscher aus Händlerschicht und anerkannter Vater der abstrakten Mengenlehre und der transfiniten Mathematiker. […] G.F.L.P. Cantor ist der bedeutendste Mathematiker des 19. Jahrhunderts und war eine sehr komplexe und leidvolle Persönlichkeit.“
Nur, Foster Wallace bespricht weder die Person Cantors, noch die Psyche, noch seine Geschichte wirklich, noch geht er ins Detail der mathematischen Ausführungen und Errungenschaften. Er schreibt im Stil einer Huldigung über das Problem einer Wissenschaft, das bis in die Gegenwart hineinreicht und das Feld der Mathematik in finite (computer- und ergebnisorientierte) und infinite (spekulative, ontologische) Mathematik teilt. Foster Wallace will davon nichts wissen und schließt:
„In der reellen Mengenlehre haben wir es mit abstrakten Gesamtheiten so vieler abstrakter Objekte zu tun, dass diese nicht gezählt oder vervollständigt oder auch nur vorgestellt werden können … und trotzdem beweisen [im Original hervorgehoben] wir deduktiv und damit definitiv Aussagen über die Zusammensetzungen und die Beziehungen zwischen diesen Gebilden.“
Ohne auf die Bedingung der Möglichkeit von Beweisen einzugehen, bspw. die Anschaulichkeit der vorgestellten Größen, die Widerspruchsfreiheit der Annahmen, bleibt David Foster Wallaces Sachbuch ein Pfeifen im dunklen Wald. Leider informiert es nicht noch belehrt es. Es feiert, aber was es feiert, wird nicht so richtig klar. Für viele hat Cantor die Mathematik in zwei Teile geteilt. Der eine glaubt, was er glaubt, solange das Gegenteil nicht bewiesen wird (aber Erfundenes, egal wie konsistent, kann nicht widerlegt werden), und der andere schaut schockiert weg und entwickelt Algorithmen zur Lösung von Problemen, die den ersteren trivial vorkommen. David Foster Wallace hat mit seinem „Die Entdeckung des Unendlichen“ keiner der beiden Seiten einen Gefallen getan.
David Foster Wallace hat drei Romane geschrieben „Der Besen im System“, „Unendlicher Spaß“ und „Der bleiche König“, der Fragment geblieben ist. Neben den Romanen und auch Kurzgeschichten schrieb er für Zeitungen, hielt Reden, rezensierte. Seine Essays sind mehrheitlich auch auf Deutsch erschienen in Sammlungen, die bunte Namen tragen wie „Am Beispiel des Hummers“ oder „Der Spaß an der Sache“. Hinzukommen auch Sachbücher. Zu ihnen gehört „Die Entdeckung des Unendlichen“:
„Sobald eine Zahl all diesen Bedingungen [der logischen Konsistenz] genügt, kann und muss sie als existent und real in der Mathematik betrachtet werden. Hier erblicke ich [Cantor] den … Grund, warum man die rationalen, irrationalen und die komplexen Zahlen für durchaus existent anzusehen hat wie die endlichen positiven ganzen Zahlen.“
Foster Wallace zeichnet in seinem Buch die Geschichte des Unendlichkeitsbegriffes in der Mathematik nach. Dieser gipfelt in Georg Ferdinand Ludwig Philipp Cantors Bemühen über die rationalen Zahlen hinaus eine Zählbarkeit einzuführen, die sich überabzählbar nennt und die Grundlage für eine völlig neue Form der Mengenlehre und der Zahlen bildet. Wie genau Foster Wallace die Sache nimmt, lässt sich sofort an dem Zitat ablesen. Dort, wo bei Foster Wallace „den … Grund“ steht, heißt es im Original: „Hierin blicke ich den in §. 4 angedeuteten Grund“. Cantor weiß also selbst, dass er keinen Beweis für das gibt, was er behauptet. Der Grund bleibt angedeutet. Foster Wallace kümmert sich um solche Nebensächlichkeiten nicht, schließlich gilt es einen Helden zu verehren:
„Die verführerischen [im Original steht sexy] Mathe-Ausdrücke spielen im Moment keine Rolle. Der Cantor der letzten Zeile ist Professor Georg F.L.P Cantor, geboren 1845, ein naturalisierter Deutscher aus Händlerschicht und anerkannter Vater der abstrakten Mengenlehre und der transfiniten Mathematiker. […] G.F.L.P. Cantor ist der bedeutendste Mathematiker des 19. Jahrhunderts und war eine sehr komplexe und leidvolle Persönlichkeit.“
Nur, Foster Wallace bespricht weder die Person Cantors, noch die Psyche, noch seine Geschichte wirklich, noch geht er ins Detail der mathematischen Ausführungen und Errungenschaften. Er schreibt im Stil einer Huldigung über das Problem einer Wissenschaft, das bis in die Gegenwart hineinreicht und das Feld der Mathematik in finite (computer- und ergebnisorientierte) und infinite (spekulative, ontologische) Mathematik teilt. Foster Wallace will davon nichts wissen und schließt:
„In der reellen Mengenlehre haben wir es mit abstrakten Gesamtheiten so vieler abstrakter Objekte zu tun, dass diese nicht gezählt oder vervollständigt oder auch nur vorgestellt werden können … und trotzdem beweisen [im Original hervorgehoben] wir deduktiv und damit definitiv Aussagen über die Zusammensetzungen und die Beziehungen zwischen diesen Gebilden.“
Ohne auf die Bedingung der Möglichkeit von Beweisen einzugehen, bspw. die Anschaulichkeit der vorgestellten Größen, die Widerspruchsfreiheit der Annahmen, bleibt David Foster Wallaces Sachbuch ein Pfeifen im dunklen Wald. Leider informiert es nicht noch belehrt es. Es feiert, aber was es feiert, wird nicht so richtig klar. Für viele hat Cantor die Mathematik in zwei Teile geteilt. Der eine glaubt, was er glaubt, solange das Gegenteil nicht bewiesen wird (aber Erfundenes, egal wie konsistent, kann nicht widerlegt werden), und der andere schaut schockiert weg und entwickelt Algorithmen zur Lösung von Problemen, die den ersteren trivial vorkommen. David Foster Wallace hat mit seinem „Die Entdeckung des Unendlichen“ keiner der beiden Seiten einen Gefallen getan.
January 2, 2020
Q:
To me Everything and More reads, rather, as a discourse from a green, gridded prairie heaven, where irony-free people who’ve been educated to a turn in those prairie schoolhouses and great-but-unpretentious universities sit around their dinner tables buttering sweet corn, drinking iced tea, and patiently trying to explain even the most recondite mysteries of the universe, out of a conviction that the world must be amenable to human understanding and that if you can understand something, you can explain it in words: fancy words if that helps, plain words if possible. But in any case you can reach out to other minds through that medium of words and make a connection. (c)
Q:
Here is a quotation from G. K. Chesterton: “Poets do not go mad; but chess players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not attacking logic: I only say that this danger does lie in logic, not in imagination.” Here also is a snippet from the flap copy for a recent pop bio of Cantor: “In the late nineteenth century, an extraordinary mathematician languished in an asylum. . . . The closer he came to the answers he sought, the further away they seemed. Eventually it drove him mad, as it had mathematicians before him.” (c) Well, this doesn't seem to be 100% true. Even though mathematicians do often get a tad odd...
