把數學烘焙成一道美味甜點
CHICAGO — We had just finished the mathematician Eugenia Cheng’s splendid demonstration of nonassociativity where the order of operations counts — as it does in, say, subtraction.
芝加哥——我們剛剛聽完了數學家尤金妮婭·鄭(Eugenia Cheng)關於非結合律的精彩講解：運算的順序會影響運算結果，比如在減法中。
Now she wanted to forge ahead with the next lesson, in knot theory.
現在，她想開始下一課：紐結理論(knot theory)。
I suggested we wait until later. “Why?” she asked.
我建議等一下。“爲什麼？”她問。
“Well, we shouldn’t eat two desserts before dinner, should we?” I said, and giggled nervously.
“這個，正餐之前總不能吃兩道甜點吧，對不對？”我不安地笑道。
“Why not?” she replied, not giggling. She tightened her apron strings and walked over to her stove.
“爲什麼不能呢？”她沒有笑，繫緊腰上的圍裙，走向烤爐。
Of course. What was I thinking? Hadn’t Dr. Cheng already made clear her conviction that in mathematics, rules are like eggs: meant to be broken, stirred, flipped over and taste-tested? And that day, we had broken a lot of eggs.
當然啦。我想什麼呢？鄭博士不是早就闡明瞭自己的理念嗎——在數學中，規則就像雞蛋一樣，就是用來打破、攪拌、翻轉和嘗試的。那一天，我們已經打破了不少雞蛋了。
“You’re absolutely right,” I said, rushing to her side for the grand unveiling of another mathematically themed confection.
“你是對的，”我快速走到她身邊，等待她鄭重展示下一道數學主題的甜點。
Dr. Cheng pulled from the oven a perfectly baked specimen of what she calls Bach pie, named for the great composer beloved by mathematicians everywhere: an oblong rectangle of creamy dark chocolate studded with banana slices and topped by an Escher-like braid of four glazed pastry plaits that followed divergent trajectories, never quite crisscrossing where you expected them to.
鄭博士從烤箱中拿出一份完美烘焙的樣本，她稱之爲“巴赫派”(Bach pie)，以那位全世界數學家都深深喜愛的偉大作曲家命名。它是一塊長方形的奶油黑巧克力蛋糕，點綴着香蕉切片，頂上是一個髮辮一樣的埃舍爾式圖案：四股糖麻花呈放射狀分佈，在看似縱橫相交的地方卻並不相交。
The filling was a clever concatenation — “BAnana added to CHocolate gives you Bach,” Dr. Cheng said. The braiding illustrated the structure of a Bach prelude and the sorts of patterns that knot theorists study “to see how looped up the braids are,” Dr. Cheng said, “and whether you can transform one braid into another by wiggling the different strings.”
餡料別具匠心——“香蕉(BAnana)和巧克力(CHocolate)的前兩個字母加起來就是巴赫(Bach)，”鄭博士說。這個編織圖案展示了巴赫一首序曲的結構，也是紐結理論所研究的那種圖形，爲了“研究麻花辮子結構是如何紐結起來的”，她說，“以及你是否可以通過扭動不同的辮股，把一條辮子變形爲另一條辮子”。
The pie was a true union of Art and math, too beautiful to besmirch, and besides, you’re not supposed to untie knots with your teeth, are you?
這個派真是藝術與數學的結合，美到讓人不敢褻玩，另外，也不應該用牙齒解開繩結呀，對不對？
Another rule, easily broken.
不過這個規則是很容易打破的。
Dr. Cheng, 39, has a knack for brushing aside conventions and edicts, like so many pie crumbs from a cutting board. She is a theoretical mathematician who works in a rarefied field called category theory, which is so abstract that “even some pure mathematicians think it goes too far,” Dr. Cheng said.
39歲的鄭博士慣於拋棄慣例與成規，就像信手拂去砧板上的糕點碎屑一樣。她是個理論數學家，研究範疇論(category theory)這個罕爲人知的領域，它非常抽象，“甚至許多純數學家都覺得它走得太遠了，”鄭博士說。
At the same time, Dr. Cheng is winning fame as a math popularizer, convinced that the pleasures of math can be conveyed to the legions of numbers-averse humanities majors still recovering from high school algebra. She has been featured on shows like “Late Night With Stephen Colbert,” and her online math tutorials have been viewed more than a million times.
