◇◇新语丝(www.xys.org)(xys.dxiong.com)(xys.3322.org)(xys.xlogit.com)◇◇ 解说员的临门一脚 ——谁证明了庞加莱猜想？ 三镜堂主 　　2006年6月3日，数学大师、菲尔兹奖得主丘成桐教授在北京宣布：在美、俄 等国科学家的工作基础上，中山大学朱熹平教授和旅美数学家、清华大学兼职教 授曹怀东已经彻底证明了庞加莱猜想。大师说：“这就像盖大楼，前人打好了基 础，但最后一步——也就是‘封顶’工作是由中国人来完成的。”为了让普通人 了解这一猜想的重要性，大师进一步解释道，“这是一项大成就，比哥德巴赫猜 想重要得多。”大师此说的根据何在？何以哥氏就不如庞氏？不得而知。但是只 证明了1+2的陈景润身材无疑是越发的见矮了。 　　由于丘大师的特殊声望，没有人怀疑这一消息的正确性。各大中文报纸网站 纷纷登出振奋人心的醒目标题：悬赏百万美金求解的数学世纪难题被中山大学教 授朱熹平和旅美数学家曹怀东彻底证明 ！ 　　然而，与喜气洋洋的中文媒体形成鲜明对比的是，对这样一条惊天动地的新 闻，国际数学界的反应冷淡到了不可思议的地步。消息传出之后许多天里，用 Google查询Poincaré Conjecture，只能查到这些中文消息的英文翻译。对中国 数学家的惊世之作，国际数学界的集体失语，令人有山雨欲来的不祥之感。难道 真如丘大师所说，前人只是给大楼打了个基础，而中国数学家完成了最后的“封 顶”工作？大家知道，任何一个现代数学难题的最终解决，无不是站在前人的肩 膀上完成的。十年前，普林斯顿大学教授瓦尔斯（Andrew Wiles）寒窗枯坐多年 所完成的费尔马大定理的证明，正是这样的一次“封顶”。 　　直到《华尔街日报》7月21日登载了一篇关于庞加莱猜想的专题文章，所谓 的“封顶”才算是被揭开了神秘的面纱。为方便读者查询，我将英文原文附于文 末，这里只摘要介绍其中与“封顶”有关的信息供读者评估。 　　2002年和2003年，俄国数学家佩瑞曼（Grigori Perelman）在一个存档网站 上贴了两篇论文，给出了庞加莱猜想的证明草稿。他甚至都没有提及庞加莱猜想， 因为他认为他证明的是一个更广泛的命题，庞加莱猜想不过是其中的一个推论而 已。他的论文不是用期刊发表所要求的严谨格式写成的，因而十分晦涩难懂。正 当数学界期待他给出更详细正规的证明时，不按常理出牌的佩瑞曼却如隐士一般 从人间蒸发，不再回应。 　　也许论文的潦草正反应了天才是如何与常人交流的。佩瑞曼可能认为他根本 不需要为那些在他看来显然的结论详加解说，读者如果愚笨到不能填补他的证明 空白，那不是他的问题，与其耗费时间纠缠于那些烦人的细节，不如去做些更重 要的事。 　　数学家们于是试图去填补佩瑞曼论文留下的空白。佩瑞曼2003年的论文只有 22页，2002年的是39页，可是由密西根大学的克莱纳(Kleiner)和劳特(Lott)逐 行加以详解的《佩瑞曼论文注释》(Notes on Perelman’s Papers)却达192页之 长。另一本将要出版的关于佩瑞曼论文的书有三百页之多。 　　文章也提到了两位中国数学家的论文：发表在上个月的《亚洲数学期刊》上、 根据佩瑞曼博士的突破（写成的）庞加莱猜想的一个“完整证明”长达328页。 米尔诺教授称此文是向功劳归属的问题扔了一个“猴子的扳手”。 　　“猴子的扳手”是一句美国俚语，相当于中文里的“搅局”。米尔诺教授可 能只是认为，这篇“完整证明”的仓促发表，意在争夺填补佩瑞曼证明空缺的首 功，破坏了游戏规则。专家们对庞加莱猜想的归属已有公论，不会想到“完整证 明”对于传媒和社会大众还可以有另外一种解释。因为“完整证明”本身就在暗 示此前所有的包括佩瑞曼的证明都是“不完整证明”(incomplete proofs)，一 面不完整的镜子也就是支离破碎的镜子是不能行使镜子的功能的。 　　顺便说一句，丘成桐教授正是《亚洲数学期刊》的主编。米尔诺(John Milnor)是纽约大学石溪分校的教授，杨振宁教授的同事，1962年的菲尔兹奖得 主。愤青们如有砖头尽管砸向jack@math.sunysb.edu 　　文章继续说，奇怪的是，这本书（注：不是指朱-曹的证明）或克莱纳-劳特 的注释却可以作为克莱数学研究院颁奖所需要的参考资料。如此一来，我们陷入 了一个怪圈，写出符合颁奖条件的论文作者们却不是发现证明的人，他们的努力 只不过将帮助佩瑞曼获得一百万美元的奖金。 　　这篇文章明白无误地告诉我们，佩瑞曼不但造好了大楼，而且封了顶。包括 朱、曹在内的数学家们不过给佩瑞曼的大楼铺平了门前的道路，好让克莱数学研 究院的专家前来验收时不至于不得其门而入。 　　如果没有确凿的证据，请不要用“种族歧视”或“妖魔化”做幌子来转移视 线。陈省声“统治”美国数学界几十年，丘成桐获菲尔兹奖，都是有力的反证。 个人愚见，这篇文章浅显易懂，用幽默风趣的语言叙述了庞加莱猜想及佩瑞曼证 明的来龙去脉，可读性极强。诸君不妨一读。 　　让我们回顾一下中国数学界的说法。著名数学家杨乐如此评价道，“这是第 一次在国际数学期刊上给出了猜想的完整证明，成果极其突出。”且不说这个 “第一次”需要佐证，即朱、曹二位的论文确实是率先发表并经专家检验无误， 而且杨院士的结论显然有严重的误导之嫌。读者不会由此想到朱、曹二位只不过 是在解读佩瑞曼的证明。杨院士进一步将庞加莱猜想这块大饼切成了三块，50% 送了汉弥尔顿，佩瑞曼25%，中国数学家得了30%。多出来的5%可能是杨院士自掏 腰包送丘院士的。根据杨教授的评价，我们只能得出中国数学家的贡献比佩瑞曼 高，克莱数学研究院的100万美元奖金非朱曹二位莫属的封顶结论。 　　大家知道，数学是超越意识形态、没有国界人种之别、放之宇宙皆准的学问。 在学科分类中，数学是独立于科学（science）之外的。数学证明的对与错，只 有黑白之分，没有模糊不清的灰色地带。除非佩瑞曼的证明有错，而且朱、曹二 位在他们的“完整证明”中改正了他的错误，否则克莱数学研究院是不大可能将 100万美元的奖金颁给他们的。 　　再以刚刚结束的足球世界杯为例。在电视机前观看比赛的观众包括许多铁杆 球迷，看到的只是封顶的临门一脚，至于球星背景、攻防转换、战略战术甚至比 赛规则其实不甚了了。因此电视解说员的讲解和点评是足球盛宴上不可或缺的一 味佐料。但无论解说员如何鼓动如簧巧舌，球场上的风云变化却不是以他的意志 为转移的。夺冠的意大利队压根儿也没想过要和狂热地爱着他们的中国解说员黄 健翔分享那份丰厚的冠军奖金。 　　可是现在偏偏有人告诉你，那惊心动魄的临门一脚，其实是解说员吼进去的， 因此解说员就是那封顶的功臣。 　　文章写到这里，心里很不是滋味。朱、曹二位都是优秀的数学人才。能成功 地解读佩瑞曼的证明，本来已经是很了不起的成就。