解說員的臨門一腳
——誰證明了龐加萊猜想？
三鏡堂主
2006年6月3日，數學大師、菲爾茲獎得主丘成桐教授在北京宣布：在美、俄
等國科學家的工作基礎上，中山大學朱熹平教授和旅美數學家、清華大學兼職教
授曹懷東已經徹底證明了龐加萊猜想。大師說：“這就像蓋大樓，前人打好了基
礎，但最後一步——也就是‘封頂’工作是由中國人來完成的。”為了讓普通人
了解這一猜想的重要性，大師進一步解釋道，“這是一項大成就，比哥德巴赫猜
想重要得多。”大師此說的根據何在？何以哥氏就不如龐氏？不得而知。但是只
證明了1+2的陳景潤身材無疑是越發的見矮了。
由於丘大師的特殊聲望，沒有人懷疑這一消息的正確性。各大中文報紙網站
紛紛登出振奮人心的醒目標題：懸賞百萬美金求解的數學世紀難題被中山大學教
授朱熹平和旅美數學家曹懷東徹底證明 ！
然而，與喜氣洋洋的中文媒體形成鮮明對比的是，對這樣一條驚天動地的新
聞，國際數學界的反應冷淡到了不可思議的地步。消息傳出之後許多天裡，用
Google查詢Poincaré Conjecture，只能查到這些中文消息的英文翻譯。對中國
數學家的驚世之作，國際數學界的集體失語，令人有山雨欲來的不祥之感。難道
真如丘大師所說，前人只是給大樓打了個基礎，而中國數學家完成了最後的“封
頂”工作？大家知道，任何一個現代數學難題的最終解決，無不是站在前人的肩
膀上完成的。十年前，普林斯頓大學教授瓦爾斯（Andrew Wiles）寒窗枯坐多年
所完成的費爾馬大定理的證明，正是這樣的一次“封頂”。
直到《華爾街日報》7月21日登載了一篇關於龐加萊猜想的專題文章，所謂
的“封頂”才算是被揭開了神秘的面紗。為方便讀者查詢，我將英文原文附於文
末，這裡只摘要介紹其中與“封頂”有關的信息供讀者評估。
2002年和2003年，俄國數學家佩瑞曼（Grigori Perelman）在一個存檔網站
上貼了兩篇論文，給出了龐加萊猜想的證明草稿。他甚至都沒有提及龐加萊猜想，
因為他認為他證明的是一個更廣泛的命題，龐加萊猜想不過是其中的一個推論而
已。他的論文不是用期刊發表所要求的嚴謹格式寫成的，因而十分晦澀難懂。正
當數學界期待他給出更詳細正規的證明時，不按常理出牌的佩瑞曼卻如隱士一般
從人間蒸發，不再回應。
也許論文的潦草正反應了天才是如何與常人交流的。佩瑞曼可能認為他根本
不需要為那些在他看來顯然的結論詳加解說，讀者如果愚笨到不能填補他的證明
空白，那不是他的問題，與其耗費時間糾纏於那些煩人的細節，不如去做些更重
要的事。
數學家們於是試圖去填補佩瑞曼論文留下的空白。佩瑞曼2003年的論文只有
22頁，2002年的是39頁，可是由密西根大學的克萊納(Kleiner)和勞特(Lott)逐
行加以詳解的《佩瑞曼論文注釋》(Notes on Perelman’s Papers)卻達192頁之
長。另一本將要出版的關於佩瑞曼論文的書有三百頁之多。
文章也提到了兩位中國數學家的論文：發表在上個月的《亞洲數學期刊》上、
根據佩瑞曼博士的突破（寫成的）龐加萊猜想的一個“完整證明”長達328頁。
米爾諾教授稱此文是向功勞歸屬的問題扔了一個“猴子的扳手”。
“猴子的扳手”是一句美國俚語，相當於中文裡的“攪局”。米爾諾教授可
能只是認為，這篇“完整證明”的倉促發表，意在爭奪填補佩瑞曼證明空缺的首
功，破壞了遊戲規則。專家們對龐加萊猜想的歸屬已有公論，不會想到“完整證
明”對於傳媒和社會大眾還可以有另外一種解釋。因為“完整證明”本身就在暗
