SURREALNUMBERS Broadview 研究之美【美】D.E.KNUTH著高博译 83 電子工業出版社 eepe.wow.p..cuo.cm PEARSON 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 传世经典书丛 Broadview www.broadvlew.com.cn 博文视点IT出版旗舰品牌技我的要思标并不是真的要教给读者Conway教技的理全，是快者一个人爱如何着来研究出这么一理论来高德纳我很高义地看到高德纳先生在38年前写成的一本全数字奇究的，能在会天被传译给中国读者，读完后，意料之中的是美本书果然反映了人师的和思路：意料之外的是，原来数学也能以小说的形式来写！张亚勤上架建议：计算机科学普及读物 PEARSON 策划编辑：郭立责任编辑：徐津平封面设计：李玲符隆美本书贴有激光防伪标志，凡没有防伪标志者，属盗版图书。 ISBN978-7-121-15553-6 17871211 1155536 定价：49.00元更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 一对学友如何启发了对纯数学的兴趣并获得了终极幸福的故事研究之美 SURR NUMBERS 電子王業出版社 D.E. Knuth 著高博译 Publishing House of Electronics Industry 北京·BEIJING 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 内容简介　本书是计算机科学大师、“算法分析之父”高德纳（DonaldE.Knuth）在20世纪70年代旅居挪威时撰写的适用于计算机科学的一种全新基础数学结构的情景小品。全书以一对追求自由精神生活的青年男女为主人公，展开了一段对于该种全新结构的发现和构造的对白。在此过程中，本书充分展示了计算机科学的从业人员进行全新领域探索时所必备的怀疑、立论、构造、证明、归纳、演绎等逻辑推理和深入反思的能力。本书可以看作是读懂高德纳的艰深著作《计算机程序设计艺术》和《具体数学》的钥匙。 Adaptation from the English language edition, entitled SURREAL NUMBERS, Authorized 1E,9780201038125 by KNUTH, DONALD E., published by Pearson Education, Inc, publishing as Addison-Wesley, Copyright C 1974 by Addison- Wesley Publishing Company, Inc. All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage retrieval system, without permission from Pearson Education, Inc. O n PUBLISHING HOUSE OF ELECTRONICS INDUSTRY CoPyTight @ 2012 ENGLISH language adaptation edition is mnanufacfured in the People's Republic of China, and is authorized for sale only in the mainland of China excluding Hong Kong and Macau. 本书中英文双语版专有出版权由培生教育出版集团亚洲有限公司授予电子工业出版社。仅限于中国大陆境内（不包括中国香港、澳门特别行政区和中国台湾地区）销售发行。未经许可，不得以任何方式复制或抄袭本书之部分或全部内容。本书中英双语版贴有Pearson Education培生教育出版集团激光防伪标贴，无标签者不得销售．版权所有，侵权必究。版权贸易合同登记号：图字：01-2011-8101 图书在版编目（CIP）数据研究之美：汉英对照／（美）高德纳（Knuth,D.E.）著；高博译．一北京：电子工业出版社，2012.1 书名原文：Surreal Numbers ISBN 978-7-121-15553-6 I．①研．I．①高·②高·III.①计算机科学普及读物V.①TP3-49 中国版本图书馆CIP数据核字（2011）第264437号策划编辑：郭立责任编辑：徐津平符隆美印刷：北京市顺义兴华印刷厂装订：三河市双峰印刷装订有限公司出版发行：电子工业出版社北京市海淀区万寿路173信箱邮编100036 开本：787×980 1/16　印张：12.75　印次：2012年1月第1次印刷　印数：4000册　定价：49.00元字数：132千字　凡所购买电子工业出版社图书有缺损问题，请向购买书店调换。若书店售缺，请与本社发行部联联系及邮购电话：（010）88254888 系，盗版侵权举报请发邮件至dbqq@phei.com.cn。质量投诉请发邮件至 zlts@phei.com.cn，服务热线：（010）88258888。更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 推荐序最近几年科学技术和全球市场的变迁，已经越来越清楚地让我们看清了一个事实,那就是持续创新不仅已经成为企业发展的重要基石，甚至已经成为了企业生存的必要前提。这对微软和其他意图在激烈竞争中立于不败之地的企业来说，意味着要拥有马拉松式的创新耐力。创新的基础是研究，主体是人才。研究机构的建设和创新人才培养，也是我们一直以来的专注和思考。没有研究人才多年如一日地甘于对未知世界的孤独探索，那些能够极大地提高生产力的科技，将不会诞生。高德纳先生是世界公认的算法大师，他在计算机基础科学方面的很多研究成果，对现在乃至未来的信息产业，已经产生和必然发生深刻影响。他洋洋四卷的《计算机程序设计艺术》，更是被全世界的数学家和工程师奉为圭桌。我很高兴地看到，他在38 年前写成的-一本讨论数学研究的书，能在今天被传译给中国读者。读完后,意料之中的是,本书果然反映了大师的智慧和思路；意料之外的是，原来数学也能以小说的形式来写！高德纳通过引人入胜的对白和推理来展开情节，为我们展现了数学、方程式、逻辑和算法之美。研究是一种揭示客观规律的行为，但研究行为自身，也有其独特规律。高德纳先生的这本书，是他几十年智力和思维活动和工业实践的总结，从更高的层次上为我们揭示了研究行为自身的客观规律。从这个意义上说，本书的中文译名《研究之美》，是恰如其分的。近些年，中国每年毕业的理工科毕业生有七八十万。这些有望成为具备研究能力、掌握研究规律的专门人才，是中国未来竞争方的关，键。但遗撼的是,我也看到不少年轻人耐不住枯燥与寂寞,而中途放弃。为此，裹心祝愿本书能为有志于投身研究事业的读者们，打开一扇趣味之门！张亚勤微软全球副总裁微软亚太研发集团主席 2011年12月1日于北京 i 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / PREFACE Mathematics is the science of patterns, and I especially like the fact that we can use mathematical reasoning to deduce amazing con-sequences from only two or three simple rules. One of the most beautiful topics in all of mathematics is the theory of surreal numbers, discovered by John Horton Conway about 1970. A few months after he told me about it, I decided that it would be. fun to base a short story around his exciting ideas. Just as an opera consists of wonderful music together with a bit of a plot, I wanted to. describe this wonderful mathematics together with a bit of a plot. I was living in Norway at the time. In January of 1973 I rented a hotel roon in downtown Oslo, near where Hendrik Ibsen once lived, in adses eso s et sed six days writing this little book. On the seventh day I rested. It was the happiest week of my life! Now, almost forty years later, I'm glad to see that people all over the world have enjoyed this story so much that they have translated it into many different languages. I have been reading many accounts these days about "Doctors Without Borders" and "Engineers With-out Borders"; I like to think of myself as a Mathematician Without Borders. For thousands of years mathematics has been a worldwide enterprise, appreciated by people everywhere, and much of this devel-opment has occurred in China. Therefore I'm especially happy that this book now appears also in the Chinese language. (Also I'm pleased resees Na', which Frances Yao gave to me in 1977 when I was first invited to visit China. I sincerely hope that Chinese readers, young and old, will en- joy discovering the beautiful mathematical patterns that Conway has bequeathed to us. Donald E. Knuth IGao De Na iv 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 序数学是模式的科学。而我则尤其喜爱这样的事实，就是我们能够运用数学推理，由两三条平凡的规则出发，最终得出令人惊喜的结果在所有的数学领域中堪称是最美妙的主题之一，就是超现实数理论。它是由 John Horton Conway 在 1970 年左右发现的。在他告诉我这个理论数月之后，我产生了一个想法：如果能以他的绝妙想法为基础写个短篇故事，那该多么有趣呀。正如歌剧就是美妙的音乐加上那么一点儿情节，我也想在讲述这样美妙的数学时加上那么一点儿情节。写作此书时我正旅居挪威。1973 年 1月,我在奥斯陆市区租了～间宾馆的客房，离易卜生1的故居很近，所以我指望能通过这种方式沾上点儿易卜生的灵气。然后，我花了六个工作日完成了这本小册子，而到了第七日，我就停下来休息。2这是我这辈子最快乐的一星期！如今，事情已经过去了近四十年。我十分高兴地看到，全世界读者如此地喜爱这个故事，所以他们将它翻译成了很多不同的语言。最近段时间，我读了不少有关“无国界医生”和“无国界工程师”3的故事，我也倾向于认为自已是一名无国界数学家。经历了千百年以后,数学其中相当一部分的进展都发生在中国。因此，这本书现在出版了简体中文版，是尤其令我欣喜的事。（同时我也愉快地发现译者的名字和我的中文名“高德纳”同姓，而我的中文名字乃是储枫4在我1977年首次访华时为我起的。）我衷心希望中国的读者,无论是否仍然年轻,都能够从 Conway 留给我们的美妙数学模式中得到乐趣。 Donald E.Knuth（高德纳）亨德利克·易卜生（Hendrik Ibsen，1828-1906),挪威戏剧大师,现代现实主义戏剧的创始人，为国人耳熟能详的代表作有《玩偶之家》、《人民公敌》等。一一译者注，全书同 2根据《圣经·创世纪》，上帝以六日创生万物，而第七日为安息日。 3无国界组织，指一些专业人士号召摆脱种族、宗教和政治立场，为全人类谋福祉而自发的组织。无国界医生（Médecins Sans Frontieres）曾获1999 年诺贝尔和平奖： 4 储枫（Frances Yao),著名学者,1973 年从麻省理工学院取得博士学位。曾在美国夫姓姚，其夫为2000年图灵奖获奖者姚期智（AndrewYao)。更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 译者序摆在诸位读者面前的,是一本可能会给您的一生带来重大转折契机的书。这并非故弄玄虚,因为它的作者一一计算机算法大师高德纳(Donald E.Knuth)，以三卷本的《计算机程序设计艺术》（The Art of Computer Programming）－举获得 IEEE 先驱奖和 ACM 图灵奖　这两个计算科学界最高奖项。以出版物的成就拿到这两个奖项的，在历史上可谓空前绝后。《计算机程序设计艺术》是无可争辩的神作，毫不夸张地说，全世界和算法分析相关的一切著作，都直接或间接地与这套书有关。但是，能够真正读懂这套书的读者，真是凤毛麟角。月不说里面的数学分析，就算是为讲解算法所使用的程序语言也全是作者自行发明的一种理想机器所使用的，这对于读者的数学功底和抽象思维都提出了非常严苛的要求。所以,有一种说法是,拥有全套《计算机程序设计艺术》的人很多，但是读过十页以上的人则屈指可数，可是，高德纳本人却一直都认为，再深入的研究也是从最简单的情况入手的。并且一旦掌握了几种做研究的固定套路，即使是普通人也不仅可以着手做研究，甚至还可能做出一流的成果来。这里所说的研究，并非一定是数学研究，而是指从一系列基本的事实或定义出发，通过若干明确的规则，推导出满足这些前提条件的有价值的结论。研究，是最能够体现人类心智的活动，也是创新乃至人类进步的根本源动力。每个人在每天的日常工作和生活中，都在进行着不同程度的研究活动但是高德纳所主张的研究,既是自觉的、有目的的研究,更是一种工作和生活的基本态度。不仅仅是把研究看作是一种高级的智力活动，更是给心灵带来深层次乐趣的生活方式。唯有如此，才能使研究活动走出象牙塔，打破原本就不存在的条条框框，让每一个人都能领略研究带来的美学享受，提升生活的品质和境界。高德纳的博士学位是数学专业，他最得心应手的技术和工具自然也是数学。在本书中，他为了展示研究活动所涉及的方法和思维，也使用了数学作为演示工具。但是，为了尽可能地不让读者陷入具体的技术细节，大师采用了不仅在学术作品中绝无仅有，并且在科普作品中也 Vii 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 绝不多见的情景小品形式。这么一来，从研究问题的提出，到研究体系的构建，再到研究思路的形式，直到研究结论的得出，都通过男女主人公的对白完成。我想，大师这样另辟蹊径的意图，是想让我们把关注的焦点，始终放在研究过程而非具体的数学讨论上。从主人公发现问题的兴奋、遭遇困境的、探索出路的苦闷、得到结论的欣喜中,我们能够从观众视角充分地体会到研究那点事儿的方方面面，既明确了研究的要素，又了解了研究的方法。更重要的是，从这些对白中我们能够逐渐地明白，对同一问题的研究能够拉近人与人心灵的距离，使得人类之升华，真正地使“终极幸福”成为可能。纯粹从数学知识的角度来看,本书的内容是建立在集合论的基础之上的。数通过集合加以定义，数的顺序映射为集合的关系，而数的运算则映射为集合的运算。定义和规则只有简单的几条，并且非常直观易懂。但是由于数的定义已经从基础上被完全颠覆了，一切有关数的顺序和运算都必须重起炉灶。本书中，男女主人公不断地从基础定义出发,研究出了一系列的中间定理,并总结出若干反复运用的研究套路。最终发现，采用新方法定义的数所组成的集合，竟然是比实数系统更加稠密的数系。并且在这种新的数系中，无穷大量和无穷小量可以像普通的数那样参与运算，而且像“无穷大的一半是多少”这样在传统的实数连续统中没有定义的量,在新的数系中有着很明确的数学意义（实际上，该集合大致相当于现在称为Grothendieck 宇集的集合，但是不了解这个背景完全不影响阅读）。更妙的是，可以发明一些不符合数的定义,却可以作为中间结果使得一些运算成立或简化的“伪数”,其地位相当于V一1这个在人类的认识尚局限于实数的时代曾经一度被排能，大推动力的复数····总之，内容精彩纷呈,欲知详情,请仔细阅读。大师的手笔，绝不会让你失望！作为本书的译者，我感觉翻译的过程本身是充满愉悦的享受，可以说是一次与高德纳大师进行心灵对话的宝贵体验。以我个人的体会来说，本书只要仔细阅读，是连高中生都应该可以完全看懂的。即使一时静不下心来去读懂具体的推导过程，看看大师在主人公的对白中有意 viii 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 插入和强调的那些研究所带来的普世价值，也是大有裨益的。本书成稿的过程中，得到了博文视点符隆美编辑的大力支持，没有她的关照和鼓励，我不可能这么快地完成全书的翻译。承美国南密西西比大学数见。北京师范大学数学系的赵钊研究员也费心费力，对译稿的可读性和技术问题作了上百处修订。在此，向他们两位表示裹心的谢意。本平教授、上海交通大学软件学院的陈平教授、SAP中国的范德成工程师、盛大创新院的刘海平研究员、微软亚洲工程院的王楠工程师、谷歌中国的龚理工程师、美光半导体的赵海源技术顾问、上海申通地铁集团的蒋振伟项目经理等都提出过若干稿件修正意见，在此一并致谢。当然，由于本人能力所限，本书的缺点和不足仍在所难免，这些理应由我一人负责。为向高德纳大师致敬，本书简体中文版采用大师本人发明的 TEX排版系统排版。我的老朋友、SAP 美国的技术咨询顾问劳佳同志在承担了排版工作的同时，亦费心费力在封面设计工帮了天忙，还对文稿内容有颇多指正之功，这里要特别致谢。我也想借此机会向在工作和生活上给了我莫大支持的父母和家人表达我内心最深处的敬意，希望本书的出版能给你们带来快乐。高博 2011年12月于盛大集团上海总部 x 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 岩符证坏进石明数展第三日发加答定缘灾平宇玄乘跋现法案理定难复宙极法更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 当 11 21 29 41 51 63 73 85 97 109 115 125 137 149 163 175 CONTENTS The Rock 2 3 5 6 7 8 9 10 11 12 13 14 15 16 Symbols Proofs Bad Numbers Progress The Third Day Discovery Addition The Answer Theorems The Proposal Disaster Recovery The Universe Infinity Multiplication Postscript 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 0 10 19 28 40 50 62 72 84 96 108 114 124 136 148 162 174 A：Bill,你觉得你已经找到自我了吗？ B：你说什么？ A：我是说－岩石海hi 一我们现在身处印度洋的边缘,远离尘。为了不被世俗的体系拖跨，我们逃离尘世来“寻找自我”到现在算起来已有好几个月。我就是想了解一下，你觉得我们已经成功地做到这点了吗？更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Actually, Alice, I've been thinking about the same thing. These past months together have been really great -- we're completely free, we know each other, and we feel like real people again in-stead of like machines. But lately I'm afraid I've been missing some of the things we've “escaped" from. You know, I've got this fantastic craving for a book to read -- any book, even a textbook, even a math textbook. It sounds crazy, but I've been lying here wishing I had a crossword puzzle to work on. Oh, c'mon, not a crossword puzzle; that's what your parents A. like to do. But I know what you mean, we need some mental stimulation. It's kinda like the end of summer vacations when we were kids. In May every year we couldn't wait to get out of school, and the days simply dragged on until vacation started but by September we were real glad to be back in the classroom. B. Of course, with a loaf of bread, a jug of wine, and thou beside me, these days aren't exactly "dragging on." But I think maybe the most important thing I've learned on this trip is that the simple, romantic life isn't enough for me. I need something com plicated to think about. A. Well, I'm sorry I'm not complicated enough for you. Why don't we get up and explore some more of the beach? Maybe we'll find some pebbles or something that we can use to make up some kind of a game. B. (sitting up) Yeah, that's a good idea. But first I think I'll take a littile swim. (running toward the water) Me, too -- bet you can't catch me! B. Hey, what's that big black rock half-buried in the sand over there? 