與此同時，鄭博士還以數學科普者而聞名。她堅信， 大批在高中數學課上留下後遺症、至今看到數字就頭痛的文科生也可以領略到數學的樂趣。她上過“科爾伯特晚間秀”(Late Night With Stephen Colbert)等電視節目，她的在線數學課訪問量超過了100萬次。
The hardcover edition of her first book, “How to Bake π: An Edible Exploration of the Mathematics of Mathematics,” has sold about 25,000 copies in this country and been translated into six languages, a surprising hit for a text visibly if judiciously seasoned with numbers, graphs and equations. The book is being released in paperback this month.
她的第一本書名爲《怎樣烘焙π：對數學中的數學的可食用探險》(How to Bake π: An Edible Exploration of the Mathematics of Mathematics)，其精裝版在美國售出了2.5萬冊，並被翻譯成六種語言。對於一本滿篇（雖說是慎重使用的）數字、圖表和等式的書籍來說，真是驚人的成功。這本書的平裝版本月也將上市。
“I spend a lot of time explaining mathematics on blogs, and I try to cut through the technicalities and make things easier to understand,” said John Baez, a professor of math at the University of California, Riverside (and yes, a cousin of Joan). Still, his posts are aimed at scientists and others with some quantitative background.
“我花費了很多時間在博客上解釋數學，試着邁過學術性，把問題弄得簡單易懂，”加州大學河濱分校(University of California, Riverside)的數學教授約翰·貝茲(John Baez)說（沒錯，他是瓊·貝茲的親戚）。不過，他在網上的帖子還是針對科學家和其他有定量研究背景的人的。
“Eugenia has gone all the way in,” he said. “She’s trying to explain math to everybody, with or without pre-existing expertise, and I think she’s doing wonderfully.”
“尤金妮婭則是徹底投入，”他說。“她試着向所有人解釋數學，不管對方是不是已經具備了專業知識，而且我覺得她乾得很棒。”
So committed is Dr. Cheng to mass math demystification that she recently left a tenured professorship at the University of Sheffield in Britain to take a position at the School of the Art Institute of Chicago, where she teaches math to art students, lectures widely and continues her research in category theory on the side.
鄭博士是如此專注於大衆數學啓蒙工作，她前不久辭去了英國謝菲爾德大學(University of Sheffield)的終身教授職位，來到芝加哥藝術學院(Art Institute of Chicago)，向學藝術的學生們教授數學，四處講座，同時繼續自己在範疇論領域的研究。
Dr. Cheng adopts a literal approach to making math more appetizing. “Math is about taking ingredients, putting them together, seeing what you can make out of them, and then deciding whether it’s tasty or not,” she said.
鄭博士採用一種直接的方式讓數學更“開胃”。“數學就是使用各種元素，把它們放在一起，看看能得到什麼結果，然後判斷它是不是美味可口，”她說。
Every chapter in “How to Bake π” offers recipes for desserts and other dishes that encapsulate mathematical themes. To demonstrate how math seeks to identify underlying similarities across a broad set of problems, for example, Dr. Cheng starts with a recipe that can be readily tweaked to make mayonnaise instead of hollandaise sauce.
《怎樣烘焙π》中的每一章都提供甜點菜譜和其他菜譜，都包含數學的主題。比如，爲了展示數學是如何在一組廣泛的問題中發現潛在的相似性，鄭博士從一份便於調整的食譜入手，不調製荷蘭醬，而代之以普通蛋黃醬。
“Books might tell you that hollandaise sauce needs to be done differently,” she writes, “but I ignore them to make my life simpler. Math is also there to make things simpler, by finding things that look the same if you ignore some small detail.”
“書本會告訴你，荷蘭醬有另一種做法，”她寫道，“但是我忽略了它們，好讓自己的生活簡單點。數學在這裏也發揮了作用，找出相同點，幫助你把小的細節忽略掉，讓一切變得簡單。”
Her recipe for lasagna illuminates the importance of context to math. Dr. Cheng lists among the basic ingredients “fresh lasagna noodles,” and then points out that another cookbook might deem the noodles not truly basic and instead describe their preparation from scratch.
她的千層麪菜譜顯示出背景條件在數學中的重要性。鄭博士把“新鮮千層麪”列爲這道菜所需的基本原料，並指出，另一本菜譜或許並不把麪條視爲做這道菜的基本原料，而是從零開始，描述了麪條的製作方法。
So, too, do numbers change their character and degree of basicness depending on context. The number 5, for example, when viewed among the natural, or counting, numbers is one of those elemental creatures: a prime number, divisible only by 1 and itself.
同理，根據背景條件，數字的特性及其基本程度也會改變。比如數字5，在自然數或計數中，它是一個基本的數字：質數，只能被1或它自身整除。
But in the context of the so-called rational numbers, which include fractions, 5 loses its prime identity and gains versatility, able to be divided into ever tinier slivers, like a cake at a dieters’ convention.