示此前所有的包括佩瑞曼的證明都是“不完整證明”(incomplete proofs)，一
面不完整的鏡子也就是支離破碎的鏡子是不能行使鏡子的功能的。
順便說一句，丘成桐教授正是《亞洲數學期刊》的主編。米爾諾(John
Milnor)是紐約大學石溪分校的教授，楊振寧教授的同事，1962年的菲爾茲獎得
主。憤青們如有磚頭儘管砸向jack@math.sunysb.edu
文章繼續說，奇怪的是，這本書（註：不是指朱-曹的證明）或克萊納-勞特
的注釋卻可以作為克萊數學研究院頒獎所需要的參考資料。如此一來，我們陷入
了一個怪圈，寫出符合頒獎條件的論文作者們卻不是發現證明的人，他們的努力
只不過將幫助佩瑞曼獲得一百萬美元的獎金。
這篇文章明白無誤地告訴我們，佩瑞曼不但造好了大樓，而且封了頂。包括
朱、曹在內的數學家們不過給佩瑞曼的大樓鋪平了門前的道路，好讓克萊數學研
究院的專家前來驗收時不至於不得其門而入。
如果沒有確鑿的證據，請不要用“種族歧視”或“妖魔化”做幌子來轉移視
線。陳省聲“統治”美國數學界幾十年，丘成桐獲菲爾茲獎，都是有力的反證。
個人愚見，這篇文章淺顯易懂，用幽默風趣的語言敘述了龐加萊猜想及佩瑞曼證
明的來龍去脈，可讀性極強。諸君不妨一讀。
讓我們回顧一下中國數學界的說法。著名數學家楊樂如此評價道，“這是第
一次在國際數學期刊上給出了猜想的完整證明，成果極其突出。”且不說這個
“第一次”需要佐證，即朱、曹二位的論文確實是率先發表並經專家檢驗無誤，
而且楊院士的結論顯然有嚴重的誤導之嫌。讀者不會由此想到朱、曹二位只不過
是在解讀佩瑞曼的證明。楊院士進一步將龐加萊猜想這塊大餅切成了三塊，50%
送了漢彌爾頓，佩瑞曼25%，中國數學家得了30%。多出來的5%可能是楊院士自掏
腰包送丘院士的。根據楊教授的評價，我們只能得出中國數學家的貢獻比佩瑞曼
高，克萊數學研究院的100萬美元獎金非朱曹二位莫屬的封頂結論。
大家知道，數學是超越意識形態、沒有國界人種之別、放之宇宙皆準的學問。
在學科分類中，數學是獨立於科學（science）之外的。數學證明的對與錯，只
有黑白之分，沒有模糊不清的灰色地帶。除非佩瑞曼的證明有錯，而且朱、曹二
位在他們的“完整證明”中改正了他的錯誤，否則克萊數學研究院是不大可能將
100萬美元的獎金頒給他們的。
再以剛剛結束的足球世界盃為例。在電視機前觀看比賽的觀眾包括許多鐵杆
球迷，看到的只是封頂的臨門一腳，至於球星背景、攻防轉換、戰略戰術甚至比
賽規則其實不甚了了。因此電視解說員的講解和點評是足球盛宴上不可或缺的一
味佐料。但無論解說員如何鼓動如簧巧舌，球場上的風雲變化卻不是以他的意志
為轉移的。奪冠的意大利隊壓根兒也沒想過要和狂熱地愛着他們的中國解說員黃
健翔分享那份豐厚的冠軍獎金。
可是現在偏偏有人告訴你，那驚心動魄的臨門一腳，其實是解說員吼進去的，
因此解說員就是那封頂的功臣。
文章寫到這裡，心裡很不是滋味。朱、曹二位都是優秀的數學人才。能成功
地解讀佩瑞曼的證明，本來已經是很了不起的成就。比如人人都能從山腳下看到
珠穆朗瑪峰，但即使沿着前人的足跡攀登珠峰，也不是人人都能做到的。朱、曹
並不需要用“封頂”來證明他們不世的才華。“封頂”論恐怕也不是二位的初衷。
這大概可以解釋為什麼兩位年輕數學家在別人發布“封頂”的消息時一致地選擇
了沉默。
附錄：華爾街日報專文
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.