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：其实吧，Alice，我也一直想着这事儿呢。这几个月过得真是开心我们彻底解放了一把，我们也了解了彼此，并且我们总算又感觉到自已活得像个人样，而不是像机器那种活法儿了。不过，最近我老是在想那些我们以前总在竭力逃避的一些东西。你也看到了，我着了魔似地想弄本书来翻翻一一什么书都成，即使是教科书，甚至是数学教科书也无妨。听上去是不是我脑子有毛病了啊，但我躺在这里的时候，真希望哪怕有个纵横学谜让我填填也好啊。 A：我说,纵横字谜就算了吧,那是老一辈的人才玩的东西了。不过我知道你是什么意思，我们得来点儿脑力刺激才行。就像我们还，是小孩子的时候，放暑假的感觉。一到五月，所有的小朋友们都迫不及待地想离开学校，盼着暑假的那段白子可真是难熬，可是到了九月开学，我们还是兴高采烈地回到教室的。 B：当然,如果啃着香喷喷的面包，又可以喝点儿小酒,还有佳人如你在侧，那段时间也不能说是“熬”过去的嘛。但是我觉得，这次出游给我带来的最重要的感受是简单浪漫的生活并不能满足我。我需要一些更复杂的东西来思考才行。 A：好吧，看来我对你来说原来还不够复杂啊。那你何不起来到海滩上多探探宝呢？说不定还能找到一些鹅卵石什么的，我们可以用它们想出什么新的游戏花样来。 B：（坐起身来）好啊，这个主意不赖。不过我想先游一小会儿泳再说。 A：（奔向海水)我也要　你背定追不上我！ B：看,那边半埋在沙中的黑色大块岩石样的是什么？更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. Search me, I've never seen anything like it before. Look, it's got some kind of graffiti on the back. B. Let's see. Can you help me dig it out? It looks like a museum piece. Unnh! Heavy, too. The carving might be some old Ara-bian script . .. no, wait, I think it's maybe Hebrew; let's turn it around this way. A. Hebrew! Are you sure? B. Well, I learned a lot of Hebrew when I was younger, and I can almost read this. . .. A. I heard there hasn't been much archaeological digging around these parts. Maybe we've found another Rosetta Stone or some thing. What does it say, can you make anything out? Wait a minute, gimme a chance. ... Up here at the top right is B. where it starts, something like "In the beginning everything was void, and ..." Far out! That sounds like the first book of Moses, in the Bible A.. Wasn't he wandering around Arabia for forty years with his followers before going up to Israel? You don't suppose . . B. No, no, it goes on much different from the traditional account. Let's lug this thing back to our camp, I think I can work out a translation. A. Bill, this is wild, just what you needed! B. Yeah, I did say I was dying for something to read, didn't I. Al- though this wasn't exactly what I had in mind! I can hardly wait to get a good look at it ---some of the things are kinda strange, and I can't figure out whether it's a story or what. There's something about numbers, and ... A. It seems to be broken off at the bottom; the stone was origi nally longer. 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：我真的不知道呀，以前还从来没有见过这样的家伙呢。你看,在它背后有一些像是涂鸦的东西呢！ B：我来看看。能帮我把它挖出来吗？这像是个老古董了。哟！还挺沉的。这些雕文像是一些阿拉伯文字.··不,等等,我觉得好像是希伯来文。我们把它翻过来这样看看。 A：希伯来文！你确定吗？ B：那个,我小时候学过不少希伯来文,这上面的文字应该差不多都能看懂 A：我并没有听说这一带有过很多考古发掘。也许我们又发现了- 块罗塞塔石碑(Rosetta Stone)l什么的。它上面写了什么，你能看出门道来吗？ B：等一下,让我试试那个上面,右上角是文字开始的地方,好像是“初,万物混沌苍茫,尔后·．.” A：好棒！这听起来像是《圣经》里摩西写的第一卷经书哦。那个不是摩西和他的追随者们在动身去往以色列之前在阿拉伯仿了四十年的事儿吗？难道·…·… B：不,不,这上面写的和古典经文可是大相径庭。咱们先把这个搬到帐篷里去,我想我可以把它翻译出来。 A：Bill,真不可思议啊,这正是你想要的！ B：对,我是说过我想找点儿什么看看都快想疯了,对吧。尽管这玩意儿和我脑子里想到的东西并不太一样！可我已经等不及想仔细对这东西研究研究一一它看起来有点儿不寻常,我还不能断定它上面写的是个故事还是什么别的东西。好像有一些和数有关系的内容,还有····· A：这石头好像从底部断开来了，它原本更长一些的。 1一块制作于公元前196 年的大理石石碑，刻有埃及国王托勒密五世（PtolemyV) 的诏书。由于这块石碑刻有同一段文字的三种不同语言版本，近代的考古学家得以有机会对照内容，解读出已经失传千余年的埃及象形文的意义与结构。 5 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. A good thing, or we'd never be able to carry it. Of course it'll be just our luck to find out the message is getting interesting, right when we come to the broken place. Here we are. I'll go pick some dates and fruit for supper while A you work out the translation. Too bad languages aren't my thing, or I'd try to help you. B. Okay, Alice, I've got it. There are a few doubtful places, a cou- ple signs I don't recognize; you know, maybe some obsolete word forms. Overall I think I know what it says, though I don't know what it means. Here's a fairly literal translation: 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 a sda s o aa o e sa 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 frst 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 6 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：好事儿，不然我们就搬不动啦。当然，如果说我们更走运一点儿的话，那就是在看到断开的地方之前，已经开始出现一些有意义的文字了呢。 A：我们到了。我去找些枣儿水果什么的当晚饭，你就在这儿做你的翻译吧。可惜我实在不是搞语言的料，要不然我就帮你一把了。-中 B：好了，Alice,我搞定了。当然还是有些不确定的地方,有些符号我没看明白是什么意思,有些可能是已经废弃不用的构词。大体上我已经知道它讲的是什么内容了，但还没弄明白这些内容是什么意思。下面是严格逐字逐句的翻译：初,万物混沌苍茫,尔后 J.H.W.H.Conway 始创诸　数。Conway 日，“创生二道,大小诸数盖由此出。其一日：凡数，皆合于前创二数之集，其位左者，无一大于或等于其位右者。其二日：甲数小于或等于乙数，当且仅当甲数之左集中无一大于或等于乙数,且乙数之右集中无一小于或等于甲数。”Conway 检视二道，连呼妙哉! 此二道真妙绝。元初之数，左有皆空。Conway 名之“零”命其为正负两界分野之符。Conway 证得，零小于或等于零,此间妙也。夜去昼来，是为零日。次日，又得二数。其一以零为左集，更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 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 egual to zero and zero is less than but not equal to one. And the evening .. That's where it breaks off. Are you sure it reads like that? A. B. More or less. I dressed it up a bit. A. But "Conway" ... that's not a Hebrew name. You've got to be kidding. B. No, honest. Of course the old Hebrew writing doesn't show any vowels, so the real name might be Keenawu or something; maybe related to the Khans? I guess not. Since I'm translating into English, I just used an English name. Look, here are the places where it shows up on the stone. The J. H. W. H. might also stand for "Jehovah." A. No vowels, eh? So it's real. ... But what do you think it means？ B. Your guess is as good as mine. These two crazy rules for num. bers. Maybe it's some ancient method of arithmetic that's been obsolete since the wheel was invented. It might be fun to figure them out, tomorrow; but the sun's going down pretty soon so we'd better eat and turn in. Okay, but read it to me once more. I want to think it over, and A. the first time I didn't believe you were serious B. (pointing)“In the beginning,:. 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 其一以零为右集。Conway 名前者日“一”，后者日“负一”。Conway 证得，负一小于而不等于零，零小于而不　等于。是夜．岩石就是从这里断开了。 A：你确信上面写的是这些东西吗？ B：多多少少吧。我稍加了点儿修饰。　 A：但那个“Conway”..··这不是个希伯来名吧。你肯定是在开玩笑。 B：不,我是实话实说。当然,纸草希伯来文没有给出任何元音,所以真实的姓名可能是Keenawu 什么的，是不是和“可汗”有点儿关系？我想大概没任么关系。因为我是捆这段话译成英文的，所以我就用了个英文名。你看，这就是名字在石头上出现的位置，J H.W、H.可能是“Jehovah”（耶和华）的缩写吧， A：没有元音？，那真是····可你觉得这些话是什么意思呢？ B：你猜的和我猜的应该是半斤八两。这是有关数的两条不可思议的规则。也许这是古代某种算术法,老早的时候就给废弃了。把它研究出来应该会很有意思，明天吧。现在太阳马上就下山了我们吃点东西打点歇息了吧。 A：好的,不过你再给我念一遍。我想再在脑子里过一过,这是我第一次觉得你在逗我玩儿 B：（指点着岩石）“初，….” 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 2 SYMBOLS A. I think your Conway Stone makes sense after all, Bill. I was thinking about it during the night. B. So was I, but I dozed off before getting anywhere. What's the secret? A. It's not so hard, really; the trouble is that it's all expressed in words. The same thing can be expressed in symbols and then you can see what's happening. 10 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 2 符号 A：我觉得你那“Conway之岩”说到底还是有理可循的,Bill。我整夜都在琢磨这个。 B：我也是啊,但是我还没琢磨明白就睡过去了。你琢磨出什么来了？ A：其实说起来也不算太难,难的是用适当的词句将它表达出来。我可以用符号来表达同样的意思，这样你一看就明白了。 11 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. You mean we're actually going to use the New Math to decipher this old stone tablet. A. I hate to admit it, but that's what it looks like. Here, the first rule says that every number α is really a pair of sets, called the left set L and the right set R: (L,R). B. Wait a sec, you don't have to draw in the sand, I think we still have a pencil and some paper in my backpack. Just a minute. Here, use this. A. =(L,R). These &L and &R are not just numbers, they're sets of numbers; and each number in the set is itself a pair of sets, and so on. B. Hold it, your notation mixes me up. I don't know what's a set and what's a number. Okay, I'll use capital letters for sets of numbers and small letters A. for numbers. Conway's first rule is that =(XL,XR), where XL±XR- (1) This means if &L is any number in XL and if R is any number in XR, they must satisfy &L + R. And that means &L is not greater than or equal to &R. (scratching his head) I'm afraid you're still going too fast for B. me. Remember, you've already got this thing psyched out, but I'm still at the beginning. If a number is a pair of sets of num-bers, each of which is a pair of sets of numbers, and so on and so on, how does the whole thing get started in the first place? Good point, but that's the whole beauty of Conway's scheme. A. Each element of Xz and XR must have been created previ- ously, but on the first day of creation there weren't any previous 12 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：你是说，我们要用当代数学来解密这段古老石板上的文字吗？ A：我真不想承认，但好像还真的就得这么办。你瞧，这第一条规则是说,所有的数实际上都是一对集合,称为左集 L 和右集 R： =(L,R B：等一等，你没必要把这些画在沙地上，我想起来我背包里还带了一支铅笔和一些纸呢,我找找…···给你,用这个写。 A: = (CL,TR) 此处,L和 R 并不仅仅是数,而是数的集合。并且,这些集合中的每一个数本身也是一对集合,依此类推。 B：慢点儿，你的记法把我搞糊涂了。我分不清哪表示集合，哪个表示数了。　 A：好吧，那我就用大写字母表示数的集合，而用小写字母表示数。 Conway的第一条规则就是说： =(XL,XR)，其中　XLXR　 (1) 这就是说,若 L 是 XL 中的任意数,而 R 是 XR 中的任意数则必须满足 L R。而这就表示 L 不大于也不等于 R B：（挠了挠脑袋）你说得有点儿太快了，我跟不上。要知道，你已经就这个问题研究了好半天了，我这才刚开始呢。如果一个数其实是一对数集，而且这些数集中的每一个数本身又都是一对数集：这样周而复始，但所有这些总得有个并始吧？ A：说得好,这正是 Conway 体系的妙处。XL 和 XR 中的每一个元素都必须“合于前创二数”，但是在诸数创生的首日，并没有任何所谓“前创”的数 13 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / numbers to work with; so both XL and XR were taken to be the empty set! B. I never thought I'd live to see the day when the empty set was meaningful. That's really creating something out of nothing, eh? But is XL XR when XL and XR are both equal to the empty set? How can you have something unequal itself? Oh yeah, yeah, that's okay since it means no element of the empty set is greater than or equal to any element of the empty set -it's a true statement because there aren't any elements in the empty set. A. So everything gets started all right, and that's the number called zero. Using the symbol 0 to stand for the empty set, we can write 0 = (0,0). B. Incredible A. Now on the second day, it's possible to use O in the left or right set, so Conway gets two more numbers -1 = (0, {0]) and 1 = ({0},0). B. Let me see, does this check out? For --l to be a number, it has to be true that no element of the empty set is greater than or equal to O. And for 1, it must be that 0 is not greater than any element of the empty set. Man, that empty set sure gets around! Someday I think I'll write a book called Properties of the Empty Set. A. You'd never finish. If XL or Xr is empty, the condition XL + Xr is true no matter what is in the other set. This means that infinitely many num- bers are going to be created. 