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ravuri's review against another edition
4.0
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
wanderinggnome's review against another edition
3.0
If I read this book five times I would still probably only understand 60% of the math involved. I imagined it would be a bit more narrative and bit less proof but it wasn't and I really struggled through the back half.
paulogonzalez's review against another edition
1.0
Excessive use of abbrevations, a lot of footnotes, some mathematical mistakes, lack of clarity, poorly organized... Wallace is a good writer, and it shows that in some passages, it's undeniable. But the peculiarities in his style simply does not suit maths.
heidibrianne's review against another edition
2.0
I seriously over-estimated the ability of my experience in high school calculus and better-than-average knowledge of math to carry me through the dense theoretical jungle of this book. I barely made it halfway through this slender volume. It is such a disappointment when a book simply goes over my head.
blackoxford's review against another edition
5.0
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.
fi_chince's review against another edition
4.0
Frankly, I'm surprised I finished this book. I sort of saw it as part of my current project to work my way up to reading [b:Infinite Jest|6759|Infinite Jest|David Foster Wallace|http://photo.goodreads.com/books/1165604485s/6759.jpg|3271542]. (Which is currently sitting on my dining room table. I'm afraid to shelve it lest the lack of a visual reminder will make me forget that I have it. It is also my hope that visitors will be impressed by the sight of the thing.)
Anyway, I figured I would read a bit of Everything and More, see what it's like, and skim through the rest when the math got too hairy. Now, I'm not claiming that I understood all of the math and that it didn't get hairy, or that I never skimmed through any of it at all, but I did follow the general gist for most of the way, or at least enough that I never felt like slamming the book closed and hurling it across the room.
This I credit to DFW's writing style. I don't think I've ever read anything where the text was so aware of its being read. There are constantly little asides and apologies (many in Wallace's trademark footnotes) about how difficult a particular section is, how you might want to re-read this or that paragraph, how it's all going to be OK in the end. These constant conversational reassurances do a lot to encourage the reader (me, at least) to keep going, despite the difficult math.
And there is a suspense to it all too. Cantor is mentioned near the beginning and is set up to be the Hero of the Story, the one whose theories are the ultimate culmination of everything I'm reading, and I genuinely felt the urge to know what Cantor did, like wanting to find out who the killer is in a mystery novel. Wallace does a good job of reminding us how each theory through history will be relevant to Cantor's transfinite numbers, while making each theory interesting to learn about on its own. And while the actual proofs and formulae are explained well, I found the most enjoyment in the connective tissue about the like societal and cultural and historical contexts around each discovery, e.g. the geometric rigidity of the Greeks, the need to develop and accept infinitesimals in physics and science during the time of Newton and Leibniz, &c. I actually wish he had focussed on those contexts more, and I think he probably could have written a thousand-page book (it amazes me how much research must have gone into this as is). I would probably have still read it all.
Anyway, I figured I would read a bit of Everything and More, see what it's like, and skim through the rest when the math got too hairy. Now, I'm not claiming that I understood all of the math and that it didn't get hairy, or that I never skimmed through any of it at all, but I did follow the general gist for most of the way, or at least enough that I never felt like slamming the book closed and hurling it across the room.
This I credit to DFW's writing style. I don't think I've ever read anything where the text was so aware of its being read. There are constantly little asides and apologies (many in Wallace's trademark footnotes) about how difficult a particular section is, how you might want to re-read this or that paragraph, how it's all going to be OK in the end. These constant conversational reassurances do a lot to encourage the reader (me, at least) to keep going, despite the difficult math.
And there is a suspense to it all too. Cantor is mentioned near the beginning and is set up to be the Hero of the Story, the one whose theories are the ultimate culmination of everything I'm reading, and I genuinely felt the urge to know what Cantor did, like wanting to find out who the killer is in a mystery novel. Wallace does a good job of reminding us how each theory through history will be relevant to Cantor's transfinite numbers, while making each theory interesting to learn about on its own. And while the actual proofs and formulae are explained well, I found the most enjoyment in the connective tissue about the like societal and cultural and historical contexts around each discovery, e.g. the geometric rigidity of the Greeks, the need to develop and accept infinitesimals in physics and science during the time of Newton and Leibniz, &c. I actually wish he had focussed on those contexts more, and I think he probably could have written a thousand-page book (it amazes me how much research must have gone into this as is). I would probably have still read it all.
kalivha's review against another edition
3.0
I was going to give this four stars - it is quite hard to digest for what it is, but overall good. Then the last 30 pages happened and contained a fair few non-justified inaccuracies and approximations that just didn't sit right with me on a different level from everything else DFW had glossed over for the sake of legibility etc. - so three stars it has to be.
peebee's review against another edition
1.0
I don't know why, maybe it's his three word name and the fact everyone seemed broken up when he died, but I figured he was some eminent figure of literature. I didn't realize he was some snarky NPR dick in the McSweeny's mold. This guy is WAY out of his league in writing a book on high level math, and from the ugly mess of footnotes, cutesy abbreviations that waste more time in the looking up than they save in the reading the non sequitur jokey jokes about people's names and how hard math is, and disjointed style of cramming math exposition into random sections, he maybe shouldn't write books period. This is the first book I've put up I haven't finished, but I am done with it, completely.
susanhert's review against another edition
3.0
It's totally amazing to me that so much about the mathematical concept of infinity was so much in flux as recently as 1900. There are still some conundrums that infinity gives birth to, of course, but that it would take so long for it to be generally accepted that there are such things as irrational numbers is amazing.
The style of this book, with its myriad of footnotes and interpolations, was both interesting and somewhat distracting. I wonder what the story would have been like if I had not read all the IYI's.
The style of this book, with its myriad of footnotes and interpolations, was both interesting and somewhat distracting. I wonder what the story would have been like if I had not read all the IYI's.
hal_incandenza's review against another edition
4.0
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.