Q:
The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That’s probably a bit overblown, at least in most cases. (c)
Q:
And of course since mathematics is a totally abstract language, one whose lack of specific real-world referents is supposed to yield maximal hygiene, its paradoxes and conundra are much more of a problem. Meaning math has to really deal with them instead of just putting them in the back of its mind once the alarm goes off. Some dilemmas can be handled legalistically, so to speak, by definition and stipulation. (c)
Q:
The real irony is that the view of ∞ as some forbidden zone or road to insanity—which view was very old and powerful and haunted math for 2000+ years—is precisely what Cantor’s own work overturned. (c)
Oh, boy! The dictionary was here:
Q:
...the other, antipodal stereotype of mathematicians as nerdy little bowtied fissiparous creatures.(с)
Q:
And at what point do the questions get so abstract and the distinctions so fine and the cephalalgia so bad that we simply can’t handle thinking about any of it anymore? (c)
Q:
The source of this pernicious myth is Aristotle, who is in certain respects the villain of our whole Story (c)
To me Everything and More reads, rather, as a discourse from a green, gridded prairie heaven, where irony-free people who’ve been educated to a turn in those prairie schoolhouses and great-but-unpretentious universities sit around their dinner tables buttering sweet corn, drinking iced tea, and patiently trying to explain even the most recondite mysteries of the universe, out of a conviction that the world must be amenable to human understanding and that if you can understand something, you can explain it in words: fancy words if that helps, plain words if possible. But in any case you can reach out to other minds through that medium of words and make a connection. (c)
Q:
Here is a quotation from G. K. Chesterton: “Poets do not go mad; but chess players do. Mathematicians go mad, and cashiers; but creative artists very seldom. I am not attacking logic: I only say that this danger does lie in logic, not in imagination.” Here also is a snippet from the flap copy for a recent pop bio of Cantor: “In the late nineteenth century, an extraordinary mathematician languished in an asylum. . . . The closer he came to the answers he sought, the further away they seemed. Eventually it drove him mad, as it had mathematicians before him.” (c) Well, this doesn't seem to be 100% true. Even though mathematicians do often get a tad odd...
Q:
The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That’s probably a bit overblown, at least in most cases. (c)
Q:
And of course since mathematics is a totally abstract language, one whose lack of specific real-world referents is supposed to yield maximal hygiene, its paradoxes and conundra are much more of a problem. Meaning math has to really deal with them instead of just putting them in the back of its mind once the alarm goes off. Some dilemmas can be handled legalistically, so to speak, by definition and stipulation. (c)
Q:
The real irony is that the view of ∞ as some forbidden zone or road to insanity—which view was very old and powerful and haunted math for 2000+ years—is precisely what Cantor’s own work overturned. (c)
Oh, boy! The dictionary was here:
Q:
...the other, antipodal stereotype of mathematicians as nerdy little bowtied fissiparous creatures.(с)
Q:
And at what point do the questions get so abstract and the distinctions so fine and the cephalalgia so bad that we simply can’t handle thinking about any of it anymore? (c)
Q:
The source of this pernicious myth is Aristotle, who is in certain respects the villain of our whole Story (c)
February 24, 2024
“Everything and More: A Compact History of Infinity” is a book by David Foster Wallace that explores the concept of infinity and its implications for mathematics, philosophy, and human understanding. A dramatic undertaking by a brilliant writer, the book covers the historical development of infinity, from the ancient Greeks to the modern era, explaining ideas and proofs in an accessible and engaging way, using examples, analogies, and humor. He also discusses the philosophical and theological implications of infinity, such as the paradoxes of the infinite and the nature of God. In it he reveals his own struggles with depression and anxiety, and how mathematics helped him cope with them—for a time—because now he’s dead. It’s difficult to read something like this and know what happens after. Nevertheless, I recommend “Everything and More” for anyone who is interested in mathematics, infinity, or Wallace’s writing.
June 3, 2020
I love David Foster Wallace, however this book was a flop. The writing is good but it fails in delivery. It was not an enjoyable read.
April 19, 2011
Well, as you might expect, this is great writing, at least the parts of it that are plain english. I hesitated to read it because it was, well, a math book, and the 7 semesters of college math i had to take was enough to last me a lifetime. Although I must say that if I had math teachers like David Foster Wallace, I probably would have liked it more. So anyway the book was a gift but sat on my shelf for a few months but I eventually sat down and read it. It was worth reading, but... I doubt it will be my favorite DFW book ( so far Infinite Jest is the only other one I've read - I like it much more). It's just lots of slogging through a subject that has no real bearing on anything in my life and is also not that entertaining other than the 5% of it that is Wallace saying hilarious little things in footnotes, for instance that Kurt Gödel is the Dark Prince of math. It's alluring in a geeky way but that's about the extent of it. It doesn't make me wiser about life, or help me be a better person, or a better anything else that I am, and it wasn't a whole lot of fun. Those are my criteria for reading a book. But I will give this one 4 stars because I think Wallace did probably about the best job possible of doing what he set out to do.
But you can pretty much summarize the book thus: "Mathematicians and philosophers kept putting off dealing with the concept of infinity for centuries. Finally some guys in Germany dealt with it. They showed that there are different kinds of greater and lesser infinities. This created more paradoxes and problems for the field, some of which still never got solved. David Foster Wallace was a really smart guy and really geeky (even though he may have screwed up some of the finer points of the math)."
But you can pretty much summarize the book thus: "Mathematicians and philosophers kept putting off dealing with the concept of infinity for centuries. Finally some guys in Germany dealt with it. They showed that there are different kinds of greater and lesser infinities. This created more paradoxes and problems for the field, some of which still never got solved. David Foster Wallace was a really smart guy and really geeky (even though he may have screwed up some of the finer points of the math)."
December 16, 2008
Reading other goodreads reviews, I decided I should write something because it seems that the other reviewers are either lazy or illiterate. "Everything and More" is unlike any other "pop" math book I've ever read. Most math books involve the personalities of these mythical math beings with some horrible math analogies sprinkled in to deceive the reader into thinking she is reading a math book rather than a poor biography. DFW does something completely different, actually writing about the intricacies of a math concept (that of infinity), while trying to break down the Hollywood notions of the mathematicians behind the work. Yes, the book is tough to read, and this is probably why it has received mixed reviews. The problem, however, is the underlying math is much harder to understand/enjoy if one decided to take a real analysis course (which is all about these type problems) instead of reading this book.* The book is not perfect (sometimes the frenetic style is a bit much, even for me), but it will be the most rewarding math book you have read.