14 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 可以拿来用,所以 XL 和 XR 只能都取空集！ B：没想到我在有生之年会看到空集有了意义的一天。这真是无中　生有啊,对吧？但是 XL 和 XR 都是空集时,XL ± XR 成立吗？你怎么能说某物和它自己不相等呢？峨哦,对的,没错儿,这一点之所以成立,是因为空集中没有元素会大于或等于空集中的任何元素一这是个真命题，因为空集中没有任何元素。 A：所以，一切各就各位,而那个数就被称为“零”。如果用符号の表示空集，我们就得到 0 = (0,0) B：真是妙不可言。 A：那么次日,就可以在左集或右集中使用 0 了，这样 Conway 就得到了两个新数 -1= (0,{0})，以及1=([0},0) B：让我瞧瞧,这些数符不符合规则？一1 作为一个数，它必须满足空集中没有任何元素大于或等于0。而对于 1来说，它必须满足.0 不比空集中的任何元素大。哇哦,空集肯定能满足这些要求！以后我一定要写本书，名字叫做《空集的性质》。 A：你写不出来的啦！若 XL或 XR为空,则另一个集合中无论有什么元素,XL±XR 皆成立。这就意味着可以创造出无穷多个数来。 15 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Okay, but what about Conway's second rule? That's what you use to tell whether XL XR, when both sets A. are nonempty; it's the rule defining less-than-or-equal. Symboli-cally， y means XL and α?YR． B. Wait a minute, you're way ahead of me again. Look, XL is a (2) set of numbers, and y is a number, which means a pair of sets of , m non a a on o e sa A. I mean that every element of XL satisfies L y. In other words, no element of X is greater than or equal to y. B. Oh, I see, and your rule (2) says also that α is not greater than or equal to any element of Yr. Let me check that with the text. A. The Stone's version is a little different, but ≤ y must mean the same thing as y ≥ a. B. Yeah, you're right. Hey, wait a sec, look here at these carvings off to the side: These are the symbols I couldn't decipher yesterday, and your. notation makes it all crystal clear! Those double dots separate the left set from the right set. You must be on the right track. Wow, equal signs and everything! That stone-age carver must A. have used to stand for --1; I almost like his notation better than mine. 16 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：明白，那Conway 的第二条规则又在说什么呢？ A：它用以在 XL 或 XR 皆不为空时,判定 XL XR 成立与否。这条规则定义了“小于或等于”的意义。可以用符号表示为 ≤y　意味着　Xy且YR． 2 B：等一下,我又跟不上你说的速度了。看看这个,XL 是个数集,而 y是个数,也就是一对数集,那你写的 XL 表示什么意思呢？ A：我的意思是 X中的每个元素 α都满足 Ly。换言之,就是 XL 没有任何元素大于或等于 y。 B：哦,我明白了。你这里写的规则（2）是说,不大于或等于YB 中的任何元素。让我对照这段文字检查一下。·· A：岩石上的原文版本略有不同,但是，α≤y 其实和 y≥ 是一回事。 B：你说得没错。嘿，等一下,你看看边上的这些雕文：这些符号我昨天还一头雾水，但你的记法让它们变得一目了然啦！这个双点符隔开了左集和右集。你肯定已经在正轨上了。 A：哇哦,等于号,以及所有的这一切！石器时代的人们一定是用一来代表一1。比起自已的符号，我简直还更喜欢他们的。 17 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. I bet we've underestimated primitive people. They must have had complex lives and a need for mental gymnastics, just like us ---at least when they didn't have to fight for food and shelter We always oversimplify history when we look back. A. Yes, but otherwise how could we look back? B. I see your point. A. Now comes the part of the text I don't understand. On the first day of creation, Conway “proves" that O ≤ O. Why should he bother to prove that something is less than or equal to itself, since it's obviously equal to itself. And then on the second day t so et st o t ea o st et aod proof, since --1 is a different number? Hmmm. I don't know about you, but I'm ready for another B. swim. A. Good idea. That surf looks good, and I'm not used to so much concentration. Let's go! 18 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：我敢打赌,我们是低估原始人了。他们一定和我们一样,有过高层次的生活，而且也需要思维的体操一一至少在他们衣食无忧时会有。我们在回顾历史时总是会过分简化。 A：不然你怎么个回顾法儿？ B：我明白你的意思。　 A：下面这段话我可就看不懂了。在诸数创生的首日,Conway“证明”了00。为什么他要费神去证明某数小于或等于它自身？而在次日,他“证明”了1不等于 0,而一1 本来就是另一个数; 这难道还用证？ B：嗯,这个嘛。我不知道你是怎么想的，反正我现在又想游会儿泳 A：好呀。这会儿冲浪条件看起来不错，而且我也不习惯集中精力那么久。走，起吧！　 19 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 20 PROOFS B. An idea hit me while we were paddling around out there. Maybe my translation isn't correct. A. What? It must be okay, we've already checked so much of it out. B. I know; but now that I think of it, I wasn't quite sure of that word I translated "equal to." Maybe it has a weaker meaning, “similar to" or "like." Then Conway's second rule 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 3 Ce 证明 B：刚才咱们在那边划水的时候,我忽然闪过一个念头，想到我的翻译可能有个地方并不正确。 A：你说什么？你那翻译肯定是八九不离十的，我们已经推导出这么多结果来了。 B：我明白，但转念一想,我就有点儿拿不准我译成“等于”的词儿了。可能它只有较弱的意义，也就是“相似于”或是“相像”那个程度。这么一来，Conway 的第二条规则　 21 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / becomes "One number is less than or like another number if and only if ... ." And later on, he proves that zero is less than or like zero; minus one is less than but not like zero; and so forth. A. Oh, right, that must be it, he's using the word in an abstract technical sense that must be defined by the rules. So of course he wants to prove that O is less than or like O, in order to see that his definition makes a number "like" itself. B. So does his proof go through? By rule (2), he must show that no element of the empty set is greater than or like O, and that O is not greater than or like any element of the empty set. . . Okay, it works, the empty set triumphs again. More interesting is how he could prove that- -1 is not like O A. The only way I can think of is that he proved that O is not less-than-or-like -1. I mean, we have rule (2) to tell whether one number is less than or like another; and if r is not less-than-or-like y, it isn't less than y and it isn't like y. B. I see, we want to show that O < --1 is false. This is rule (2) with = 0 and YR = {0), so 0 ≤ -1 if and only if 0 ± 0. But 0 is ≥ O, we know that, so 0 ± --l. He was right. A. I wonder if Conway also tested -1 against 1; I suppose he did. although the rock doesn't say anything about it. If the rules are any good, there should be a way to prove that -l is less than 1 Well, let's see: -1 is (0, {0)) and 1 is ({0), 0), so once again B. the empty set makes -1 < 1 by rule (2). On the other hand 1 ≤ -1 is the same as saying that 0 ± -1 and 1 ± 0, according to rule (2), but we know that both of these are false. Therefore 1 ± -1, and it must be that -1 < l. Conway's rules seem to be working. A. Yes, but so far we've been using the empty set in almost every argument, so the full implications of the rules aren't clear yet. 22 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 就变成了“甲数小于或相似于乙数，当且仅当·”再后面，他证明了零小于或相似于零，负一小于而不相似于零，依此类推。 A：哦，没错，背定是这样了，他在使用这个词的时候，实际上是采取了种抽象观念，而这种抽象观念必须用法则来加以定义。所以理所当然地，他需要证明0小于或相似于0，这样才可以确定在他的定义中一个数相似于它本身。 B：那末,他的证明行得通吗？由规则（2),他必须证明空集中没有元素大于或相似于0，且0不大于或相似于空集中的任何元素···· 好吧，这显然成立，空集又赢了。 A：更意味深长的是，他可以证明一1不相似于0。要做到这一点，我能想到的唯一办法是他要去证明0 不小于也不相似于一1。我是说，既然由规则（2）可以判定某数是否小于或相似于另一个数，那末若小于或相似于 y不成立,则不小于 y,且不相似于 y。 B：我明白了,我们要证明 0 ≤ －1 不成立。也就是在 = 0 和 YR = {0} 时,应用规则(2),亦即 0 ≤－1 当且仅当 0 ±0。但是显然有0≥0,这是已知的。所以 0 ±一1 成立,他的证明没错。 A：我想知道 Conway 有没有也对－1和 1 的关系也做过同样的检验,我认为他肯定做了,尽管岩石上没有提及此事。如果这些规则真的管用，那背定能找出办法来证明一1 小于 1。 B：嗯,咱们来试试：一1就是（0,{0}),而1就是（{0},0),我们可以用空集定义，借助规则（2）来证得一1<1。反过来说，由规则(2),说 1≤－1成立等于是说 0±－1 和 1±0,但我们已知后面两个命题不成立。因此 1 ±一1,即 1 < 1。Conway 的规则看起来的确管用。 A：对，可是直到目前,我们好像几乎在每一次论证时都用到了空集：所以这些规则完全的涵义我们尚不清楚。 23 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / Have you noticed that almost everything we've proved so far can be put into a framework like this: "If X and Y are any sets of numbers, then = (0, X) and y = (Y,) are numbers, and < y." B. It's neat the way you've just proved infinitely many things, by looking at the pattern I used in only a couple of cases. I guess that's what they call abstraction, or generalization, or something. But can you also prove that your r is strictly less than y? That was true in all the simple cases and I bet it's true in general. A. Uh huh ... I Well no, not when X and Y are both empty, since sa r sn s e e Let's look at the case when X is the empty set and Y is not empty; is it true that 0 is less than (Y, 0)? B. If so, then I'd call (Y,) a "positive" number. That must be what Conway meant by zero separating the positive and nega. tive numbers. A. Yes, but look. According to rule (2), we will have (Y,0) ≤ 0 if and only if no member of Y is greater than or like O. So if, for example, Y is the set {-1}, then (Y,0) ≤ 0. Do you want positive numbers to be ≤ o? Too bad I didn't take you up on that bet. B. Hmm. You mean (Y, 0) is going to be positive only when Y contains some number that is zero or more. I suppose you're right. But at least we now understand everything that's on the stone. A. Everything up to where it's broken off. B. You mean ...? A. I wonder what happened on the third day. 24 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 你发现了没有，我们到目前为止所证明的一切都可以套用同一一个框架：“若 X 和 Y 为任意数集,则 =(O,X)和 =(Y,の)为数,且≤y。” B：真的是很简明扼要的办法，你只是观察了一下我举的几个例子，就推出了无穷多种情形。我觉得这就是人们讲的抽象、概括之类的行为吧。但是，你能否证明你这里的是严格地小于y的？这个命题在所有的平凡案例中都成立,我打赌它应该在一般情况下也都成立。 A：·…··好吧,不过,在 X 和 Y皆为空集时就并非如此,因为那样一来就变成 0 ±0了。不过在其他情形中,就很有意思了。我们先看一下 X 为空集,而 Y 不为空集的情形,这里 0 小于(Y,0) 成立吗？ B：如果成立的话,那我就会将(Y,の)称作“正"数。这肯定是Conway 以零为“正负两界分野之符”想要表达的意思。 A：话是没错,但是你瞧。由规则（2）我们得到，(Y,の）≤0 当且仅当 Y 中没有元素大于或相似于 0。所以,假设,举例来说,Y 表示集合「-1}的话,可就有（Y,の）≤0了。你想要正数≤0 成立吗？真后悔当时没跟你商量打赌的赌注。 B：唔。你是说（Y,の）是正数这个命题仅在 Y包含一些或为零、或比零大的数时才成立。我觉得你是对的。但至少我们现在已经理解石板上的所有内容了。 A：只能说是理解了所有在它断开之前的内容。 B：你的意思是··． A：我想知道在第三日发生了什么事， 25 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Yes, we should be able to figure that out, now that we know the rules. It might be fun to work out the third day, after lunch A. You'd better go catch some fish; our supply of dried meat is getting kinda low. I'll go try and find some coconuts. 26 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：这个应该可以推得，因为我们已经掌握规则了。推算出第三日的情形背定有趣，我们吃完午饭再说吧。 A：你最好去抓几条鱼来,咱们的干肉储备可是一天比一天见少了。我去试着找几个椰子来。 27 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / snin Pr monsd BADNUMBERS B. I've been working on that Third Day problem, and I'm afraid it's going to be pretty hard. When more and more numbers have been created, the number of possible sets goes up fast. I bet that by the seventh day, Conway was ready for a rest. A. Right. I've been working on it too and I get seventeen numbers on thethird day. 28 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 坏数 B：我一直在琢磨那个第三日的问题,我觉得那怕是比较棘手。