比如人人都能从山脚下看到 珠穆朗玛峰，但即使沿着前人的足迹攀登珠峰，也不是人人都能做到的。朱、曹 并不需要用“封顶”来证明他们不世的才华。“封顶”论恐怕也不是二位的初衷。 这大概可以解释为什么两位年轻数学家在别人发布“封顶”的消息时一致地选择 了沉默。 附录：华尔街日报专文 The Wall Street Journal Home Page Friday, July 21, 2006 SCIENCE JOURNAL By SHARON BEGLEY Major Math Problem Is Believed Solved By Reclusive Russian July 21, 2006; Page A9 For six years, $7 million in prize money has lay unclaimed at the Clay Mathematics Institute in Cambridge, Mass., waiting for someone to solve any of the seven "millennium prize problems," the oldest of which has been kicking around since 1859. Despite periodic claims, it looked like the institute would hold on to the cash until after the sun burned out. But the math world is abuzz over the very real possibility that one millennium problem, the Poincaré conjecture, has been proved by a mathematician in Russia. After nearly four years of scrutiny by other mathematicians, the work holds up, even though Grigori Perelman's work is decidedly unusual. In 2002 and 2003, he posted two papers to an online archive. Usually, a posting serves a flag-planting function -- "I solved this first!" -- until the paper is published in a journal, which can take years. But as the math community waited for him to follow up his postings, a realization set in. Dr. Perelman, long affiliated with the Steklov Institute of Mathematics in St. Petersburg, apparently has no intention of saying more. He probably feels he proved the Poincaré conjecture, mathematicians surmise, and has no interest in the $1 million bounty. (He did not respond to emailed requests for comment.) Dr. Perelman's style is reminiscent of the Sid Harris cartoon of a board filled with equations and, at a key step, the words, "then a miracle occurs." One mathematician tells the other, "I think you should be more explicit here in step two." The conjecture Henri Poincaré posited in 1904 is the most famous problem in topology, the branch of math that analyzes the shape of objects and space. He claimed, "if a closed 3-dimensional manifold has trivial fundamental group, [it must be] homeomorphic to the 3-sphere," as John Milnor of Stony Brook University puts it. Translated, that means that if you wrap one rubber band around the surface of an orange and another around a doughnut, and shrink down both, the rubber bands act differently. The one around the orange keeps shrinking without tearing or leaving the surface. The one around the doughnut can't, without breaking itself or the doughnut. This difference says something profound about the structure of space itself. Many mathematicians have claimed to prove Poincaré, but the claims flamed out immediately, their fatal flaws obvious. Dr. Perelman's proof has survived. The dilemma for the Clay Institute is that, according to its rules, a proof must be published in a refereed math publication. The