* IYI(If you're interested) - I suffered through a real analysis course for a while before finding it completely boring and useless. After reading this work, I've decided that these questions are deep and beautiful and I will take another shot at learning this material
* IYI(If you're interested) - I suffered through a real analysis course for a while before finding it completely boring and useless. After reading this work, I've decided that these questions are deep and beautiful and I will take another shot at learning this material
October 1, 2014
I've now read everything that David Foster Wallace published in book form, which became a goal of mine back on 09/15/08 when I heard that he'd hanged himself on 09/12/08. At that time, this book and "Signifying Rappers" were the only two I hadn't yet read. I wouldn't otherwise have read "Everything and More," given that I'm not all that strong a math student.
With that happy preface, let me tell you that "Everything and More: A Compact History of ∞" is very technical, and its reader should ideally possess a medium to strong math background. This reader, mathematically anemic at best, did however enjoy the good old DFW rhetorical japes and games and general good times, which are also present in this work. Also, I did enjoy learning the basic rough outlines of such concepts as Zenos's Paradox, Vicious Infinite Regress, number theory, etc. ("Rough outlines" not because DFW doesn't devote considerable rhetorical- and word-count attention to the concepts, but "rough" because so much of it went over my head.)
Here's what the back of the book says, "[DFW:] brings his intellectual ambition and bravura style to the story of how mathematicians have struggled to understand the infinite, from the ancient Greeks to the nineteenth-century mathematical genius Georg Cantor's counterintuitive discovery that there was more than one kind of infinity."
Okay. Not exactly ADD medicine here, but the blurb is at least generally accurate. But not "sexy," which is DFW's operative term (that and synonyms like "eros-laden", "zaftig", etc.) for interesting or exciting concepts. Sexy is this quote from the little booklet's text (which quote will be this review's conclusion):
"The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That's probably a bit overblown, at least in most cases. (FN2: Although, so is the other, antipodal stereotype of mathematicians as nerdy little bowtied fissiparous creatures. In today's archetypology, the two stereotypes seem to play off each other in important ways.) But Cantor fits the template better than most. And the reasons for this are a lot more interesting than whatever his problems and symptoms were. (FN3: In modern medical terms, it's fairly clear that G.F.L.P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. Of course, this makes for less interesting flap copy than Genius Driven Mad By Attempts To Grapple With ∞. The truth, though, is that Cantor's work and its context are so totally interesting and beautiful that there's no need for breathless Prometheusizing of the poor guy's life. The real irony is that the view of ∞ as some forbidden zone or road to insanity -- which view was very old and powerful and haunted math for 2000+ years -- is precisely what Cantor's own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George's loss to the dragon: it's not only wrong but insulting.)"
Please recall or here be informed (as wikipedia just informed me) that St. George actually first wounded then tamed and finally slew the dragon. Which of course underscores DFW's point about Georg F.L.P. Cantor first (figuratively) taming then slaying the ∞.
DFW: requiescat in pace.
With that happy preface, let me tell you that "Everything and More: A Compact History of ∞" is very technical, and its reader should ideally possess a medium to strong math background. This reader, mathematically anemic at best, did however enjoy the good old DFW rhetorical japes and games and general good times, which are also present in this work. Also, I did enjoy learning the basic rough outlines of such concepts as Zenos's Paradox, Vicious Infinite Regress, number theory, etc. ("Rough outlines" not because DFW doesn't devote considerable rhetorical- and word-count attention to the concepts, but "rough" because so much of it went over my head.)
Here's what the back of the book says, "[DFW:] brings his intellectual ambition and bravura style to the story of how mathematicians have struggled to understand the infinite, from the ancient Greeks to the nineteenth-century mathematical genius Georg Cantor's counterintuitive discovery that there was more than one kind of infinity."
Okay. Not exactly ADD medicine here, but the blurb is at least generally accurate. But not "sexy," which is DFW's operative term (that and synonyms like "eros-laden", "zaftig", etc.) for interesting or exciting concepts. Sexy is this quote from the little booklet's text (which quote will be this review's conclusion):
"The Mentally Ill Mathematician seems now in some ways to be what the Knight Errant, Mortified Saint, Tortured Artist, and Mad Scientist have been for other eras: sort of our Prometheus, the one who goes to forbidden places and returns with gifts we all can use but he alone pays for. That's probably a bit overblown, at least in most cases. (FN2: Although, so is the other, antipodal stereotype of mathematicians as nerdy little bowtied fissiparous creatures. In today's archetypology, the two stereotypes seem to play off each other in important ways.) But Cantor fits the template better than most. And the reasons for this are a lot more interesting than whatever his problems and symptoms were. (FN3: In modern medical terms, it's fairly clear that G.F.L.P. Cantor suffered from manic-depressive illness at a time when nobody knew what this was, and that his polar cycles were aggravated by professional stresses and disappointments, of which Cantor had more than his share. Of course, this makes for less interesting flap copy than Genius Driven Mad By Attempts To Grapple With ∞. The truth, though, is that Cantor's work and its context are so totally interesting and beautiful that there's no need for breathless Prometheusizing of the poor guy's life. The real irony is that the view of ∞ as some forbidden zone or road to insanity -- which view was very old and powerful and haunted math for 2000+ years -- is precisely what Cantor's own work overturned. Saying that ∞ drove Cantor mad is sort of like mourning St. George's loss to the dragon: it's not only wrong but insulting.)"
Please recall or here be informed (as wikipedia just informed me) that St. George actually first wounded then tamed and finally slew the dragon. Which of course underscores DFW's point about Georg F.L.P. Cantor first (figuratively) taming then slaying the ∞.
DFW: requiescat in pace.
March 7, 2011
I bought this book despite the strong criticism it got from mathematicians who found pretty egregious mistakes in some of the math. But I'd never read David Foster Wallace before (aside from some of his journalism) and I wanted to try him out.
I suspect the criticism is largely unwarranted - DFW provides enough forewarning that he has "dumbed down" much of the math in order to bridge the gap to the difficult and abstract math he is describing. Doing so comes with the sacrifice of some accuracy. Richard Feynman once explained that there is no real substitute to getting down and dirty in the math - no amount of summarizing and translation into layman's terms will ever do. So for those who want a complete and 100% correct understanding of these ideas, well, caveat emptor.
Then again, for those who want a complete understanding, none of these kind of books will really do.
DFW keeps a very conversational tone throughout the book - peppering words like "stuff" around concepts like Fourier series and uniform convergence, which helps keep your attention without blunting the fidelity (can you blunt fidelity???) of the explanation. I also really enjoyed his extensive foot-noting, which I understand turns a lot of people off. DFW defended his footnotes in Infinite Jest on Charlie Rose, and I think the defense works very well for this book also:
"There is a way, it seems to me, that reality is fractured right now (at least the reality that I live in) and the difficulty of writing about that reality is that text is very linear, and I am constantly on the lookout for ways to fracture the text that aren't totally disorienting." http://www.charlierose.com/view/inter...
The history of our grasp of infinity from a mathematical perspective isn't linear, so why should the telling of it be? DFW's constant use of IYI (his invented acronym for "if you're interested") allows for the reader to delve deeper into the history, if they're interested.