当更多的数被创造出来的同时,数集的可能状态数目增长非常快。我敢打赌到了第七日，Conway 也该收手歇息了。1 A：没错，我也一直在琢磨这个问题哩，而且我推算出在第三日可以得到七个数。见作者序注2。 29 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Really? I found nineteen; you must have missed two. Here's my list: A. I see you're using the Stone's notation. But why did you include e as a eae aea e e B. Well, we have to test the new numbers against the old, in order to see how they fit in. A. But I only considered new numbers in my list of seventeen, so there must actually be twenty different at the end of the third day. Look, you forgot to include <> in your list. B. (blinking) So I did. Hmm ... 20 by 20, that's 400 different cases we'll have to consider in rule (2). A lot of work, and not much fun either. But there's nothing else to do, and I know it'll bug me until I know the answer. A. Maybe we'll think of some way to simplify the job once we get started. B. Yeah, that would be nice. Well, I've got one result, 1 is less than ({1}, 0). First I had to prove that 0 ± ({1),0). A. I've been trying a different approach. Rule (2) says we have to test every element of XL to see that it isn't greater than or like y, but it shouldn't be necessary to make so many tests. If 30 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：真的吗？我得到了十九个，你肯定是漏掉了两个。这是我得到的　数列成的表： A：我明白，你用的是岩石上的记法。但是为什么把《：》也包括进去呢？这个数不是在首日就创造出来了吗？ B：嗯，我们必须经由旧数来验证新数，以此来决定它们应该填到什么位置。 A：但是我在我的十七个数构成的数表中只考虑了新数，所以在第三日结束时，肯定总共有二十个不同的数了。瞧，你把这个数漏掉 ( B：（舵巴巴眼晴）嗯,我还真是把这个给忘了。这么来20　乘以20，根据规则（2），总共就有400种不同的情况需要我们去考虑了。这里面有大量的工作要做，也没啥乐趣可言。但是在这里又没有别的事情可做，而且我要是不能了解答案的话，心里会，一直不舒服的。 A：也许我们在动手之前就得想点儿办法，来简化这个工作。 B是啊，能简化的话就好了· 好,我已经得到了一个结果,1小于（{1},0)。首先我要证明的是 0({1), 0)。 A：我是从另外一个角度来考虑的。规则（2）说，我们必须检测X 中的每一个元素是否不大于或相似于 y,但实际上并不需要做那么多检测。如果 31 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / any element of XL is ≥ y, then the largest element of XL ought to be ≥ y. Similarly, we need only test against the smallest element of YR. B. Yeah, that oughta be right. ... I can prove that 1 is less than ({0, 1}, 0) just like I proved it was less than ({1}, 0); the extra"o" in X didn't seem to make any difference. A. If what I said is true, it will save us a lot of work, because each number (XL, Xr) will behave in all ≤ relations exactly as if XL were replaced by its largest element and Xr by its smallest. We won't have to consider any numbers in which X or Xr have two or more elements; ten of those twenty numbers in the list will be eliminated! B. I'm not sure I follow you, but how on earth can we prove such a thing？ What we seem to need is something like this: A. if and ‘z then 。 (T1） I don't see that this follows immediately, although it is consis-tent with everything we know. B. At any rate, it ought to be true, if Conway's numbers are to be at all decent. We could go ahead and assume it, but it would be neat to show once and for all that it is true, just by using Conway's rules. A. Yes, and we'd be able to solve the Third Day puzzle without much more work. Let's see, how can it be proved? .. B. Blast those flies! Just when I'm trying to concentrate. Alice can you- no, I guess I'll go for a little walk. Any progress? 32 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / XL 中的任意元素都满足 ≥,则 XL 中的最大元素就应该满足 ≥。相似地，我们只需要将与 YR 中的最小元素比较大小即可。 B：嗯,这应该没错儿···我可以证明1小于（{0,1}，の）,就像我刚才证明1小于（{1},の）那样。那个在 XL中多出来的 0似乎没有造成任何结果上的不同， A：如果我所说的方法成立,那真的可以省不少事儿，因为（XL，XR）在 ≤ 关系下的表现,与将 XL 替换为其最大元素、XR 替换为其最小元素时一模一样。这么一来,我们就用不着考虑那些 XL 或 XR 包含两个或两个以上元素的任何数了。这么一来,现有数表中的二十个数中，一下子能去掉十个！ B：我不确定是不是跟上了你的思路，但是到底怎样才能证明这么- 件事呢？ A：我们所需要的，看起来是类似于这样的论断：若，则．（T1）我还看不出如何能够立即推得这个结论，但它与我们已知的知识是相一致的。 B：无论如何，这个都应该成立，我是说如果Conway 的数概念从根本意义上合理的话。我们当然可以不管三七二十一，直接假定它成立，但还是仅使用 Conway 的规则一次性地证明它对所有情况都成立，会更合理一些。 A：那肯定啊,这么一来解决第三日的问题也就不需要花费太多额外的力气了。那么,这个该怎么证明呢·……·· B：该死的苍蝇！我刚想集中精力就飞来捣乱。Alice,你能不能· 啊呵，我想我得出去溜哒一圈儿。............ 你搞出点儿名堂了吗？ 33 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. No, I seem to be going in circles, and the ± versus ≤ is confus-ing. Everything is stated negatively and things get incredibly tangled up. B. Maybe (Tl) isn't true. A. But it has to be true. Wait, that's it! We'll try to disprove it. And when we fail, the cause of our failure will be a proof! B. Sounds good -- it's always easier to prove something wrong than to prove it right. Suppose we've got three numbers , y, and z for which A.1 ≤， and ， and z. l sn e e s t () saop e B. It says that XL， and and and YR， Y， y?ZR， and then also r ± z, which means what? A. It means one of the two conditions fails. Either there is a num- n st o r for which ≥ zR. With all these facts about &, y, and z, we ought to be able to prove something. B. Well, since rl is in XL, it can't be greater than or like y. Say. it's less than y. But y ≤ z, so L must be ... no, sorry, I can't use facts about numbers we haven't proved. Going the other way, we know that y ≤ z and z ≤ L and y ± L; so this gives us three more bad numbers, and we can get more facts again. But that looks hopelessly complicated. 34 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：没有，我觉得好像在兜圈子。又是±又是<的，搞得我十分头大。而且所有的内容都是负向表述的，简直一团糟啦！ B：也许（T1）并不成立呢？ A：但是它肯定成立。等一下,我明白了！我们先试着证伪它。如果　我们发现无法证伪，那么导致这个结果的原因即为所证！ B：听上去不错 -证明某个命题是错的，总比证明它是对的要容易吧。 A：设有三数 α、y和 z满足 y，且 y，具规则（2）有没有就这种“坏数”让我们讨论呢？ B：它只指明了　　 XLy，且且且 “ YL ， yZR，并且还有车，这说明什么呢？ A：这意味着两个条件中的有一个不成立。不是 XL中的某个元素满足了≥,就是 ZR 中的某个元素满足了 ≥R。在有了关于、和的这么多题设的前提之下,我们应该能够证明出一些什么吧。 B：嗯，αL 既然是 XL中的某个元素，它就不能大于或相似于 ,即它小于。而又 y≤ ,所以 L 肯定……·不,不好意思,我不能利用我们还没有证明过的数的性质。换个思路,我们有≤,且≤，且。这么一来我们又得到了三人坏数，并且我们还可以重复这个过程以得到更多的条件。但这看起来真是复杂到了令人绝望的程度。 35 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. Bill! You've got it. B. Have I? A. If (r, y, z) are three bad numbers, there are two possible cases. Case 1, some &L ≥ z: Then (y, z, &L) are three more bad num-bers. Case 2, some zR ≤ z: Then (zR, &,y) are three more bad num. bers. B. But aren't you still going in circles? There's more and more bad numbers all over the place. A. No, in each case the new bad numbers are simpler than the. original ones: one of them was created earlier. We can't go on and on finding earlier and earlier sets of bad numbers, so there can't be any bad sets at all! (brightening) Oho! What you're saying is this: Each num-B. ber r was created on some day d(). If there are three bad numbers (r, y, z), for which the sum of their creation days is d() + d(y) + d(z) = n, then one of your two cases applies and gives three bad numbers whose day-sum is less than n. Those, in turn, will produce a set whose day-sum is still less, and so on; but that's impossible since there are no three numbers whose day-sum is less than 3. Right, the sum of the creation days is a nice way to express A. the proof. If there are no three bad numbers (α, y, z) whose day-sum is less than n, the two cases show that there are none whose day-sum equals n. I guess it's a proof by induction on the day-sum. B. You and your fancy words. It's the idea that counts A. True; but we need a name for the idea, so we can apply it more easily next time. B. Yes, I suppose there will be a next time. 36 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：Bill！你成功啦。 B：我哪有啊？ A：设（α,y,z）是三个坏数,则有两种可能情况情况 1,某些 ≥:这种情况下，（y,,）又成为三个坏数。情况 2,某些 R≤:这种情况下，（zR,,）又成为三个坏数 B：可是即便这样，你不是仍然在兜圈子吗？这样会产生越来越多的坏数,越来越乱了。 A：并非如此,两种情况中的每一种,新的坏数都比原先的更简单,并且其中一个比其他两个更先创造出来。我们寻找坏数的重复过程是不能像这样永无止境地进行下去的，所以坏数集合根本就不存在！ B：（眼前一亮）哦！你想说的是：每个数都是在某日 d(α）被创造出来的。如果存在三个坏数（,,z），它们的创造日之和为 d（α）+d（）十d（）= n,然后根据你说的两种情况之一,得到三个坏数的创造日之和小于 n。而这个结果又会产生一个新的创造日之和更小的集合，依此进行。但是,这里面存在矛盾，因为不存在这样的三个数,使得它们的创造日之和小于3。 A：对,通过创造日之和来证明是一种很妙的方法。若没有三个坏数 (α,y,2）的创造日之和小于n,则可能的两种情况说明,也没有创日数之和等于 n 的坏数能够满足题设。我猜想这应该是一种基于创造日之和的归纳法来证明的。 B：你说得太棒了。这就是它得以运作的思想。 A：没错，但我们需要为这个思想起个名字，这样下次就可以直接应用它了。 B：对,我相信它一定还会被再次用到的···· 37 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / Okay, I guess there's no reason for me to be uptight any more about the New Math jargon. You know it and I know it, we've just proved the transitive law. A. (sigh) Not bad for two amateur mathematicians! B. It was really your doing. I hereby proclaim that the transitive law (T1) shall be known henceforth as Alice's Theorem. A. C'mon. I'm sure Conway discovered it long ago. great mathematician started by rediscovering a bunch of “well knownresults. Gosh, let's not get carried away dreaming about greatness! Let's A. just have fun with this. 38 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 好了，我觉得不必为这个当代数学的行话而绞尽脑汗汁了。你我都知道这是什么，我们刚才证明的是传递律。 A：（叹了口气）对于两个业余数学家来说，这也就颇为不易了！ B：这都是你的功劳。在这里我隆重宣布，传递律（T1）今后将被称为Alice定理。 A：得了吧。Conway 肯定早就发现这个定理了。 B：但是那对你探索的创新性有丝毫的减损吗？·我敢说任何伟大的数学家都是从重新发现一堆“众所周知”的结果起步的。 A：啊呀，我们就不要被这种成为伟人的白日梦弄得神魂颠倒啦。我们只是从这里面找些乐子罢了。 39 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 5 a PROGRESS B. I just thought of something. Could there possibly be two num bers that aren't related to each other at all? I mean y and y， like one of them is out of sight or in another dimension or some-thing. It shouldn't happen, but how would we prove it? 40 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 5 进展 B：我刚刚想到-一件事。有没有可能存在两个彼此毫无关系的数我的意思是 ry 有 y车，就好像一个数对另一个视而不见，仿佛身处异次元空间似的。这种情况理应是不存在的，但我们又该如何证明它的确不存在呢？ 41 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. I suppose we could try the same technique that worked before. If and y are bad numbers in this sense, then either some EL ≥ y or ≥ some yR. B. Hmm. Suppose y ≤ &L. Then if L ≤ , we would have y ≤ by our transitive law, and we have assumed that y . So&L . In the other case, yR ≤ , the same kind of figuring would show that y yR. A. Hey, that's very shrewd! All we have to do now, to show that such a thing can't happen, is prove something I've suspected for a long time. Every number must lie between all the elements of its sets XL and XR. I mean, XL≤ and ≤XR. B. That shouldn't be hard to prove. What does L ± say? (T2) A. Either there is a number LL in XLL, with LL ≥ , or else there is a number &R in XR with &L ≥ &R. But the second case can't happen, by rule (i). B. I knew we were going to use rule (1) sooner or later. But what can we do with LL? I don't like double subscripts. A. Well, &LL is an element of the left set of rL. Since &L was cre-ated earlier than r, we can at least assume that LL ≤ L, by induction. B. Lead on. A. Let's see, LL ≤ &L says that LLL + &L and .. B. (interrupting) I don't want to look at this -your subscripts are getting worse. A. You're a big help. B. Look, I am helping, I'm telling you to keep away from those. hairy subscripts! 