然而，如果把“5”放在包括分數在內的有理數中考慮，它就失去了質數的特性，有了更多用途，可以被劃分爲更小的部分，就像節食者的蛋糕。
The number 1 in its multiplicative identity is practically bedridden, leaving other numbers unchanged: 6 times 1 equals 6. In its additive capacity, however, 1 is unstoppable: if you keep adding 1 to itself, Dr. Cheng noted, you can generate all the natural numbers, out to infinity.
數字“1”在乘法中起一種限制作用，就是讓其他數字保持不變：6乘以1還等於6。而在加法中，1的作用是不可遏制的：鄭博士指出，如果在1上面持續再加1，就會得到所有的自然數，直到無窮大。
Context can prod numbers to defy grade-school verities: 2 plus 2 equals 4, and that’s that. But not if you’re talking about a clock face with only three numbers: 1, 2 and 3. In that case, 2 plus 2 equals 1 – if you start at the 2 and move clockwise by 2, you reach 1.
背景條件可以令數字違背學校裏教的“2加2等於4”之類公理。如果一個錶盤上只有1、2、3這3個數字，在這種情況下，2加2就等於1——如果你從2開始，把指針順時針移動2次，你就可以得到1。
“I admit I was skeptical at first about her analogies to cooking, but I ended up being completely sold,” said Steven Strogatz, a professor of applied mathematics at Cornell University who also writes popular books.
“我承認，對於她把數學和烹飪做類比的方法，我一開始感到懷疑，但最後我完全被她說服了，”同樣撰寫通俗書籍的康奈爾大學(Cornell University)應用數學教授史蒂芬·斯特朗蓋茨(Steven Strogatz)說。
“She conveys the spirit of inventiveness and creativity in math that all mathematicians feel but do a very poor job communicating when teaching math. Refreshing is the word that keeps coming to mind.”
“她傳達了數學中的創新精神與創造性，所有數學家都能體會到，但是在教授數學的時候，卻很難同學生溝通這一點。看她的書不斷讓人感覺耳目一新。”
Dr. Cheng insists that the public has it all wrong about math being difficult, something that only the gifted mathletes among us can do. To the contrary, she says, math exists to make life smoother, to solve those problems that can be solved by applying math’s most powerful tool: logic.
鄭博士堅持說，公衆認爲數學很難、只有天才才能搞數學的看法是錯誤的。相反，她說，數學就是爲了讓生活更簡單；憑藉數學當中最強大的工具：邏輯，可以解決各種問題。
Science may depend on forming hypotheses, doing experiments and gathering evidence that support or refute your hypothesis, but math is simply a matter of stating the terms of your argument and then defending those statements using logic.
科學或許要依靠提出假設、做實驗、收集證據，以此支持或否定自己的假設，但數學就只需要擺出論點的條件，然後使用邏輯，支持自己的論述。
“The great thing about math is you don’t need much to start exploring it,” Dr. Baez said. “No expensive equipment, just pencil and paper, and you can start fiddling around with patterns and numbers.”
“數學最棒的一點，就是探索它不需要很多條件，”貝茲博士說。“不需要昂貴的設備，只需要紙和筆，你就可以在各種模型與數字之中摸索。”
Dr. Cheng recognizes that people can feel uncomfortable with some of the abstractions required by mathematical thinking, by the need to ignore the particulars of, say, this green round pillow and that square purple pillow in favor of an abstract ideal of a pillow that you’re going to call x.
鄭博士發現，有些需要數學思維的抽象概念可能會讓人們感覺不舒服，它們需要人們忽略事物的特殊性，比如說這個綠色的圓枕頭，那個紫色的方枕頭，在數學中，它們都是抽象概念的枕頭，可以管它們叫做“x”。
But it’s just a matter of practice, she said, before the idea starts to feel like a real object that you can manipulate with ease. “You become very good at separating what’s relevant from what isn’t, and that can be very useful in daily life,” she said.
但這只是個實踐問題，她說，漸漸地，抽象概念就變得好像真實存在的物體，你可以輕易操縱它。“你開始擅長把重要的事物從不重要的事物中分辨出來，這在日常生活中非常有用，”她說。
Sometimes, she finds it “oddly satisfying” to mentally shave a bearded man or imagine how a furry dog would look like after a swim in a lake. “That’s what abstraction is,” she said. “You reveal the structure underneath.”
有時候，她覺得在想象中給一個留鬍子的男人剃鬚，或是想象一隻毛茸茸的狗從湖裏溼淋淋地爬上來，會有一種“奇異的滿足感”。“這就是抽象，”她說，“揭示出深層的結構。”