In this way, DFW writes Everything And More mimics the way that I amble wikipedia. Or perhaps I go too far, the asides are always brief and are seldom interconnected.
I enjoyed it, and I suspect that I'll pick it up again.
I suspect the criticism is largely unwarranted - DFW provides enough forewarning that he has "dumbed down" much of the math in order to bridge the gap to the difficult and abstract math he is describing. Doing so comes with the sacrifice of some accuracy. Richard Feynman once explained that there is no real substitute to getting down and dirty in the math - no amount of summarizing and translation into layman's terms will ever do. So for those who want a complete and 100% correct understanding of these ideas, well, caveat emptor.
Then again, for those who want a complete understanding, none of these kind of books will really do.
DFW keeps a very conversational tone throughout the book - peppering words like "stuff" around concepts like Fourier series and uniform convergence, which helps keep your attention without blunting the fidelity (can you blunt fidelity???) of the explanation. I also really enjoyed his extensive foot-noting, which I understand turns a lot of people off. DFW defended his footnotes in Infinite Jest on Charlie Rose, and I think the defense works very well for this book also:
"There is a way, it seems to me, that reality is fractured right now (at least the reality that I live in) and the difficulty of writing about that reality is that text is very linear, and I am constantly on the lookout for ways to fracture the text that aren't totally disorienting." http://www.charlierose.com/view/inter...
The history of our grasp of infinity from a mathematical perspective isn't linear, so why should the telling of it be? DFW's constant use of IYI (his invented acronym for "if you're interested") allows for the reader to delve deeper into the history, if they're interested.
In this way, DFW writes Everything And More mimics the way that I amble wikipedia. Or perhaps I go too far, the asides are always brief and are seldom interconnected.
I enjoyed it, and I suspect that I'll pick it up again.
September 23, 2018
Partiamo dal principio. La nuova copertina della nuova edizione del libro è favolosa. La traduzione è quella corretta (ma lo era già nelle ristampe del 2011), salvo gli eventuali svarioni di DFW e due o tre tecnicalità di cui non vi accorgerete se non sapete già di che si sta parlando. Ma la cosa più bella del libro è la sua genesi. L'edizione originaria faceva infatti parte di una collana "Great Discoveries", pubblicata da Norton, nata come serie di biografie tecniche di scienziati. DFW non era un matematico, anche se aveva studiato abbastanza per poterne parlare con cognizione di causa, e questo lo si vede dal modo in cui approccia il tema, con una forte componente metafisica che in genere viene trascurata quando si arriva all'Ottocento. Ma era per l'appunto DFW, il che significa che il testo non è per niente lineare e parte per la tangente con le note NCVI ("Nel Caso Vi Interessi", in originale IYI, If You"re Interested) che naturalmente sono imprescindibili, e una serie di rimandi incrociati. Poi lo stile è al solito scanzonato, il che darà al lettore la falsa impressione che tutto sia facile nonostante i mille caveat nel testo. Diciamo che non credo che nessuno imparerà qualcosa sull'infinito leggendolo, ma tanto non era quello il suo scopo.
Il libro non è un classico di DFW ma sicuramente è nel suo stile, con le 408 (!) note a piè di pagina e il misto di linguaggio elevato, quasi triviale e in questo caso pieno di abbreviazioni. Wallace non ci prende sempre del tutto con la teoria matematica - secondo me se ne era dimenticata un po' - ma la lettura è sicuramente indicata per capire cosa noi esseri umani abbiamo fatto per venire più o meno a patti con l'infinito matematico. Aggiungo una chicca per chi è interessato alla storia della matematica: questo è forse l'unico libro che io abbia letto nel quale viene spiegato perché Cantor si sia interessato ai numeri transfiniti, che non erano certo il suo campo di studi. Se poi siete davvero interessati, su DFW e la matematica ho scritto un miniebook liberamente scaricabile...
Il libro non è un classico di DFW ma sicuramente è nel suo stile, con le 408 (!) note a piè di pagina e il misto di linguaggio elevato, quasi triviale e in questo caso pieno di abbreviazioni. Wallace non ci prende sempre del tutto con la teoria matematica - secondo me se ne era dimenticata un po' - ma la lettura è sicuramente indicata per capire cosa noi esseri umani abbiamo fatto per venire più o meno a patti con l'infinito matematico. Aggiungo una chicca per chi è interessato alla storia della matematica: questo è forse l'unico libro che io abbia letto nel quale viene spiegato perché Cantor si sia interessato ai numeri transfiniti, che non erano certo il suo campo di studi. Se poi siete davvero interessati, su DFW e la matematica ho scritto un miniebook liberamente scaricabile...
March 1, 2019
I DID IT. I FINISHED IT. Phew, the last half was a slog.
This was basically a history of math, with the bent of focusing on how/why we got to certain calculations about infinity. I can easily recommend the first 100 pages to everybody who has a passing interest in philosophy or David Foster Wallace. I have read exactly two things by him (now three): Infinite Jest and This is Water, both things I bend over backward to recommend to people. The first 100 pages of Everything and More were like a confluence of everything I love about those two pieces and it was seriously blowing my mind.
Then it got into math, and my eyes started to glaze over. Math has never been a strong suit of mine, and there are some page-long proofs that I straight up skipped. It got harder and harder to understand what was happening because I didn't really understand the formative/underlying principles, so when he started to build on them, I really didn't get it.
Loved the first part though and I can recommend that with abandon.
This was basically a history of math, with the bent of focusing on how/why we got to certain calculations about infinity. I can easily recommend the first 100 pages to everybody who has a passing interest in philosophy or David Foster Wallace. I have read exactly two things by him (now three): Infinite Jest and This is Water, both things I bend over backward to recommend to people. The first 100 pages of Everything and More were like a confluence of everything I love about those two pieces and it was seriously blowing my mind.
Then it got into math, and my eyes started to glaze over. Math has never been a strong suit of mine, and there are some page-long proofs that I straight up skipped. It got harder and harder to understand what was happening because I didn't really understand the formative/underlying principles, so when he started to build on them, I really didn't get it.
Loved the first part though and I can recommend that with abandon.
Read
November 5, 2011I'm on page 109, and I think that's where I'll stop. It's not that I haven't enjoyed it, I have. In fact it's quite soothing to try to see how many layers of abstraction you can hold in your mind at once. However, I only seem to be able to read 2-5 pages at a time before the soothingness of it puts me to sleep, and my mind really is somewhat math resistant. I've gotten to a point in the book where the equations are just meaningless to me. One of my best friends loved this book intensely, and actually kept a note pad at hand so she could work out the math problems for herself, so she could follow more closely. Maybe that would have helped me, but I didn't want to! So, Dan Newton, I'll be handing this off to you!
March 29, 2024
A supposedly hilarious science book which had me pulling my hair!
Why it was picked?
Borrowed from a friend, it satisfied the recurrent urge to read other's books instead of my own!