42 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：我觉得我们可以尝试一下之前发挥过作用的技术。如果 α 和 y 是这种意义下的坏数,那末,要么有某些 L≥,要么有 ≥某些YR。 B：唔。假设 ≤好了,若 ≤,根据我们的传递律,有≤。但我们已经假定了±,那就是说 ±。再看另一种情况， yR,同理可得 yyR。 A：嘿，这个结果可真是富有洞见呀！这下子，我们要做的工作就变成了这样：为了证明某个命题不成立，我们就要证明某个我很久，以来都疑心为真的命题。任何数都必须位于它的左集和右集的所有元素之间。即， Xr≤α且　≤Xr. B：那个应该不难证明吧。L ± 实质上是在说什么呢？ (T2） A：或在 XLL 中有某个数 LL 满足 LL ≥ ,亦或在 XR 中有某个数 R 满足 L≥ R。但是根据规则(1),后一种情况是不可能出现的。 B：我就知道早晚还是会用到规则（1）的。但是我们能对LL做点儿加工么？我讨厌双重下标。　 A：那个嘛,LL是 L 的左集中的一个元素。由于 L 比先创造出来，我们至少可以假设LL≤，运用归纳法。 B：你说下去。　 A：我们继续,L≤ 表明 LLL ,还有..． B：（打断）我可不想看到事情变成这样一一你的下标变得更不像话了。 A：你可是帮了大忙了。　 B：对，我是在帮忙,我是想让你远离这种纠缠不清的下标形式！ 43 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. But I ... Okay, you're right, excuse me for going off on such a silly tangent. We have α ≤ &LL and LL ≤ &L, so the transitive law tells us that ≤ L. This probably gets around the need for extra subscripts. B. Aha, that does it. We can't have < L, because that would mean XL ± &L, which is impossible since L is one of the ele-ments of XL. A. Good point, but how do you know that ≤ L. B. What? You mean we've come this far and haven't even proved that a number is like itself? Incredible ... there must be an easy proof. A. Maybe you can see it, but I don't think it's obvious. At any rate, let's try to prove (T3) This means that XL ± a and a + XR. B. It's curiously like (T2). But uh-oh, here we are in the same spot again, trying to show that ≤ L is impossible. A. This time it's all right, Bill. Your argument shows that r ≤ & implies aL ± L, which is impossible by induction. B. Beautiful! That means (T3) is true, so everything falls into place. We've got the " XL ≤ " half of (T2) proved -- and the other half must follow by the same argument, interchanging left and right everywhere. A. And like we said before, (T2) is enough to prove that all num- bers are related; in other words if 44 y， then h 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / (T4） A：但是我··好吧，你是对的,请原谅我在这丛乱象中纠缠了那么久。我们有 ≤和,根据传递律可以得出。这样做，也许就可以绕过额外的下标了吧。 B：啊哈,结果出来了。≤ L 是不成立的,因为这个不等式意味着 XL,而后者是矛盾的,因为 L 是 X 中的一个元素呀。 A：说得好，但是你又如何知道 L≤L。 B：什么？你是说我们做了这么多工作，却还连一个数相似于其自身都还没证出来？真是难以置信····这个证明肯定易如反掌吧． A：大概你已经想到证法了，但在我看来这并不是显而易见的。无论如何，我们来试证一下 . 这就是说,X 且εXR。 (T3） B：这和（T2）惊人地相似啊。但是，啊呀,我们又绕回来了，还是要证明 ≤不成立。 A：这回没问题了,Bill。你的论据显示 ≤ L 蕴涵了 L αL,而这一点可以由归纳法证明不成立。 B：漂亮！这就意味着（T3）成立,现在一切各就各位了。我们终于证明了（T2）中“XL≤α”这一半，而另一半肯定遵循着相同的论据，只是在各处将左右对调即可。 A：然后就像我们先前所说的，（T2）的成立已经足以证明所有的数都相关，亦即若 y，则 y. 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / (T4） 45 B. Right. Look, now we don't have to bother saying things so indi- rectly any more, since "r ± y" is exactly the same as "r is less than y. A. I see, it's the same as "a is less than or like y but not like y." We can now write >ε in place of ± y, and the original rules (1) and (2) look much nicer. I wonder why Conway didn't define things that way? Maybe it's because a third rule would be needed to define what"less than" means, and he probably wanted to keep down the number of rules. B. I wonder if it's possible to have two different numbers that are like each other. I mean, can we have both ≤ y and ≥ y when Xl, is not the same as YL? Sure, we saw something like that before lunch. Don't you re- A member, we found that 0 ≤ y and y ≤ 0 when y = ({-1),0) And I think ({0, 1), 0) will turn out to be like ({1), 0) B. You're right. When r ≤ y and r ≥ y, I guess α and y are effec-tively equal for all practical purposes, because the transitive law tells us that α ≤ z if and only if y ≤ z. They're interchangeable. A. Another thing, we've also got two more transitive laws. I mean if if > ≤ and and y， <， then then > (T5）(T6) Very nice ---in fact, these both follow immediately from (Ti), if B. we consider " < y" equivalent to "r y." There's no need to use (T2), (T3), or (T4) in the proofs of (T5) and (T6). A. You know, when you look over everything we've proved, it's really very pretty. It's amazing that so much flows out of Con-way's two rules. 46 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：对的。你瞧,现在我们用不着啰啰嗪嗪、吞吞吐吐地说话了，因为 “”和“ 小于 ”是一模一样的意思。 A：我懂了,也就是说“α 小于或相似于 y,但是又不相似于 ”。我们现在尽可以把“”直接写作 > 并且最原始的规则（1）和(2)看起来也顺眼多了。我在想： Conway 为什么不直接那样给出定义呢？也许因为这么一来就需要第三条规则来定义何为“小于”，而他想尽可能压低规则的数目。 B：我在想,是否有可能存在两个不同的数彼此相似。就是说 ≤y 和 ≥ y 都成立,但 X不同于 Y呢？ A：肯定有的，我们在午饭前看到过这种现象的。你不记得了吗，我　们曾发现当 y =（{-1},の）时,有 0 ≤ y 且 y ≤ 0,并且我想（0,1},の）和（{1},0）也是这样的关系。 B：你说得对。当 ≤y和 ≥ 都成立时,它们在实践上完全可以说是相等的,因为由传递律可以推出≤当且仅当≤,两者在这里能够互换的。 A：还有，我们能够得到另外两条传递律。我是说　若若 B：真不错 > 且 y， y<，则则。 (T5）(T6）一事实上,如果我们考虑到“:0:>I so seven of the numbers are placed already and it's just a matter of fitting the others in. You know, now that it's getting easier, this is much more fun than a crossword puzzle. A. We also know, for example, that 《> lies somewhere between -- and I. Let's check it against the middle element, zero. Hmm, it's both < and ≥ O, so it must be like O, according to B.. rule (2). As I said yesterday, it's effectively equal to O, so we might as well forget it. That's eight down and twelve to go. A. Let's try to get rid of those ten cases where Xr. or Xp have more than one element, like I tried to do yesterday morning. I had an idea during the night, which might work. Suppose r - (XL, Xr) is a number, and we take any other sets of num-bers YL and YR, where YL<< <0:>1<1 所以，已经七个数各就各位了，而问题就在于将其他的数放入合适的位置。这么一来,现在要完成这件事已经容易多了，这可比纵横字谜要好玩。 A：我们知道，举例来说 <:> 位于一和之间的某个位置,我们就把它和中间的元素,也就是零,进行一下比较吧。 B：嗯,它既满足<0 又满足>0,根据规则（2）,它肯定相似于0。正如我昨天说的那样，它实际上就等于0，那未我们也把它忘掉就可以了。现在有八个数已经安排妥当，还有十二个。 A：我现在试着把那些XL或XB包含多于一个元素的种情况处理掉，就像我们昨天早上尝试过的那样。我昨晚产生了一个想法，也许会有用。假设=（XL，XR）是个数,那么我们任意另取两个数集YL或YR满足 YL< But that's easy, now, since we know that XL < , YL < , and z =<0:1> and so on. B. So we're left with only two cases: > and <: A. Actually, (T7) applies to both of them, too, with r = O! B. Cle-ver. So the Third Day is now completely analyzed; only (T7) sers iaas s 54 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 那么，我认为在这种情况下有相似于，如果有 Z = (XL UYL,XRUYR) 换句话说,采用将数字加到适当一一侧的手法,对 XL 和 XR 进行扩张,其实改变不了。 B：我来瞧瞧,这听起来似是而非。无论如何,根据规则（1),z的确是一个数,所以它迟早会被创造出来的。 A：为了得到 ≤,我们必须证明 XLUYL< Ｚ=<1:0> 依此类推。川 B：这么一来,我们只剩下两种情况了：《一：》和《:》 A：实际上,这两种情况也都可以适用（T7),只要令 =0! B：真—一聪明。这么一来,现在第三日的情形已经完全分析好了只有我们前面列出的那七个数才是本质上有所区别的。 55 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. I wonder if the same thing won't work for the following days, too. Suppose the different numbers at the end of n days are i2··m Then maybe the only new numbers created on the (n + 1)st day will be (0,[α1]),({a1],{α2]), ...,({αm-1],{cm]),({cm],0) B. Alice, you're wonderful! If we prove this, it will solve infinitely many days in one swoop! You'll get ahead of the Creator him. self. A. But maybe we can't prove it. B. Anyway let's try some special cases. Like, what if we had the number ({ci-1}, [zi+i); it would have to be equal to one of the others. A. Sure, it equals i, because of (T7). Look, each element of XiL is ≤ &i-1, and each element of XiR is ≥ i+1. Therefore, by (T7) we have ri=(XiLU[ci-i},XiR U{ai+i}) And again by (T7) ({Ci-1}, {i+i) = ({ri-1}UXiL, [ai+1}U XiR) By the transitive law, ci = ({αi-}},{ci+1}) B. (shaking his head) Incredible, Holmes! A. Elementary, my dear Watson. One simply uses deduction B. Your subscripts aren't very nice, but I'll ignore them this time. What would you do with the number ({zi-i),{αi+i}) if i < j? 56 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：我想,会不会同样的规则在后续的日子中行不通了。假设经历了 n日时彼此相异的数有 1<2<-.- 6= A. Brilliant! You must be right, because Conway's next rule is -2 = (-XR, -XL). (4) B. So it is! Okay --now we can start adding these numbers. Like. what's 1 + I, according to rule (3)? A. You work on that, and I'll work on 1 + a. B. Okay, I get (10 + 1, 0 + 11, 0). And 0 + 1 is ({0 + 01, 0); 0 + 0 is (0, 0) = 0. Everything fits together, making 1 + 1 - ({1}, 0) = a. Just as we thought, a must be 2! A. Congratulations on coming up with the world's longest proof that 1 + 1 is 2. B. Have you ever seen a shorter proof? A. Not really. Look, your calculations help me too. I get f + 2 - ({2), 0), a number that isn't created until the fourth day. 78 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：好吧，我真是跃跃欲试，看看它会如何运作。我看到，你把它称为规则（3）了。 A：现在,过了第三日时,我们已经知道一共有七个数，也许可以称它们为 0、1、一1、a、b、c,以及 d。 B：不,我有个主意,是不是可以采用一个左右对称的办法来表示 -α<1<-b<0 res'm se m e Such cases would be ruled out by (Ti3), but ... A. But we need (T13) to prove (T14), and (T14) to prove (T13). And (T13) to prove (T12). B. We're going around in circles again. 100 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 因为这一条定理可以立即证明（T12)。 B：我明白了,因为当且仅当 α ≤ y且 y≤ 时,α 三。（T13)看起来也会比较有用。我们不也可以更进一步的结论吗？我是说　若 ≤y且则 +w≤y+z? ≤， A：你说的这个可以直接从（T13）推出,因为 ε ＋ w≤ y＋w w＋y＋y=y＋。 B：好吧,那这样就很好,因为（T13）形式更简单。那么，你既然是公式达人，那你说说（T13）等价于什么？ A：已知 XL<且， and and for all aL in XL and all rr in Xn 102 (-CR)+<0,(-)+>0, 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：啊哈,看起来有一条出路哩，我们可以一举两得地证明这两条定　理！我们可以通过对（,y,2）运用创造日之和的归纳法来证明组合命题“(T13）且（T14)”。 B：（喜不自胜）Alice，你真是人天才！一个超级美丽又动人的天才！ A：慢着，还没大功告成呢。我们最好能证明 -=0. (T15) B：那个减号是什么意思？我们还从来没有把Conway有关减法的规则写下来过呢。 A: -y=+(-y). (5) B：我发现你在（T15）中放入了一个三。好吧,显然 +（一）不可能与 0完全等同，我是说,不可能左集和右集都是空集，除非是 0。 A：根据规则（3)、（4）和（5)，（T15）等价于这个式子： ((X+(-))U((-XR)+)， (XR +(-) U((-XL)+a)) = 0 B：哦,这个看起来可有点儿头疼啊。我们到底怎么才能证明某个数三 0 呢？·…..·根据（T8),y 三0 当且仅当 YL <0 且 0< YR,因为 0 是所有数中首个被创造的。 A：这些都可以从规则（2）立即推得,我的意思是，≤0 当且仅当 YL <0,而 0≤ 当且仅当 0，直且 (R)+<0，(-)+ > 0, 103 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Hmm. Aren't we allowed to assume that L + (-L) = 0 and R+(-R)=O？ A. Yes, since we can be proving (T15) by induction. B. Then I've got it! If &L + (-α) were ≥ O, then (-X)R + L would be > O, by definition. But (-X)R is -(XL), which con tains -L, and (-L)+L is not > O. Therefore &L+(-) must be < O, and the same technique works for the other cases too. A. Bravo! That settles (T15). B. What next? A. How about this? (-a) = 。 B. Sssss. That's trivial. Next? A. All I can think of is Conway's theorem. (+y)-y=ε. B. What's that equivalent to? (T16)(T17） A. It's a real mess. ... Can't we prove things without going back to the definitions each time? B. Aha! Yes, it almost falls out by itself: (+y) -y= (α+y) +(-y) = r+ (y+ (-y) =+(yy) =+0 = by (5） by（Tii） by (5) by (T12) and (T15) by (T10)． We've built up quite a pile of useful results - even the associa-tive law has come in handy. Thanks for suggesting it against my better judgment. 104 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：唔。我们可不可以假定＋（-）三0且（-）三O？ A：可以吧，只要我们可以运用归纳法把（T15）证明出来。 B：那我就明白了！如果 L ＋（-α）是 ≥0 的,那么根据定义： (-X)R十 L就应该是 >O 的。但是（一X)R 就是－(XL),后者包含－,而（）并非>的。所以,＋（）肯定是 < 0 的,运用同样的技术,另一种情况也可以得到证明。 A：太棒了,这么一来（T15）就得证了． B：下一条定理是什么? A：这一条怎么样？ =() B：切,这个不值一证嘛。下一条呢？ A：我能想到的就是Conway 的定理： (α+y) -y=. B：它等价于什么？ (T16）(T17） A：这真是一塌糊涂了·我们难道就不能不要每证一条定理都要回到原始定义去吗？ B：啊哈！可以的,这条定理几乎都不证自明了： (+y)-y=(α+y)＋(-y) = ε+(y+(-y)) =+0 = 我们已经建立了颇一整套实用的规则根据（5）根据（T11）根据（T12）和（T15）根据（T10)．即使是结合律也凑了个数帮上了忙。多亏你的建议加上了这一条，才避免让我自作聪明了回。 105 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. Okay, we've probably exhausted the possibilities of addition, negation, and subtraction. There are some more things we could probably prove,like (α+y)(-r)+(-y)， if≤y， then y， (T18）(T19） but I don't think they involve any new ideas; so there's little point in proving them unless we need 'em. B. Nineteen theorems, from just a few primitive rules A. Now you must remember your promise: This afternoon we take a vacation from mathematics, without looking at the rest of the stone again. I don't want that horrible multiplication jazz to rob you of any more sleep. B. We've done a good day's work, anyhow -all the problems are resolved. Look, the tide's just right again. Okay -the last one into the water has to cook supper! 106 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：好，我们很可能已经穷尽了加法、求相反数和减法的所有可能的定理了。也许还有更多我们可能可以证明的，比如 -(α＋y)=(-α)+(-y) 若 y 则 ‘-- (T18)(T19) 但我不觉得这些定理中包含任何新的思想，所以除非很有必要给出证明就是没多大意义的事了。 B：九条定理，仅仅从寒寒几条规则就可以导出来。 A：现在你得兑现你的承诺：今天下午我们彻底从数学中解放一个下午，并且不再朝岩石的其余部分多看一眼。我可不想让那段有关乘法的鬼画符再让你不得安眠，啊阿。 B：无论如何，我们今天的成果很不错　一一所有的问题一扫而光。你　看，一切再次运行如仪。好了— 晚下水的那个今天晚上做饭！ 107 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / THEPROPOSAL A. That sure was a good supper you cooked. (lying down beside her) Mostly because of the fresh fish you B. caught． What are you thinking about now? (blushing) Well, actually I was wondering what would happen if A. I got pregnant. 108 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：你做的晚饭真香啊。缘定 B：（躺在她旁边）还不多亏你抓来的新鲜鱼儿。你现在想什么呢？ A：（脸红）那个,其实我在想假如我有了孩子会是怎么样的。 109 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. You mean, here we are, near the Fertile Crescent, and . -.?. A. Very funny. And after all our work to prove that 1 + 1 - 2 we'll discover that 1 + 1 = 3. B. Okay, you win, no more jokes. But come to think of it, Con-way's rules for numbers are like copulation, I mean the left set meeting the right set, -.. A. You've got just one thing no, two things--on your mind. But seriously, what would we do if I really were pregnant? B. Well, I've been thinking we'd better go back home pretty soon anyway; our money's running out, and the weather is going to get bad. Actually, I really want to marry you in any case, whether you're pregnant or not. If you'll have me, of course. A. That's just what I feel too. This trip has proved that we're ready for a permanent relationship. I wonder ... When our children grow up, will we teach them our theory of numbers? B. No, it would be more fun for them to discover it for themselves. A. But people can't discover everything for themselves; there has to be some balance. Well, isn't all learning really a process of self-discovery? Don't B. the best teachers help their students to think on their own? A. In a way, yes. Whew, we're getting philosophical B. I still can't get over how great I feel when I'm doing this crazy mathematics; it really turns me on right now, but I used to hate it. A. Yes, I've been high on it, too. I think it's a lot better than drugs; I mean, the brain can stimulate itself naturally. B. And it was kind of an aphrodisiac, besides. 110 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：你不会是说,在这里,在这新月沃土1附近····吧? A：那就有意思了。我们费了这么大的劲儿去证明1＋1=2,到头来　却发现 1＋1=3。　 B：你可真行，不开玩笑了啊。不过回头想想,Conway 的这些有关数的规则还真有点儿像繁衍后代。你看，当左集遇上右集． A：我看你脑子现在只有一个东西一一哦,就算是两个东西吧。但是，说正经的，如果我真有了孩子的话，该怎么办呀？ B：那样的话，我们无论如何就得快点儿回家才行了。我们带出来的钱已经所剩无几，而且天气眼看着也要变糟了。说真的，无论如何我都想和你永结同好，不管你有没有孩子。当然前提是你也想要嫁我。 A：我当然也有同样的感觉。这次旅行已经证明了我们已经为永久的关系做好了准备。我在想······如果我们的孩子长大以后,我们会把这套有关数的理论教给他们吗？ B：不，也许让他们自已去发现会带来更多乐趣。 A：但是人们不可能自已来发现所有-一切，必须要把握好平衡， B：好吧，可是所有的学习不都是一种自我发现的过程吗？最好的老师不都是去培养学生自行思考的吗？ A：某种意义上来说,的确如此。哎呀,我们怎么都哲学兮兮的了。 B：我仍然沉浸在这些不着边际的数学工作所带来的成就感中，它现在给我了如此的启发，但过去我可是很讨厌它的。 A：是的,我也很为它兴奋,这可比嗑药管用。我的意思是，大脑靠它天生的力量就可以模拟它自身。 B：而且也让人心生情呢！ 1新月沃土（Fertile Crescent）为中东两河流域沿岸之大片垦地，是人类的重要发源地之一。 111 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / (gazing at the stars) One nice thing about pure mathematics. A. the things we proved today will never be good for anything, so nobody will be able to use them to make bombs or stuff like that. Right. But we can't be in an ivory tower all the time, either B.. There are lots of problems in the world, and the right kind of math might help to solve them. You know, we've been away from newspapers for so long, we've forgotten all the problems. A. Yeah, sometimes I feel guilty about that ... Maybe the right kind of mathematics would help solve some of these problems, but I'm worried that it could also be misused. oeeoe e without tools, but tools can be used for bad things as well as good. If we stop creating things, because they might be harmful in the wrong hands, then we also stop doing useful things. A. Okay, I grant you that pure mathematics isn't the answer to everything. But are you going to abolish it entirely just because it doesn't solve the world's problems? Oh no, don't misunderstand me. These past few days have B. e e saeae st soeae e os like poetry or painting or music, and it turns us on. Our natural curiosity has to be satisfied. It would destroy us if we couldn't have some fun, even in the midst of adversity. A. Bill, it's good to talk with you like this. B. I'm enjoying it too. It makes me feel closer to you, and sort of peaceful. 112 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：（凝望星空)纯数学的一个好处在于一一我们今天证明出来的成果没有任何的实际用途,所以没有人会用它来制造炸弹或是类似的东西。 B：没错。但是我们也无法一直呆在象牙塔里。这个世界有着种种的问题，而某种正确的数学就有可能为解决它们提供帮助。你也知道，我们好长时间不读书看报，已经忘却了好多问题了。 A：是啊，我有时会对此感觉一些愧· 也许某种正确的数学可以有助于解决这些问题中的一部分，但是我也担心它同样可能被误用。 B：这就是矛盾，这就是两难。没有工具，什么都干不了，但是工具既可以用来千好事，也可以用来干环事。如果我们仅仅因为可能被居心不良的人掌握而被用来干坏事，就停止创造新事物，那未我们同时也就停止制造有用的东西了。 A：OK，我敢向你保证，纯数学不是解决所有问题的办法。但是你不会因为它不能解决这个世界的问题，就把它完全抛弃了吧！ B：哦，不，请不要误解我的意思。过去的几天里，我已经感受到了纯数学的美一一它是一种类似于诗歌、绘画和音乐的艺术形式,并且启发了我们。我们天生的好奇心得到了满足。如果一直不能找到用来取乐的事情，这会毁了我们的，即使在苦难中也不例外。 A：Bill,能和你这样交谈我感觉好开心。 B：我也很开心。这种谈话拉近了我们的距离，而且让我感觉到了内心的安宁。 113 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 12 B. Are you awake already? DISASTER A. About an hour ago I woke up and realized that there's a big: gaping hole in what we thought we proved yesterday. B. No! A. Yes, I'm afraid so. We forgot to prove that r + y is a number 114 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 12 B：这么早你就醒啦？灾难 A：一小时前，我醒来时意识到，在我们以为昨天已经证明好了的东西里有个巨大的漏洞。 B：不是吧！　 A：恐怕是的。我们忘记证明 ε十y是个数了。　 115 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. You're kidding. Of course it's a number, it's the sum of two numbers! Oh wait, I see ... we have to check that rule (1) is satisfied. A. Yes, the definition of addition isn't legitimate unless we can prove that XL +y < XR +y, and XL +y< YR + , and YL +< XR+y, and YL+ 0, and 2r = 2 for all pue‘ l of pe“ B. I see. For all positive integers m and n, m n is the minimum of m and n. That's commutative and associative, all right. So your operation did turn out to be interesting 116 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B：你在说笑吧。这当然是个数，它不是两个数的和吗？等一下,我　明白了..我们得看一下它是否满足规则（1)。 A：对,加法规则不一定合法,除非我们可以证明 XL＋0 的数,有 1④ = 1;以及对于所有满足 α ≥1 的数,有 2甲 = 2;以及对于所有满足 ≥2 的数,有 3④=3;以及· B：我懂了。对于所有的正数 m 和 n 而言,m④n 是 m 和 n 中较小的那个数。这也是满足交换律和结合律的。所以，你发明的这个田好像真的还有点儿意思。 117 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A. Yes, and = . But then I tried (-), and I was stopped cold. B. You mean ...? I see, (-)由= ({(-1)由0],[0由1]), which is ({0], {0]) A. And that's not a number. It breaks rule (1). B. So your definition of 田 wasn't legit A. And I realized that you can't just go making arbitrary defini- tions; they have to be proved consistent with the other rules too Another problem with was, for example, that (f-1, 0) = 0 but ({-1),0)由10由1. B. Okay, is out, but I suppose we can fix up the real definition of +. A. I don't know; what I've just told you is as far as I got. Except that I thought about pseudo-numbers. B. Pseudo-numbers? A. Suppose we form (XL, Xr) when XL is not necessarily < XR. Then rule (2) can still be used to define the < relation between such pseudo-numbers. B. I see ... like ({1], {0]) turns out to be less than 2. A. Right. And I just noticed that our proof of the transitive law (T1) didn't use the ± part of rule (1), so that law holds for pseudo-numbers too. B. Yes, I remember saying that the full rule (1) wasn't used until (T2). That seems like a long time ago. Now get ready for a shock. The pseudo-number ({1], [0]) is A. neither ≤ o nor ≥ o! B. Far out! A. Yes, I think I can prove that ({1}, fo)) is ≤ a number y if and s. 'o > J! ruo pue J! aqunu z st pue ' < f J! ruo not related at all to any numbers between O and 1. 118 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / A：对,还有是=。