| Title | : | Surreal Numbers |
| Author | : | |
| Rating | : | |
| ISBN | : | 0201038129 |
| ISBN-10 | : | 9780201038125 |
| Format Type | : | Paperback |
| Number of Pages | : | 128 |
| Publication | : | First published January 11, 1974 |
Surreal Numbers Reviews
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Also Sprake JHWH
All words in a language, all symbols, all operations that can be performed in the language are contained only within the language itself. That is to say, languages are self-defined; their elements are constituted by other elements of the language not by not-words outside the language. Language is therefore a circular affair. Or, more optimistically, language is helical. It refers to itself endlessly and gets more expressive as it does so as it builds upon itself.
Expressive of what? Of itself of course. This is what makes language so astounding. It makes the most amazing things out of... well out of nothing but sounds and marks. And it envelopes those who use it in a potentially infinite artificial universe. This infinite character of its creation is temporal as well as spatial. No one is sure when language started. Although we know it must have had a beginning in the development of life we call evolution, it nevertheless appears that it has always been there, waiting to be discovered. Before its discovery, nothing is thinkable. And it could exist forever, like viruses continuing in an inanimate state until encountering some organism which has been prepared for it. If language is ultimately annihilated, we will never know about it. There is literally no future, as there is no past or present, without it.
This appearance of eternal existence promotes the idea of language as divine. Indeed, for many it seems to logically necessitate the idea of god itself. And language can make a credible claim to creating the world. But it does so, as in other aspects of the evolutionary process, out of the chaotic material which is already available. The point at which words emerged from the chaos of not-words is indeterminate. But the Book of Genesis is probably as good an account of the event as any. Knuth thinks so anyway.
By analogy with the emergence of DNA, perhaps language appeared as a consequence of a chemical combination as a simple instruction set from which a linguistic edifice could be built. And perhaps it is within this instruction set that the first ‘word’ was chemically contained. The instruction set operated on this strange new chemical object and self-referentially produced other words that congealed around the first, analogously to a planetary system, into an extensive language.
All this is only a theory of course. But its a theory that fits very well with Knuth’s tale of the development of surreal numbers. These numbers are not surreal because of their properties, which are more or less the same as other numbers, but because of their origins. They are produced (or explained, the distinction is largely irrelevant) by a self-referential process in which all numbers are defined in terms of not axioms, or elemental definitions of ‘units’ but in terms of each other.
Knuth lays out two rules in his fictional account of surreal numbers:“This shall be the first rule: Every number corresponds to two sets of previously created numbers, such that no member of the left set is greater than or equal to any member of the right set. And the second rule shall be this: One number is less than or equal to another number if and only if no member of the first number's left set is greater than or equal to the second number, and no member of the second number's right set is less than or equal to the first number."
Thus surreal numbers simulate the development of all language. They are the result of combinations of yet more primal numbers, which are in turn generated by yet more primal numbers, which may well include the numbers one started with. Circularity for sure but with that distinctive helical twist of increasing complexity characteristic of all languages. The key of course is in the instruction set, the algorithm, which ‘bootstraps’ its own development by ‘positing’ that which it then develops. And what it posits is, quite literally ‘nothing,’ the null set {0,0}.
Knuth’s two rules are exceptionally clever, but they may require a particular mathematical set of mind to grasp fully. The novel is clearly meant to help in easing the reader into that state. Even then the basic self-referentiality of Knuth’s rules may seem disconcerting. Because they are exactly that. Recursive logic shakes the foundations of all foundational thinking. It tears the fundament out of fundamentals.
This might be easier for most folk to appreciate in words rather than numbers. Take the word ‘fact.’ In simple terms ‘a fact is a thing known or proved to be true.’ If one were to follow each of these definitional terms back through the Oxford English Dictionary, and make substitutions along the way, the definition of ‘fact’ that would emerge is as something along the lines of: ‘a fact is that which is not contradicted by any other fact.’
This derived definition is no more or less correct than the initial one. It merely demonstrates the necessary circularity of language, a circularity which the derivation of surreal numbers in the language of mathematics makes obvious as a sufficient condition for all languages. It also indicates why an appeal to facts in an argument always begs the question of what constitutes such a thing.
Knuth’s fictional explication of surreal numbers is based on the original mathematics of John Horton Conway (JHWH, or Jehovah of the novel) who also wrote a dozen or so serious books, as well as at least that many very serious games including the cellular automaton called the Game of Life. -
Good introduction to the Surreals and overall a good description of the way new mathematical ideas are developed. I like that Knuth decided to stick with Conway's construction of the surreals as cuts between sets, as opposed to other approaches like Gonshor's ordinal-length sequences of +'s and -'s, since this approach makes it easier to build intuition about many important ideas used in later proofs.