Being a devout student of Mathematics, a subject which kept (keeps) amusing me at times, found the subheading enticing enough to take a look - A Compact History of ♾
The blurb was simple and light - thought it will be a fun read to relax at the end of a stressful work day.
Here's a pic of the "Necessary Foreword"
So far so good. Except.....
The Foreword never ends! Keeps on going for pages, tending to infinity!!
It was still fine, with dissection of one random mathematical fact after the other, all related to infinity.
Then this came - which became the last straw that broke the camel's back!
I couldn't read further and decided to DNF after a couple of more gibberish pages.
P.S: The ******* foreword doesn't end till the last page of the book, when suddenly acknowledgements are called out!!
Verdict:
This one went a bit too far.... beyond infinity... to make any sense.
January 12, 2021
Blake told me to read this book because it was one of his favorites from last year and funny. I have to disagree, I have been trying to finish it for 9 months and it was quite literally a book about math and the only reason it’s 2 stars is the part about the concept of zero was pretty interesting except I read that part in April and can no longer recall so maybe it wasn’t as interesting as I thought.
June 7, 2025
Read this until it got beyond my comprehension level. men sure love calling things schizoid idk
July 24, 2007
Love him or hate him, DFW is a prodigious talent. Except for the disturbing "Conversations with Hideous Men" I have found his previous material to be so hilariously, intelligently, on-target that I was willing to overlook a multitude of stylistic transgressions (chiefly, the overly cutesy tone, gratuitous flaunting of the author's erudition, the footnote fetish).
So I was reasonably disposed to like this book and was looking forward to reading it. Sadly, it turns out that this was a case where DFW's various idiosyncrasies combine to produce a book which is fundamentally unreadable. Normally, once I start a book, I feel enormous guilt if I don't finish. No guilt here - just exasperation. One can reasonably argue that DFW's enormous talent might justify certain peculiarities of style, but every author needs the discipline of a good editor. W.W. Norton seems to have dispensed with editors altogether, certainly with the sentient kind. A pity, because somebody should have explained to DFW that prefacing any section of text with the title
"Soft-news interpolation, placed here ante rem because this is the last place to do it without disrupting the juggernaut-like momentum of the pre-Cantor mathematical context"
is not just completely unhelpful. It is an irritating distraction, the sorry result of the inability of this talented writer to vanquish the demons which continue to plague his undisciplined style. Unfortunately, this kind of self-indulgent stylistic mannerism recurs with infuriating frequncy.
As a result, this book is a complete train wreck.
So I was reasonably disposed to like this book and was looking forward to reading it. Sadly, it turns out that this was a case where DFW's various idiosyncrasies combine to produce a book which is fundamentally unreadable. Normally, once I start a book, I feel enormous guilt if I don't finish. No guilt here - just exasperation. One can reasonably argue that DFW's enormous talent might justify certain peculiarities of style, but every author needs the discipline of a good editor. W.W. Norton seems to have dispensed with editors altogether, certainly with the sentient kind. A pity, because somebody should have explained to DFW that prefacing any section of text with the title
"Soft-news interpolation, placed here ante rem because this is the last place to do it without disrupting the juggernaut-like momentum of the pre-Cantor mathematical context"
is not just completely unhelpful. It is an irritating distraction, the sorry result of the inability of this talented writer to vanquish the demons which continue to plague his undisciplined style. Unfortunately, this kind of self-indulgent stylistic mannerism recurs with infuriating frequncy.
As a result, this book is a complete train wreck.
October 25, 2010
I’m going to describe the one person I can possibly imagine whom I would recommend this book to. His name is Andy; he was a contemporary of mine during my undergraduate days. Andy was a math major who at one point scheduled (or maybe just invited a bunch of people to?) a talk in a library conference room about how he found math to be beautiful, and in fact in some way divine.
Andy left the study of mathematics after several months teaching remedial algebra in a public school on Chicago’s South Side. I suppose the episode, the most memorable aspect of which revolved around the nickname “Mr. Mayo” (which oddly was bestowed upon him by a student in the hallway who wasn’t in any of his classes), taught him that what he lacked was not mathematical acumen, but rather patience and possibly quite a bit of compassion. As I write this Andy is in seminary.
Which is all to say as I read David Foster Wallace’s Everything And More, I was able to vividly imagine the ideal audience for the work. Conspicuously I was not part of this ideal audience.
The point of this digression, if there is one, is to answer the only “big question” that I really understood as I read Everything, which is: who is this for?
It is a work that, perhaps quixotically and much like Andy in the library conference room, seems to be trying to bring the complex tangle of math’s centuries-long tangle with the concept of infinity out of the dry math classroom environment and into a broader philosophical and historically placed context, for the public good/enjoyment. A celebration of mathematics for all to join. This is a quick and messy definition, that I wouldn’t advise adhering to too closely, lest the book too swiftly be dismissed as an utter and total failure. Because as I tried to convey with my Andy-anecdote, this might be a bit of niche thing. I imagine most people’s interest in and knowledge of calculus won’t be adequate to make this book reach the status of “page-turner.”
Wallace begins in ancient Greece, where the questions are raised, and in his narrative, placed aside until the Renaissance, and not really grasped for another 300 years give or take. The hero of the story is Georg Cantor who lurks in the background until the late 19th century, who eventually comes along, resolves things in a way that confused me (though by this point the dilemma posed by infinity was too obscured by hundreds of years of dramatic advances), and ultimately led to more questions for mathematicians. By all accounts Cantor is a big deal and I believe it, but I couldn’t begin to explain why, or even quote why, due to not really knowing where to find Greek letters on my keyboard.
Is this due to my math background, or is this due to the author? Did this inability to really grasp the material impair my ability to enjoy myself?
Well, as must be obvious at this point, the attraction leading me to this book was to the author, not to the subject, per say. DFW casts a long shadow in the contemporary literature world. His massive masterwork, Infinite Jest, is regarded as one of the most important pieces of (at least) American literature of the last twenty years. He committed suicide in 2008, and his death was probably the first celebrity passing that actually affected me.
His work revolves around a profound unironic enthusiasm, and often characters digress and discuss philosophy for pages. Not that he is ever too dense, often times those very philosophical discussions are followed by a gag, or punctuated with a joke.
Those looking for the same in Everything and More, should be warned that while the book is unmistakably DFW, it is on the subject of infinity viewed through the lens of math. Pure and simple. Conspicuously, while DFW got a degree in modal logic (!), “mathematician” is not on his resume.
So clearly affection for the author may be causing me to spare him the rod of having written a book on a subject that he may not grasp-- at least not well enough to find the terms to explain it to a layman. A colleague of mine found out what I was reading and said that a few legitimate mathematicians had come forward to critique DFW’s work, to point out holes in his retelling of the grand old tale.
Indeed DFW admits that the legitimate mathematician is going to find his explanations either too swift, or too fraught with his idiomatic prose. So he’s painted himself into a corner. “How can the discussion be pitched so that it’s accessible to the neophyte without being dull or annoying to someone who has had a lot of college math?” he asks at the end of the foreword.