但是接下来,我尝试了（一）,顿时心灰意冷了。　 B：你的意思是·？　我明白了，（-)由=（{(-1)田0},[0由1}),也就是（{0},{0)。 A：但这并非一个数。它不满足规则(1)。 B：所以你这个④ 的定义并不合法。 A：然后我意识到，并不能任意地发明定义。他们必须和其他规则保持相容才可以。关于 ④,还有一个问题,比如，（{-1},0）三 0,可是（{-1},0)10田。 B：好吧，④ 出局了,但是我觉得我们肯定能研究出十的真正定义： A：我不确定。我刚才告诉你的就是我刚才研究出来的所有东西了除此之外,我还冒出了个叫做伪数的想法。 B：伪数？ A：假设我们造出一个数对（XL,XR),但是 XL 不一定 1,以及满足 ≥ 数当且仅当 <0。但它与 0 和 1之间的任何数都没有关系。 119 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / B. Where's the pencil? I want to check that out. ... I think you're right. This is fun, we're proving things about quantities that don't even exist. isian seo ssel e sia saqopnasd op m What you mean is, we're proving things about quantities that are purely conceptual, without familiar real-world counterparts as aids to understanding. . .. Remember that V-i was once thought of as an "imaginary" number, and V2 wasn't even con sidered to be “rational." B. Conway's rule for adding ordinary numbers also gives us a way to add pseudo-numbers. I wonder what this leads to? If = ({1},{0]), then 1 + is ... ({2},{1}). A. And α + r is ({1 + a),{c)), a second-order pseudo-number. B. Pure mathematics is a real mind-expander. But did you notice that ({1}, {0}) isn't even ≤ itself? A. Let's see; ≤ r means that XL < r < XR, so this could only be true if XL ly+yL alyL? This is like ... yeah, it's equivalent to (α - L)(y -- yL) > 0. A. That's it, the product of positive numbers must be positive! The other three conditions for y to lie between its left and 164 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 为什么我们不学学他在定义加法时采取的做法呢。我的意思是， y 应该位于 XLyUYL 和 XRyUYR 之间。至少,不考负数的话应该这样做吧。 A：但这个定义不就和加法一模一样了吗？那么这样算出来的乘积　就等于加和了。 B：喔,还真是这样...·那好吧，那就只能试着去理解Conway的方　法了，我们来看看那张纸片吧。 A：别嫌不好意思呀，你所持的态度完全正确。还记得我们说过的：有关首先去自已尝试的那些话么？ B：哈哈，那个呀，是我们习得的宝贵一课 A：我最多能够看懂到这个程度：Conway为 cy 的左集选取的是所有形如 ThT- Th+ 的数，而右集则选取形如 &LY+ &YR -- CLYR 或Ry+yR－RR，或 CRY+ CYL - &RYL 的数。你看,左集选取的是“同部”而右集选取的是“异部”。这个定义是不是挺合理的？ B：我想想，这看上去有点儿怪。那个，&y按理说应该是要大于它的左集的,那未，以下不等式是否成立？ y > Ly+yL yL 有点儿眼熟···对啦,这不就等价于 ( - L)(y yL) > O. A：这就对了,正数之积为正嘛。另外三种说明 y 位于其左 165 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / right sets are essentially saying that (R -)(yR y) > O,( L)(yR -y) > O,(αR - x)(y - YL) > 0. Okay, the defnition looks sensible, although we haven't proved anything. ser r ee ae an multiplication, I want to check out a few simple cases just to make sure. Let's see . ay=ya; Oy=0； ly=y． Those were all easy. A. Good, zero times infinity is zero. Another easy result is -(sy) = (a)y。 B. Right on. Look, here's a fun one: =(X(X),(X)X） A. Hey, I've always wondered what half of infinity was. B. Half of infinity! ... Coming right up w 兰 ({1,2,3,4, ...}, [w - l,w - 2,w - 3,w - 4,...})． (T20）(T21）(T22) (T23）(T24） It's interesting to prove that w + w = w. ... Wow, here's another neat result: Ew 三 1.　 166 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 右两集之间的描述本质上就是说 (αR )(yR -y) > O,(α L)(yR y) > O,(αR )(y yL) > O. 哇，定义看上去挺合理，尽管我们还什么都没证明呢 B：在我们将注意力转移到乘法的主要定理证明上去之前,我想先把它在一些平凡的例子上试用一下以确认。比如 ay = ya; Oy = 0； 1y = y．这些都很简单了。 (T20）(T21）(T22） A：不错，零乘以无穷大的结果是零。另一个马上就能得到的结果是 -(ry) = (-r)y B：完全正确。你看，下面这个就有意思了　 (()“())= A：嘿,我一直想不明白，无穷的一半是多少。 B：无穷的一半·马上就算。 2=({1,2,3,4,...}， [w - 1,w 2,w- 3,w - 4,..})．另一个优雅的结果： Ew 1. 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / (T23）(T24） 167 Our infinitesimal number turns out to be the reciprocal of infin ity! While you were working that out, I was looking at multiplica-A. tion in general. It looks a little freaky for pseudo-numbers-I found a pseudo-number p for which ({1l, O)p is not like ((0, 11, 0)p, even though ({1], 0) and ({0, 1), 0) are both like 2. In spite of this difficulty, I applied your Big Picture method and I think it is possible to prove r(y+z)=y+z, r(yz) 三 (ry)z for arbitrary pseudo-numbers, and if> then and y>y, (yy)>o (T25)(T26） (T27) for arbitrary numbers. It will follow that ry is a number when-ever r and y are. B. Theorem (T27) can be used to show that if 三， then (T28） for all numbers. So all of these calculations we've been making are perfectly rigorous. I guess that takes care of everything it says on the tablet. Ex-cept the vague reference to “"series, and quotients, and roots." A. Hmm ... What about division? ... I bet if r is between 0 and l, it'll be possible to prove that + ({, - , 168 · tr rs、 ... 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 我们的无穷小数原来是无穷大数的倒数！ A：在你做出那些结果的时候,我在考虑乘法的一般规律。看上去　它用于伪数时显得比较反常一-我发现对于某伪数 β 而言，([0,1},0)p 并不相似于({0,1},の)p,尽管（{1},の)和（{0,1},0）都相似于2。但先不管这个困难所在,我运用了一下你的全景框　架法以后觉得，对于任何伪数我们可能都可以证明 (y+) =y+z， r(yz) 三 (y)z, 且对于任何数都可以证明若>且>y 则(')(y-y)>0． (T25)(T26） (T27）从这个可以得出推论：y一定是数,而无论和 y是否是数。 B：定理（T27）可以用来证得　若 ε三y，则=z (16.1）对于所有数成立。所以,我们进行过的一切计算在严格性方面堪称完美。　我想，现在石板上的一切都已经完成了。除了那句含糊地提到的“为级数、为商数、为方根”尚无着落。 A：唔··除法是什么样的？··我敢打赌，如果位于0和1之间的话，可能可以证明 r+t ({-2, - 2 +3-, .. {, ²+3, ²+34+ 169 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / At least, this is how we got 号, for = . Perhaps we'll be able to show that every nonzero number has a reciprocal, using some such method. B. Alice! Feast your eyes on this! Vw= ({1,2,3, 4, ..}, rwww{i'2'3'4,... (fe,2e, 3e, 4e,, Ve= (falling into his arms) Bill! Every discovery leads to more, and A. more! (glancing at the sunset) There are infinitely many things yet B.1 to do ... and only a finite amount of time ... 170 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 至少,这是我们如何通过α=计算出的过程。也许我们就用这样的办法，可以证明任何非零数都有倒数。 B：，Alice！看看这个会不会让你大饱眼福！ Va= ((1,2,3,4,小,2,3,4 mr ii１1 Ve= （{e,2e,3e, 4e,小, 24 A：（投入他的怀抱)Bill！每个发现都会导致更多的发现,然后又是更多！ B：（凝视着夕阳）需要做的工作是无穷的·而时间却是有穷的呢 +.. 171 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 172 The reader may have guessed that this is not a true story. How. ever, "J. H. W. H. Conway" does exist - he is Professor John Horton Conway of Cambridge University. The real Conway has established many remarkable results about these "extraordinal" numbers, besides what has been mentioned here. For example, every polynomial of odd degree, with arbitrary numbers as coef-ficients, has a root. Also, every pseudo-number p corresponds to a position in a two-person game between players Left and Right. where the four relations p>o, p= o, “o>d pllo correspond respectively to the four conditions Left wins， Second player wins， Right wins， First player wins sao oo soso es Games (London: Academic Press, 1976), as well as the two-volume sequel by Berlekamp, Guy, and Conway, Winning Ways (London: Academic Press, 1982).1 The theory is still very much in its infancy, and the reader may wish to play with some of the many unexplored topics: What can be said about logarithms? continuity? multiplicative properties of pseudo-numbers? generalized diophantine equations? etc. 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 读者们也许已经猜到,这里所讲述的并不是一个真实的故事。但是，“J.H.W.H.Conway”却实有其人一他就是剑桥大学的 John Horton Conway 教授。现实世界中的 Conway 教授在研究这种具有　“额外数序”之数的方面得到了许多引人注目的结果,包括本书中提到的部分。比如,每个带任意系数的奇次多项式,都有方根。还有,每个伪数 β,都对应着在一对左右玩家之间进行的二人博奔中的一个位置,其中四种关系 p>o， p=o，分别对应四种条件左胜，后者胜， o>d plo 右胜：前者胜而博奔起始于位置β。【参见他的精彩著作《论数字和博奔》（On Numbers and Games, London: Academic Press, 1976), 以及由 Berlekamp、Guy 和Conway 合著的两卷本《取胜之道》（Winning Ways, London: Academic Press, 1982)。 ↓ 整套理论仍然处在非常初步的阶段，读者可能会有兴趣研究大量未知问题中的若干个：在其中如何建立对数概念和连续性概念？伪数在乘法下会表现出哪些性质？还有广义丢番图方程1呢？等等。其中 an，c E Z 的不定方程。方程左边的表达式可能为有限项，也可能为无限项。有关这个主题,可以参阅 W. M. Schmidt, Diophantine approximations and Diophan-tine equations, Springer Verlag, 2000 等。 173 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / POSTSCRIPT The late Hungarian mathematician Alfred Renyi composed three stimulating "Dialogues on Mathematics," which were published by Holden-Day of San Francisco in 1967. His first dialogue, set in ancient Greece about 440 B.C., features Socrates and gives a beautiful description of the nature of mathematics. The second, which supposedly takes place in 212 B.C., contains Archimedes's egually beautiful discussion of the applications of mathematics. Renyi's third dialogue is about mathematics and science, and we hear Galileo speaking to us from about A.D. 1600 I have prepared Surreal Numbers as a mathematical dialogue of the 1970s, emphasizing the nature of creative mathematical explo-rations. Of course, I wrote this mostly for fun, and I hope that it will transmit some pleasure to its readers, but I must admit that I aiso had a serious purpose in the back of my mind. Namely, I wanted to provide some material that would help to overcome one of the most serious shortcomings in our present educational system, the lack of training for research work; there is comparatively little opportunity for students to experience how new mathematics is invented, until they reach graduate school. I decided that creativity can't be taught using a textbook, but that an “anti-text" such as this novelette might be useful. I there- fore tried to write the exact opposite of Landau's Grundlagen der Analysis; my aim was to show how mathematics can be "taken out of the classroom and into life," and to urge readers to try their own hands at exploring abstract mathematical ideas. The best way to communicate the technigues of mathematical research is probably to present a detailed case study. Conway's recent approach to numbers struck me as the perfect vehicle for illustrating the important aspects of mathematical explorations. because it is a rich theory that is almost self-contained, yet with close ties to both algebra and analysis, and because it is still largely unexplored. In other words. my primary aim is not really to teach Conway's theory, it is to teach how one might go about developing such a theory. Therefore, as the two characters in this book gradually explore and build up Conway's number system, I have recorded 174 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 跋话”发表在1967年的旧金山学术期刊Holden-Day上。他写的第一篇对话，背景取自古希腊，时间大约在公元前440年，彰显了苏格拉底这个人物，并出色地描述了数学的本质。第二篇对话，据称发生在公元前 212年，包括了阿基米德同样出色的对数学应用的描述。Rényi的第三篇对话是有关数学和科学的，我们从中聆听到了伽利略在公元1600年向我们发表的高论。我把本书设定为发生在20世纪70年代的一场数学对话，主要强调的是带有创新性的数学探索的本质。当然，我写作这篇东西主要是出于兴趣，并且我也希望它给读者诸君带来快乐，但我必须承认，我的内心深处有着更为严肃的目的。