The characterization is weak, but that hardly matters. It's a nice attempt to both humanize math and deliver new ideas organically. And I can forgive Knuth a lot of subpar characterization when he does things like make the Surreals appear on a "Conway stone", equal parts Rosetta Stone and Ten Commandments, and the work of the omnipotent deity J. H. W. H. Conway, who created the universe in aleph days, then kept right on creating. -
Okay, if you don't rack your brain too much with the dialogues of mathematical explanations & derivations, then I guess it's for fun !
The book was about how students should be taught to learn the interesting aspects of the problem rather than to solve it first, how they could train their mind to learn the creative aspects of proofs rather than the proof itself given in textbook.
But after being in a system of 17 year's of examsmanship, I would say I enjoyed his concrete mathematics book better than this one -_- -
mathematical & mind challenging at the first place which makes it even more exciting and interesting .
actually , I haven't put so much concentration and thought in a book or a novel since Islam between the east and the west which I have not finished so far . but anyway that's not something weird knowing that Donald Knuth of the art of programming is the same gut who wrote it ^_^ it actually adds up :D -
È stato molto bello vedere tradotto in italiano questo libretto, che ha inaugurato la collana Scienza FA di Franco Angeli. Io avevo letto l'edizione originale una ventina d'anni fa e l'ho sempre ricordata con piacere. Il libro tecnicamente è una "novelette", ma non si può leggere come un romanzo, perché di matematica ce n'è parecchia. Ma d'altra parte non è nemmeno un manuale di matematica, anche se alla fine surrettiziamente Knuth ha aggiunto qualche esercizio. In realtà è un modo per vedere come si fa davvero matematica. Si parte da una serie di regole - quelle di JHWH Conway sono solo due, proprio un insieme minimale - e si verifica da un lato che siano coerenti e dall'altro che creino una teoria interessante. Nel testo si vedono delle false strade, dei ripensamenti e a un certo punto ci si accorge di un errore fondamentale dovuto al pensare troppo per analogia: tutte cose che capitano, ma che i libri di testo nascondono accuratamente. Non è probabilmente un libro per tutti, ma gli appassionati di matematica non dovrebbero farne a meno. La traduzione di Francesco Oliveri è molto ben fatta dal punto di vista matematico, ma a mio parere avrebbe dovuto osare un po' di più nelle parti più discorsive, che sono rimaste un po' piatte. Sicuramente meglio così che una traduzione fiammeggiante ma incomprensibile, intendiamoci!
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Aww, this is so cute. Knuth is selling us number theory within a romantic plot. This is about a couple having fun in a beach discovering the fundamental laws of numbers. They even discuss why mathematics was profoundly boring at school but so exciting now; specially figuring things out by themselves. Knuth is therefore writing to young mathematicians igniting their curiosity by showing off the pleasures/frustrations of independent work. Be prepared to fire up a great deal of your 100 billion neurons while handling infinities.
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Chyba jednak straszny ze mnie konserwatysta... O ile kupiłbym taką anegdotyczną, opowiastkową formę, jaką proponuje Knuth, gdyby trzeba było wprowadzić elementy arytmetyki czy teorii mnogości w szkole, o tyle takie ujęcie bardziej złożonych zagadnień kompletnie do mnie nie przemawia. Jakkolwiek to zabrzmi: chcę poznać liczby nadrzeczywiste z nudnego i poważnego podręcznika!
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Może do tego wrócę jak zobaczę te dzikie liczby na studiach🤠
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So glad I came across this. Berlekamp & Conway's series on Winning Ways for your Mathematical Plays has been staring at me for years. But Conway works heavily with the concept of surreal numbers there, with hardly any explanation. So, maybe this will be the breakthrough.
Knuth suggests this book for an undergraduate seminar, but I found it above that level. More than once I had to put the book aside, once for over a month, before taking it up again. I worked through every point quite thoroughly, but I'm still at a loss on a couple of points.
I loved the way DK highlighted the exploratory process, rather than just presenting results. There really should be more of this. The tablet written by JHWH Conway was certainly a clever twist. As far as the love story goes, the less said, the better. Overall, a very satisfying book that had me screaming in frustration about as often as howling with delight. There were moments when I wanted to stop random strangers on the street to tell them about the surreal numbers.
If you've seen John Conway's charming YouTube videos, you know he is very proud of the Surreals, maybe more so than anything else he had done. And, I think, he is right to in his opinion. -
I worked along with it in the beginning and then kinda stopped, but working through the math is obviously the best way to read this book - it's certainly not for the dumb romance!
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This is one of those books that will blow your mind. Although it is a work of fiction it served as the introduction to the world of John Conway's surreal numbers. This is a small book, barely a novelette, I read it in less than an hour.
The entire book is presented as a dialog between a couple apparently stranded on an island. They find an ancient rock inscribed with:
"In the beginning, everything was void, and J. H. W. H.