Apparently the answer wasn’t giving a lot of biographical or historical context to the mathematicians in question, though each time he did, it was riveting.
Neal Stephenson wrote a foreword for my copy of the book (which supplemented the DFW foreword nicely), and he noted that one could interpret the entire effort of Everything as sort of ostentatious. But Stephenson asked for our charity in reading “one of the other smart kids trying to explain some cool stuff.”
So the book is probably bad journalism. It pains me to say it, but if journalism writes with an audience in mind, and if the author, the author of the foreword, and the reviewer all sort of wonder who could like this book, the odds are it needed a good hard look at the concept (of the book) before proceeding, or at the very least a good stiff edit.
Yet herein lies the lesson. At no point did it feel like DFW was taking a page or section off. His profound love of the subject is catching, even if his comprehension isn’t. It would be easy to dismiss this whole effort as too insular, but DFW’s tone is always to catching, too inviting. I read 300 pages and was enthusiastic whenever I thought the narrator was.
As journalism starts to structure itself to more and more specific audiences, it will become easier and easier to say, “That really doesn’t apply to me,” and go about your day. Already RSS feeds, customizable news aggregators like Google News, and other technologies are making it easier to block out that which we know doesn’t interest us (like 300 page books about not only calculus, but the history of calculus), in favor of that which does.
But the journalist and the science writers who can make you believe you do care, even when you can’t really grasp the point because they show such utter care and diligence are the ones who can buck this trend. Journalism that has the power to wake you up to the complexity of the world, to shake your suppositions, is worth the effort to make, and worth the effort to read.
Andy left the study of mathematics after several months teaching remedial algebra in a public school on Chicago’s South Side. I suppose the episode, the most memorable aspect of which revolved around the nickname “Mr. Mayo” (which oddly was bestowed upon him by a student in the hallway who wasn’t in any of his classes), taught him that what he lacked was not mathematical acumen, but rather patience and possibly quite a bit of compassion. As I write this Andy is in seminary.
Which is all to say as I read David Foster Wallace’s Everything And More, I was able to vividly imagine the ideal audience for the work. Conspicuously I was not part of this ideal audience.
The point of this digression, if there is one, is to answer the only “big question” that I really understood as I read Everything, which is: who is this for?
It is a work that, perhaps quixotically and much like Andy in the library conference room, seems to be trying to bring the complex tangle of math’s centuries-long tangle with the concept of infinity out of the dry math classroom environment and into a broader philosophical and historically placed context, for the public good/enjoyment. A celebration of mathematics for all to join. This is a quick and messy definition, that I wouldn’t advise adhering to too closely, lest the book too swiftly be dismissed as an utter and total failure. Because as I tried to convey with my Andy-anecdote, this might be a bit of niche thing. I imagine most people’s interest in and knowledge of calculus won’t be adequate to make this book reach the status of “page-turner.”
Wallace begins in ancient Greece, where the questions are raised, and in his narrative, placed aside until the Renaissance, and not really grasped for another 300 years give or take. The hero of the story is Georg Cantor who lurks in the background until the late 19th century, who eventually comes along, resolves things in a way that confused me (though by this point the dilemma posed by infinity was too obscured by hundreds of years of dramatic advances), and ultimately led to more questions for mathematicians. By all accounts Cantor is a big deal and I believe it, but I couldn’t begin to explain why, or even quote why, due to not really knowing where to find Greek letters on my keyboard.
Is this due to my math background, or is this due to the author? Did this inability to really grasp the material impair my ability to enjoy myself?
Well, as must be obvious at this point, the attraction leading me to this book was to the author, not to the subject, per say. DFW casts a long shadow in the contemporary literature world. His massive masterwork, Infinite Jest, is regarded as one of the most important pieces of (at least) American literature of the last twenty years. He committed suicide in 2008, and his death was probably the first celebrity passing that actually affected me.
His work revolves around a profound unironic enthusiasm, and often characters digress and discuss philosophy for pages. Not that he is ever too dense, often times those very philosophical discussions are followed by a gag, or punctuated with a joke.
Those looking for the same in Everything and More, should be warned that while the book is unmistakably DFW, it is on the subject of infinity viewed through the lens of math. Pure and simple. Conspicuously, while DFW got a degree in modal logic (!), “mathematician” is not on his resume.
So clearly affection for the author may be causing me to spare him the rod of having written a book on a subject that he may not grasp-- at least not well enough to find the terms to explain it to a layman. A colleague of mine found out what I was reading and said that a few legitimate mathematicians had come forward to critique DFW’s work, to point out holes in his retelling of the grand old tale.
Indeed DFW admits that the legitimate mathematician is going to find his explanations either too swift, or too fraught with his idiomatic prose. So he’s painted himself into a corner. “How can the discussion be pitched so that it’s accessible to the neophyte without being dull or annoying to someone who has had a lot of college math?” he asks at the end of the foreword.
Apparently the answer wasn’t giving a lot of biographical or historical context to the mathematicians in question, though each time he did, it was riveting.
Neal Stephenson wrote a foreword for my copy of the book (which supplemented the DFW foreword nicely), and he noted that one could interpret the entire effort of Everything as sort of ostentatious. But Stephenson asked for our charity in reading “one of the other smart kids trying to explain some cool stuff.”
So the book is probably bad journalism. It pains me to say it, but if journalism writes with an audience in mind, and if the author, the author of the foreword, and the reviewer all sort of wonder who could like this book, the odds are it needed a good hard look at the concept (of the book) before proceeding, or at the very least a good stiff edit.
Yet herein lies the lesson. At no point did it feel like DFW was taking a page or section off. His profound love of the subject is catching, even if his comprehension isn’t. It would be easy to dismiss this whole effort as too insular, but DFW’s tone is always to catching, too inviting. I read 300 pages and was enthusiastic whenever I thought the narrator was.
As journalism starts to structure itself to more and more specific audiences, it will become easier and easier to say, “That really doesn’t apply to me,” and go about your day. Already RSS feeds, customizable news aggregators like Google News, and other technologies are making it easier to block out that which we know doesn’t interest us (like 300 page books about not only calculus, but the history of calculus), in favor of that which does.
But the journalist and the science writers who can make you believe you do care, even when you can’t really grasp the point because they show such utter care and diligence are the ones who can buck this trend. Journalism that has the power to wake you up to the complexity of the world, to shake your suppositions, is worth the effort to make, and worth the effort to read.
February 18, 2019
Very fun and occasionally existentially terrifying. I appreciate Dave's confidence in my mathematical/logical acuity but I would not have been insulted if he had dumbed it down just a little bit more.
March 6, 2023
Good math stuff
January 5, 2025
As a mathematician, fan of DFW, and a pursuer of books that tell the history of a broad subject from the viewpoint of a single problem, this was the ideal book for me.
It is well-written, dripping with enthusiasm and respect for the subject, and offers insightful interpretations on the history of mathematics and the role of abstraction.
This said, the book is lacking in mathematical rigour. There are many errors, clear confusions, and unnecessary complications. This only serves to obscure the points being made, and will confuse newcomers more than the topic requires.