这就是，我想提供一些材料，它们会有助于克服在我们当前的教育系统中存在着的最严重的毛病，即缺乏对于研究工作的训练。相对而言，学生们在研究生阶段之前体验新的数学如何被发明出来的可能，是微乎其微的。能小说体图书却可能奏效。因此，我尝试着采用了一种与Landau的《微积分基础》（Grundlagen derAnalysis）完全背道而驰的方式来写作。我的目标是说明数学家如何能够“走出教室，融入生活”，并且鼓励读者亲手尝试去探索抽象的数学思想。传授数学研究技术的最佳方法，可能就是给出一个详细的案例研究。Conway教授最近有关数的思路触发了我的灵感，让我觉得这就是美同时，又和代数与分析有着紧密的联系，还因为它尚有一大部分是未经探索的。换言之，我的主要目标并不是真的要教给读者Conway教授的理论，而是想让读者学到，一个人要如何着手来研究出这么一套理论来。因此，当本书中的两个主角一点点地进行探索,并建立起 Conway 的数系时，我会记下 175 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / their false starts and frustrations as well as their good ideas and triumphs. I wanted to give a reasonably faithful portrayal of the important principles, techniques, joys, passions, and philosophy of mathematics, so I wrote the story as I was actually doing the. research myself (using no outside sources except a vague memory of a lunchtime conversation I had had with John Conway almost a year earlier). I have intended this book primarily for college mathematics students at about the sophomore or junior level. Within a tradi tional math curriculum it can probably be used best either (a) as supplementary reading material for an "Introduction to Abstract Mathematics" course or a “Mathematical Logic" course; or (b) as the principal text in an undergraduate seminar intended to develop the students' abilities for doing independent work. Books that are used in classrooms are usually enhanced by exercises. So at the risk of destroying the purity of this "novel" approach, I have compiled a few suggestions for supplementary problems. When used with seminars, such exercises should preferably be brought up early in each class hour, for spontaneous . class discussions, instead of being assigned as homework. 1. After Chapter 3. What is "abstraction," and what is "general- ization"？ 2. After Chapter 5. Assume that g is a function from numbers to numbers such that ≤ y implies g(r) ≤ g(y). Define f(α) = (f(XL)U (g(α)), f(XR)) Prove that f(α) ≤ f(y) if and only if ≤ y. Then in the special case that g() is identically O, evaluate f(r) for as many num-bers as you can. [Note: After Chapter 12, this exercise makes Lsaas aeda re sai a se asa 3. After Chapter 5. Let and y be numbers whose left and right n eap g a a s fL: XL Yi， 9L: YL →XL， 176 fR: XR -+ YR. 9R: YR -→XR 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 他们失败的起点以及情绪的焦虑，但也会记下他们有价值的思想以及胜利的喜悦。我想给出一张在相当程度上可信的图景，以展示数学的重要原理、技术、乐趣、热情和哲学，所以我是在-一边写着这个故事的同时，一边自已在做着研究（完全没有参考外部文献，仅凭着对于约一年前我和 John Conway 共进午餐时所进行的一番对话所留下的模糊印象）。本书的初衷主要是为了数学专业的本科生而写，大致对应的年级是大二或大三。如果按照美国传统基础课程的说法,本书既可用作（a)“近世代数导论”课程或“数理逻辑”课程的补充读物，也可用作（b）本科阶段意在培养学生独立工作能力的研讨班所使用的主要教科书。凡用作教学的书，通常都会以练习来增强其功效。所以，冒着破坏这种“小说手法”的纯粹性之大不，我汇集了-一些供大家参考的补充习题。如果将其用于研讨班，则最好能够在每课时前提出，用子课堂讨论，而不要当成作业布置下去。 1．第3章读后。什么是“抽象”，什么是“概括”？ 2．第 5 章读后。设 9 为一函数,会将作为自变量之数映射至所有满足≤y蕴涵g(α)≤g(y)的数。定义 f(α) = (f(XL) U {g(a)),f(XR) 证明 f(α)≤f(y)当且仅当 ≤y。在 g(α)恒等于 o 特殊情形下,对尽可能多的数计算f(c）的值。【附记：第 12 章读后,本练习在把“数”换成“伪数”时仍然有意义。」 3.第5 章读后。令和 y为数，耳其左部和右部“相似”但并不相同。形式上,令 fR : XR -→ YR, fL:XLYL， gL : YL -→ XL, 9R:YR→XR 177 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / be functions such that fL(&L) 三 &L, fR(R) 三 R, gL(yL) = YL， gr(yr) = yR. Prove that = y. [Alice and Bill did not real-ize that this lemma was important in some of their investiga-tions; they assumed it without proof. The lemma holds also for pseudo-numbers.J 4. After Chapter 6. When we are developing the theory of Con-way's numbers from these few axioms, is it legitimate to use the properties we already "know" about numbers in the proofs?(For example, is it okay to use subscripts like i -- 1 and j + 1?) 5. After Chapter 9. Find a complete formal proof of the general pattern after n days. [This problem makes an interesting ex-ercise in the design of notations. There are many possibilities, and the students should strive to find a notation that makes a rigorous proof most understandable, in the sense that it matches Alice and Bill's intuitive informal proofJ 6. After Chapter 9. Is there a simple formula telling the day on which a given binary number is created? 7. After Chapter I0. Prove that = y implies - = -y. 8. After Chapter 12. Establish the value of α y for as many and y as you can. 9. After Chapter 12. Change rules (1) and (2), replacing by < in all three places; and add a new rule: α 当且仅当 y且y．然后使用这些定义,将 Conway 数论从头再建立一遍。［这个问题会弓[发对于第三章材料的一次仔细检讨。论证过程要在若干处加以修订。主要的障碍在于证明对于所有的数,≤ 都成立。有一个相当短小的证明，却不那么容易找到，我也不想在这里公布答案。要鼓励学生们去发现,这个新的<关系和 Conway 的那个对于伪数来说并不完全等同（尽管对于所有的数来说是完全：样的）。在新的前提下,≤这条定律并不总是成立。并且若 = ({({0],[0])),0), 179 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / we have = 0 in Conway's system but r = 1 in the new one! Conway's definition has nicer properties but the new relation is instructive. 10. After Chapter 13. Show how to avoid Alice and Bill's circu- larity problem another way, by eliminating IHI(z, ZL, y) and III(ZR, z, ) from the requirements needed to prove II(r, y, z). In other words, prove directly that we can't have z + y ≤ zL + y for any zL. 11. After Chapter 14. Determine the "immediate neighborhood" of each real number during the first few days after day. 12. After Chapter 15. Construct the largest infinite numbers you can think of, and also the smallest positive infinitesimals. 13. After Chapter 15. Does it sufice to restrict XL and X to countable sets? [This question is difficult but it may lead to an interesting discussion. Instructors can prepare themselves by boning up on ordinal numbers.J l4. Almost anywhere. What are the properties of the operation defined by Toy = (XL NYL, XRnYR)? [The class should discover that this is not min(r, y). Many other operations are interesting to explore, for example when a o y is defined to be (XLoYL,XRUYR) or ((XL Oy) U(C O YL), XR UYR) 15. After Chapter 16. If X is the set of all numbers, show that (X, 0) is not equivalent to any number. [There are paradoxes in set theory unless care is taken. Strictly speaking, the class of all numbers isn't a set. Compare this problem with the “set of all sets paradoxes.J 16. After Chapter 16. Call r a generalized integer if = ({α -1), [α + 1)) 180 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 那么在 Conway 的数系中有三 O,而在新的数系中却成了 α 三 1！Conway 的定义有着更好的性质，但是新的关系却是很有启发性的。」 10．第 13章读后。用另-一种方法说明怎样规避 Alice 和 Bill所遭　遇的循环论证，通过从证明（,，）的前提条件中去除（z,Z，）和（Z,)。换言之,直接证明对于任何，＋≤＋都不成立。 11.第14章读后。推算出在第日后的数日，每个实数的“最近邻数”是多少。 12．第 15章读后。构造你能够想到的最大无穷数，以及最小的正无穷小数。 13.第 15 章读后。将 XL 和 XR 限制为可数集合，是否已经完备？「本问题很有难度，但是可能会引发有意思的讨论。讲师可通过补习序数知识来做课前准备。」 14．几乎在任何章节处。下式定义的运算具有何种性质？ Oy=(XL nYL,XRNYR) ［研讨班应该会发现这个运算并非 min(α,y)。还有很多其他运算探索起来令人兴致大增,比如，当采取如下定义时。 (XLＯYL,XRUYR) 或 ((XL Oy)U (C O YL), XR UYR) 15.第16章读后。若X 是所有数构成的集合,说明（X,0）不等价于任何数。「若不加小心，集合论中就会出现悸论。严格地说。全体数构成的等价类并非集合。将本问题与“全体集合的集合”悖论作一对比。」 16．第 16 章读后。称为一广义整数,若 ({}{})= 181 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / Show that generalized integers are closed under addition, sub- traction, and multiplication. They include the usual integers n, as well as numbers like w ± n, w, etc. [This exercise is due to Simon Norton.l 17. After Chapter 16. Call α a real number if --n < < n for some (nongeneralized) integer n, and if =({,,,..,{+l,+,+,...) Prove that the real numbers are closed under addition, subtrac tion, and multiplication, and that they are isomorphic to real numbers defined in more traditional ways. [This exercise and those that follow were suggested by John Conway.] 18. After Chapter 16. Change rule (1), allowing (XL, Xr) to be a number only when XL Xr and the following condition is satisfied: XL has a greatest element or is null if and only if XR has a least element or is null. Show that precisely the real numbers (no more, no less) are created in these circumstances. 19. After Chapter 16. Find a pseudo-number p such that p + p = ({0], {0]). [This question is surprisingly diffcult, and it leads to interesting subproblems.J 20. After Chapter 15 or 16. The pseudo-number ({0}, {({0}, [0}))) is > O and < for all positive numbers . Thus it is really infinitesimal! But ({0}, {({0}, {-1])]) is smaller yet. And any pseudo-number p > 0 is greater than ({0}, {({0}], {-α})]) for some suitably large number r. 21. After Chapter 16. For any number define w* - ([0)U[nw*L I aL E XL, n = 1,2,3, ...} wER aRe XR, n = l,2,3,... 2n Prove that w'wy - w*+y. 182 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 说明广义整数在加法、减法和乘法下封闭。包括通常意义下的整数 n,还有像u± n 和 u这样的数,等等。［本习题由 Simon Norton 提供。了 17.第16 章读后。当对某些(非广义)整数 n,取使得n<0 并且对所有正数都有< 的。综上，它的确是无穷小的。但是({0),{({0},{一1))）还要更小些。并且，总有充分大的使得任({(-}{o})‘{o}) o 19. Perhaps the simplest answer is p =- ({({1}, {1})}, {-1}). Conway has proved in fact that we can solve the equation p + p = r for any given x by taking p = ({q), [α - q)) where = (XLLU(α+ XRR)U(+XL XR),O) 22. Considering just the members of S created before N day, the unlike numbers are precisely go, g1, 92, 93, , where 9n =({90,...,9n-1), {g90,...,9n-1)) Moreover, the elements (go, : . : , gn-i} form a finite field (under Con-way's addition and multiplication operations), for n = 2, 4, 16, 256. 65536.22，. 188 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 在《研究之美》跋中承诺给出的解法提示 9.欲证明对新数系中的所有数 α,有 α≤ α 成立,则须首先证明辅助引理若YR且则 αy，方法是对于所有确定的 y,对 d(α)进行归纳。 19．也许最简单的答案是 p =（[({1},{1})),{-1})。Conway 业已证明,事实上我们可以对任意的 ,取 p=（{q},{α－q}）作为方程 p+p= 的解,其中 q= (XLL U(+- XRR)U(α+XL - XR),) 22.仅考虑S在第日之前创造的成员,不符合要求的数只能是90， 91，92，93，,其中　 gn =({go,...,9n-1),{go,...,9n--1}) 更进—步,当n= 2,4,16, 256,65536, 22，时,元素{90,···,9n1}（在Conway 的加法和乘法运算下）构成了一个有限域！ 189 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / 反侵权盗版声明电子工业出版社依法对本作品享有专有出版权。任何未经权利人书面许可，复制、销售或通过信息网络传播本作品的行为；歪曲：篡改、剽窃本作品的行为，均违反《中华人民共和国著作权法》其行为人应承担相应的民事责任和行政责任，构成犯罪的，将被依法追究刑事责任。为了维护市场秩序，保护权利人的合法权益，我社将依法查处和打击侵权盗版的单位和个人。欢迎社会各界人士积极举报侵权盗版行为，本社将奖励举报有功人员，并保证举报人的信息不被泄露。举报电话：（010）88254396；（010）88258888 传真：（010）88254397 E-mail:： dbqq@phei.com.cn 通信地址：北京市万寿路173信箱电子工业出版社总编办公室邮编：100036 更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com / [ G eneral I nf orm at i on] 书名=研究之美汉英对照作者=（美）高德纳著页数=190 SS号=12888607 D X号=000008237867 出版日期=2012. 01 出版社=电子工业出版社更多资料请关注我的网易博客ht t p: //nt ccy2009. bl og. 163. com /