Conway began to create numbers. Conway said, "Let
there be two rules which bring forth all numbers large
and small. This shall be the first rule: Every number
corresponds to two sets of previously created numbers,
such that no member of the left set is greater than or
equal to any member of the right set. And the second rule
shall be this: One number is less than or equal to another
number if and only if no member of the first number's left
set is greater than or equal to the second number, and no
member of the second number's right set is less than or
equal to the first number." And Conway examined these
two rules he had made, and behold! They were very good.
And the first number was created from the void left set
and the void right set. Conway called this number "zero,"
and said that it shall be a sign to separate positive num-
bers from negative numbers. Conway proved that zero was
less than or equal to zero, and he saw that it was good.
And the evening and the morning were the day of zero.
On the next day, two more numbers were created, one
with zero as its left set and one with zero as its right set.
And Conway called the former number "one," and the
latter he called "minus one." And he proved that minus
one is less than but not equal to zero and zero is less than
but not equal to one. And the evening •.."
Then the couple proceed to decipher Conway's numbers. My favorite quote from the book is:
"B. ...Someday I think I'll write a book called Properties of
the Empty Set.
A. You'd never finish. ..."
It's a math joke, I know. Even though it's a small book and written in a easy and clear style don't expect to understand everything the first time through. It's that kind of subject. -
From the standpoint of being a mathematical text, this book is awful.
Fortunately, that is not at all the point of this novellette, nor does it pretend in any capacity that it is the point. Knuth states, in no uncertain terms, that the book is designed to give the impression of what it is like to do research-level mathematics, where the answers to questions are totally unknown, and there are no resources to research from. Everything must be tried, and sometimes failure is inevitable. It is in this context that the book shines. Despite being short enough to be read in an afternoon, one comes away thinking that they could indeed be Alice or Bill, if only they were brave enough to play around with the rules, just to see what pops out.
If you are looking for a more formal, complete introduction to the theory of surreal numbers, read "On Numbers and Games," by John Conway (yes, the same "J.H.W.H. Conway" named in this book!), but be warned that it is VERY dense. -
There are a lot of numbers in this book. Enough so that I panicked when I opened it, and wondered what I'd just gotten myself into. I am not good with numbers.
I stuck it out.
Knuth - anyone who knows him will attest to this - is good at what he does. Even for someone (me) whose last year of formal math was grade 11, many many years ago, the book was a pleasure. I followed the logic, if not the notation, without too painful an effort (though it was definitely an effort). And the payoff was easily worth it. If I'd had teachers like him back then, I might've taken my math pursuits further! Hit the spot. -
Yes, this is a pretty accurate portrayal of how math and research is done. (minus the beach and kissing.) It does a decent job of portraying how enjoyable it can be to explore and figure things out for yourself, and the author correctly points out that this is sort of "how it should be done," when it comes to education. (If only, if only.) But reading about it just doesn't capture the magic of doing it yourself. So I'm not sure if this book really did what it was supposed to--it was like watching people have fun. It made me want to have the fun, but it wasn't particularly fun in itself.
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I found this book very endearing; a lovely fusion of math and (an admittedly skeletal) story. If you enjoy doing math proofs, I recommend reading this book. I wish there were more math books that attempted a similar structure. If you aren't interested in math at all, unfortunately, I have to recommend that you skip this book. There isn't enough body to the story to make it an entertaining read. It's worth noting that Knuth suggested this book as a text for a course on "mathematical creativity".
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If you want to learn the surreal numbers, there are better ways, and if you want to learn to think mathematically, there are better ways. This book does not succeed at the author's goal of inspiring creativity in the reader. However, it was worth my time, for the thing Knuth was not emphasizing: it is a sufficiently complete and mildly entertaining description of the surreal numbers.
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This is by my intellectual crush, Don Knuth. I understand where he's coming from, but wonder if the maieutic method would really work her. In any case, I should go back to it in a more leisurely way, but for now, it's definitely entertaining. I wonder hoe many things can be done in this way to popularize science this century.
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A beautifully written (using LaTex, of course) novelette that showcases Alice and Bill, a couple in an island vacationing with enough time to exercise the mind in a mathematical “mistery”.
It is playful and invites readers to follow the steps of Alice and Bill and play with mathematical proofs from scratch, creating a whole universe day by day. -
The math was ok
But the story and background dialogue of A and B's romantic relationship was passable at best
Unless there's like some hidden message in their quick romantic quips, it wasn't really endearing to me -
An entertaining little novella about theoretical mathematics and the joy of discovery. Recommended for mqth students or people otherwise familiar with the concepts of set theory, algebra etc. Others are bound to find this book a bit challenging.
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Just the Maths would have helped a lot more than injecting it with a fictitious plot. It makes it way too verbose and I lost all my interest by the third chapter. It didn’t work for me, may be other night like it.
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One of the most amazing abstract algebra childrens book I've ever read!
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A prima vista un divertissement matematico, ma in realtà un modo per vedere come si può creare una teoria, tra idee e false partenze.