The book feels like talking to someone at a party who did mathematics at University, clearly liked it, but has taken on a corporate job and only has a vague recollection of what was going on. The shadows of the ideas are all there, which I will admit is rather entertaining, but for the type of book this pretends to be it does not quite cut it. Almost every pop maths books I've read offers a proof of the irrationality of sqrt(2). The present book is the only case I know of to offer an incorrect proof of this fact. Later, when explaining the continuum hypothesis, he accidentally assumes the continuum hypothesis. As is often the case with intellectual arrogance, the author's is incorrectly placed, and he does not know the things he pretends to know.
The philosophical points are good and many will be new to working mathematicians. Reading about the history of mathematics often just feels like hearing gossip, people telling amusing stories of weird guys, but the historical notes here are serious and the author gives the subject the required amount of respect. We hear about the flawed trope of the mad mathematician, and how in DFW's mind it's not logic that drives people mad but abstraction itself. The points about the steps in abstraction needed to see infinity are illuminating. The author paraphrases Hilbert and says: "Take the single most ubiquitous and oppressive feature of the concrete world—namely that everything ends, is limited, passes away—and then conceive, abstractly, of something without this feature." The book would have done better by fleshing those ideas out a bit more.
The work feels rushed and should have been read through before publishing. I am saddened as this was a promising book which did not deliver.
It is well-written, dripping with enthusiasm and respect for the subject, and offers insightful interpretations on the history of mathematics and the role of abstraction.
This said, the book is lacking in mathematical rigour. There are many errors, clear confusions, and unnecessary complications. This only serves to obscure the points being made, and will confuse newcomers more than the topic requires.
The book feels like talking to someone at a party who did mathematics at University, clearly liked it, but has taken on a corporate job and only has a vague recollection of what was going on. The shadows of the ideas are all there, which I will admit is rather entertaining, but for the type of book this pretends to be it does not quite cut it. Almost every pop maths books I've read offers a proof of the irrationality of sqrt(2). The present book is the only case I know of to offer an incorrect proof of this fact. Later, when explaining the continuum hypothesis, he accidentally assumes the continuum hypothesis. As is often the case with intellectual arrogance, the author's is incorrectly placed, and he does not know the things he pretends to know.
The philosophical points are good and many will be new to working mathematicians. Reading about the history of mathematics often just feels like hearing gossip, people telling amusing stories of weird guys, but the historical notes here are serious and the author gives the subject the required amount of respect. We hear about the flawed trope of the mad mathematician, and how in DFW's mind it's not logic that drives people mad but abstraction itself. The points about the steps in abstraction needed to see infinity are illuminating. The author paraphrases Hilbert and says: "Take the single most ubiquitous and oppressive feature of the concrete world—namely that everything ends, is limited, passes away—and then conceive, abstractly, of something without this feature." The book would have done better by fleshing those ideas out a bit more.
The work feels rushed and should have been read through before publishing. I am saddened as this was a promising book which did not deliver.
March 2, 2017
Fantastic! And I'm not even a huge DFW fan. But man do I like this non-fiction.
To all the naysayers who say this is full of mistakes. Yes. Yes, of course. He's simplifying things in order to get the message across. But DFW is like an obsessive-compulsive, who is both trying to simplify but isn't happy with hand-waving... so you get a complex mess. I love it.
I will say that his description of Dedekind's schnittzing to prove irrationals has me completely bamboozled. But at least after reading this I am VERY interested in it.
I've read about Cantor's diagonalizing before, and once again was delighted to learn about it. It is seriously delightful.
I think the story that I loved the most, running through this, was the intuitionists vs the platonists (vs the formalists?) and I have no idea where I stand on this issue. Intuitionism is seriously lovely, and when you consider how all math is essentially done on computers (discrete)... then what does it matter if we don't allow transfinite math into existence? Anyways, it makes me really excited to learn more about discrete math and computability etc. etc.
I now want to read What is Mathematic, Really? by some guy... I can't remember.
And Ian Hacking's new book on math.
Which is to say: DFW's book on infinite has given me a boner for math. A boner I have not had for a long, long time. I missed this boner.
:)
To all the naysayers who say this is full of mistakes. Yes. Yes, of course. He's simplifying things in order to get the message across. But DFW is like an obsessive-compulsive, who is both trying to simplify but isn't happy with hand-waving... so you get a complex mess. I love it.
I will say that his description of Dedekind's schnittzing to prove irrationals has me completely bamboozled. But at least after reading this I am VERY interested in it.
I've read about Cantor's diagonalizing before, and once again was delighted to learn about it. It is seriously delightful.
I think the story that I loved the most, running through this, was the intuitionists vs the platonists (vs the formalists?) and I have no idea where I stand on this issue. Intuitionism is seriously lovely, and when you consider how all math is essentially done on computers (discrete)... then what does it matter if we don't allow transfinite math into existence? Anyways, it makes me really excited to learn more about discrete math and computability etc. etc.
I now want to read What is Mathematic, Really? by some guy... I can't remember.
And Ian Hacking's new book on math.
Which is to say: DFW's book on infinite has given me a boner for math. A boner I have not had for a long, long time. I missed this boner.
:)
March 17, 2018
This wasn't written by a mathematician; math specialists seem to notice its flaws. It was written by a literary golden boy; literati seem to like its style. Some people seem to believe these aspects roughly balance out, resulting in a somewhat pleasing and somewhat unsatisfying read.
I hoped that Wallace's treatment would be at least as much about the philosophical concept as about the mathematical description of infinity. Since it wasn't and since almost all of the math was too hard for me, i couldn't really dig it. The dude seems to know way more than the average bear about this topic ... or maybe the 1-star reviewers are on point ... i really have no way to assess. Good thing you're not looking to me for the answers.
All i can say is that if calculus was a stopping point for your mathematical education then this book's math might be too much for you also.
I hoped that Wallace's treatment would be at least as much about the philosophical concept as about the mathematical description of infinity. Since it wasn't and since almost all of the math was too hard for me, i couldn't really dig it. The dude seems to know way more than the average bear about this topic ... or maybe the 1-star reviewers are on point ... i really have no way to assess. Good thing you're not looking to me for the answers.
All i can say is that if calculus was a stopping point for your mathematical education then this book's math might be too much for you also.
Want to read
January 15, 2011I think I'm going to have to return this to the library and try to read it at another time. I can't read any of Wallace's work right now, it makes me really sad. Because when I've read it in the past I've always been like: THIS IS SO BRILLIANT and I think of how amazing it is that someone so genius is alive. But.. he's not. Anymore. I realize whining about his death is not a review. This is a review placeholder.
September 23, 2022
This math book was as compelling as only David Foster Wallace could make it, but halfway through it lifted off well above my understanding and from there on out it was all just kazoo music.
August 2, 2025
Ah, thank God, I finished reading the book! It is sort of unputdownable but, at the same time, deserves to be dashed against the wall! The only reasons I didn’t toss it is because of my own respect for books and my sympathies towards David. Let me explain.
