❴PDF❵ ✅ Surreal Numbers Author Donald Ervin Knuth – Serv3.3pub.co.uk Shows how a young couple turned on to pure mathematics and found total happiness This title is intended for those who might enjoy an engaging dialogue on abstract mathematical ideas and those who migh Shows how a young couple turned on to pure mathematics and found total happiness This title is intended for those who might enjoy an engaging dialogue on abstract mathematical ideas and those who might wish to experience how new mathematics is created.

Also Sprake JHWHAll 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 optimistically language is helical It refers to itself endlessly and gets expressive as it does so as it builds upon itselfExpressive 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 itThis 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 anywayBy analogy with the emergence of DNA perhaps language appeared as a conseuence 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 languageAll 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 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 otherKnuth 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 eual to any member of the right set And the second rule shall be this One number is less than or eual to another number if and only if no member of the first number's left set is greater than or eual to the second number and no member of the second number's right set is less than or eual to the first number Thus surreal numbers simulate the development of all language They are the result of combinations of yet primal numbers which are in turn generated by yet 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 uite literally ‘nothing’ the null set 00Knuth’s two rules are exceptionally clever but they may reuire 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 fundamentalsThis 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 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 uestion of what constitutes such a thingKnuth’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 seuences of 's and 's since this approach makes it easier to build intuition about many important ideas used in later proofsThe 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 eual 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

Interesting presentation on an alternative number scheme It has a lot of cool features ordinary numbers don't like the ability to add and multiply infinities and other cool features with infinitesimals I like that math has a free range we don't find in physics being bound by reality and all

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 textbookBut 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 exciting and interestingactually 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

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 pleasuresfrustrations of independent work Be prepared to fire up a great deal of your 100 billion neurons while handling infinities

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 hourThe 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 HConway began to create numbers Conway said Letthere be two rules which bring forth all numbers largeand small This shall be the first rule Every numbercorresponds to two sets of previously created numberssuch that no member of the left set is greater than oreual to any member of the right set And the second ruleshall be this One number is less than or eual to anothernumber if and only if no member of the first number's leftset is greater than or eual to the second number and nomember of the second number's right set is less than oreual to the first number And Conway examined thesetwo rules he had made and behold They were very goodAnd the first number was created from the void left setand the void right set Conway called this number zeroand said that it shall be a sign to separate positive num bers from negative numbers Conway proved that zero wasless than or eual to zero and he saw that it was goodAnd the evening and the morning were the day of zeroOn the next day two numbers were created onewith zero as its left set and one with zero as its right setAnd Conway called the former number one and thelatter he called minus one And he proved that minusone is less than but not eual to zero and zero is less thanbut not eual to one And the evening •Then the couple proceed to decipher Conway's numbers My favorite uote from the book isB Someday I think I'll write a book called Properties ofthe Empty SetA 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

Knuth explains Number theory with a romantic plot as a conversation between two lovers The book has many wordplays and very enjoyable read proving many basic proofsIt's the idea that countsAll we have is a bunch of objects ordered neatly in a linebut we haven't got anything to do with themI guess the excitement and the beauty comesin the discovery not the hearingRubbish Wait until you get to infinite sets What a miserable night I kept tossing and turning and mymind was racing in circles I dreamed I was proving thingsand making logical deductions but when I woke up they wereall foolishness Maybe this mathematics isn't good for us after all We were sohappy yesterday Well when we were going around in circleslike this before how did we break out? The main thing wasto use induction I mean to show that the proof in one casedepended on the truth in a previous case which depended ona still previous case and so on where the chain must eventuallycome to an end

From the standpoint of being a mathematical text this book is awfulFortunately 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 uestions 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 outIf you are looking for a formal complete introduction to the theory of surreal numbers read On Numbers and Games by John Conway yes the same JHWH 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