archives aren't refereed. Putting his proof online rather than in a journal is only one example of Dr. Perelman's iconoclasm. He admits that he gives only "a sketch of an eclectic proof of" a more general conjecture from which Poincar é's follows; he never mentions Poincaré. The papers are difficult to understand, and sketchy in the extreme. He asserts that one can prove something by a variation on an earlier argument, but it isn't clear what the variation is. "Perelman's papers are written in a style rather different from what would appear in a journal," says mathematician Bruce Kleiner of Yale University. The sketchiness may reflect how a genius interacts with mortals. Dr. Perelman may believe some things are so obvious he needn't bother to explain them step by step, say mathematicians. If readers are too dumb to fill in the blanks, he doesn't care. Or, he has better things to do than justify every tortuous step, as proofs must. Others have taken it upon themselves to explicate his work -- and find no major flaws. Like Torah commentaries, they dwarf the original. Dr. Perelman's 2003 paper is 22 pdf pages; the 2002 paper is 39. But "Notes on Perelman's Papers," in which Prof. Kleiner and John Lott of the University of Michigan explain them almost line-by-line, is 192 pages. A book on the papers is expected to top 300 pages. A "complete proof" of Poincaré, based on Dr. Perelman's breakthrough and published last month in the Asian Journal of Mathematics (which Prof. Milnor describes as throwing "a monkey wrench" into the question of who gets credit), is 328 pages long. Oddly, either the book or the Kleiner-Lott paper might count as the "refereed" work the Clay Institute demands. If so, we would have the weird situation in which authors of the work that satisfies the prize requirement aren't the people who figured out the proof. But their efforts could win Dr. Perelman $1 million. "It's definitely an unusual situation, but what's important is that the person who made the breakthrough put it out there so the community could scrutinize and analyze it," says institute president, James Carlson. Dr. Perelman shuns the limelight, but is known through lectures in the U.S. and for getting a perfect score at the 1982 International Mathematical Olympiad, at age 16. He isn't expected at the quadrennial meeting of the International Congress of Mathematicians, in Madrid. There, the Fields Medal, math's Nobel Prize, will be awarded to the "outstanding" mathematician 40 or under. Dr. Perelman is the odds-on favorite. And the millennium prizes? "I don't think the other six will be solved in my lifetime," says Dr. Carlson. "But then, I didn't think the Poincaré conjecture would be solved either." ? Email me at sciencejournal@wsj.com. (XYS20060725) ◇◇新语丝(www.xys.org)(xys.dxiong.com)(xys.3322.org)(xys.xlogit.com)◇◇