The glaring problem which we encounter as soon as we open the book is that there is no Table of Contents or even Chapters! There is not even the index at the back of the book! Sure, there are seven “chapters” but they and their sections (which Wallace refers to as §) flow continuously like one single thread from page one to last page, with lots of footnotes and embedded glossaries and interpolations. It is all a new and unique style of writing—appreciable—but the nub is that it is very difficult to find stuff because—remember—there is no index or even chapter names! The book is filled with good information about the history, and even the math of infinity is presented very well, but when I needed to quickly go back and reread certain items, it was difficult as hell! Although, I must admit, however difficult it was, I could find what I wanted to reread by flipping back and searching and, in fact, I felt I gained something by working hard to find what I was looking for instead of they being given—in the index/Table of Content—on a platter. Maybe that was David’s real intention but, believe me, it was painfully frustrating! The other side-effect of this one continuous drool of a writing—despite quote-unquote chapter divisions—is that the various topics that are being delivered don’t seem to make a coherent whole; it becomes difficult to see the forest for the trees by connecting everything that’s being said. The solution—for me—was to take breathers in between, lift my head, cuss, and think about what was being said, and how it connects together, all within the context in which it is being said. In other words, the writing style coerced me to think hard, work hard, and thereby decipher the contents which, again, may have been David’s intention, but it could also be my own prior knowledge of infinity and its history. I sincerely believe that some knowledge of math is required to enjoy this book.
The book starts about 2400 years ago, with Zeno’s paradox, talks about Greek’s mistrust and difficulty to accept infinity even in the face of knowledge of irrational numbers staring at them, Aristotle’s misleading potential-infinity vs. actual-infinity (and how the Church grabbed his theories of infinity, and of course his geocentric world too, in order to support God centric universe), introduces number line of integers, how rational number line is not continuous and has holes, and how transcendental irrational numbers fill in these holes, converges quickly on to Dedekind’s schnitt—or cut—to demonstrate irrationals, Cantor’s Set Theory, diagonalization method, his Dimension Theory, levels of infinites, power sets, cardinality and ordinality, touches Gödel’s incompleteness theorems, opening up to outstanding paradoxes in the Axiomatic Set Theory’s dealings with infinities. In other words, we come full circle: over the centuries, one set of mathematical paradoxes gets resolved only to open up new ones and mathematical research keeps chugging along! The punch of real Set Theory of infinities, you’ll see, is mostly in the last third of the book—that is, if you prevail! Difficult, but doable!
In the first few pages, David mentions that many people who worked on infinity turned mad or became depressed and were confined to mental hospitals, or were “on the spectrum,” many died very young, and some committed suicide. Sadly, David himself suffered from depression and committed suicide. I have a sneaking suspicion that not only he knew it was coming but also this book was meant to drop hints, or maybe a call for help. His thinking—we can easily see in his writing style—is recursively looping around, and sometimes jumps around from one thing to another but, in the end, finds its way back to whatever he was trying to say in the first place, something like how a musician improvises by going off and off and eventually unwinds back to fit into the original piece, or how a recursive computer program unwinds from the LIFO stack. Yes, Hofstadter’s Gödel, Escher, Bach all over again! He has tried to fit in a lot of information in this book (with few mistakes which most laypeople will not notice) and the historical perspective and the math—more math than history, which was good for me—are presented, all in one place. However, personally, since I am more interested in the math of infinity, I have to jump ship to explore the many "Roads to Infinity," (by John Stillwell). I am glad, though, for completing the book and would probably use it as a quick reference for historical perspective.
The glaring problem which we encounter as soon as we open the book is that there is no Table of Contents or even Chapters! There is not even the index at the back of the book! Sure, there are seven “chapters” but they and their sections (which Wallace refers to as §) flow continuously like one single thread from page one to last page, with lots of footnotes and embedded glossaries and interpolations. It is all a new and unique style of writing—appreciable—but the nub is that it is very difficult to find stuff because—remember—there is no index or even chapter names! The book is filled with good information about the history, and even the math of infinity is presented very well, but when I needed to quickly go back and reread certain items, it was difficult as hell! Although, I must admit, however difficult it was, I could find what I wanted to reread by flipping back and searching and, in fact, I felt I gained something by working hard to find what I was looking for instead of they being given—in the index/Table of Content—on a platter. Maybe that was David’s real intention but, believe me, it was painfully frustrating! The other side-effect of this one continuous drool of a writing—despite quote-unquote chapter divisions—is that the various topics that are being delivered don’t seem to make a coherent whole; it becomes difficult to see the forest for the trees by connecting everything that’s being said. The solution—for me—was to take breathers in between, lift my head, cuss, and think about what was being said, and how it connects together, all within the context in which it is being said. In other words, the writing style coerced me to think hard, work hard, and thereby decipher the contents which, again, may have been David’s intention, but it could also be my own prior knowledge of infinity and its history. I sincerely believe that some knowledge of math is required to enjoy this book.
The book starts about 2400 years ago, with Zeno’s paradox, talks about Greek’s mistrust and difficulty to accept infinity even in the face of knowledge of irrational numbers staring at them, Aristotle’s misleading potential-infinity vs. actual-infinity (and how the Church grabbed his theories of infinity, and of course his geocentric world too, in order to support God centric universe), introduces number line of integers, how rational number line is not continuous and has holes, and how transcendental irrational numbers fill in these holes, converges quickly on to Dedekind’s schnitt—or cut—to demonstrate irrationals, Cantor’s Set Theory, diagonalization method, his Dimension Theory, levels of infinites, power sets, cardinality and ordinality, touches Gödel’s incompleteness theorems, opening up to outstanding paradoxes in the Axiomatic Set Theory’s dealings with infinities. In other words, we come full circle: over the centuries, one set of mathematical paradoxes gets resolved only to open up new ones and mathematical research keeps chugging along! The punch of real Set Theory of infinities, you’ll see, is mostly in the last third of the book—that is, if you prevail! Difficult, but doable!
In the first few pages, David mentions that many people who worked on infinity turned mad or became depressed and were confined to mental hospitals, or were “on the spectrum,” many died very young, and some committed suicide. Sadly, David himself suffered from depression and committed suicide. I have a sneaking suspicion that not only he knew it was coming but also this book was meant to drop hints, or maybe a call for help. His thinking—we can easily see in his writing style—is recursively looping around, and sometimes jumps around from one thing to another but, in the end, finds its way back to whatever he was trying to say in the first place, something like how a musician improvises by going off and off and eventually unwinds back to fit into the original piece, or how a recursive computer program unwinds from the LIFO stack. Yes, Hofstadter’s Gödel, Escher, Bach all over again! He has tried to fit in a lot of information in this book (with few mistakes which most laypeople will not notice) and the historical perspective and the math—more math than history, which was good for me—are presented, all in one place. However, personally, since I am more interested in the math of infinity, I have to jump ship to explore the many "Roads to Infinity," (by John Stillwell). I am glad, though, for completing the book and would probably use it as a quick reference for historical perspective.
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