科學路上，數學不是絆腳石

似乎很多人小時候都有一個“長大要當科學家”的理想。如今“長大”已經實現了，說好的“科學家”呢？也許你早就發現了，要想成為科學家并不是那么容易，要為科學奉獻一生需要多么濃厚的興趣和刻苦鉆研的精神啊！什么？你是因為數學不好才斷了當科學家的念想？那還來得及，趕快重新投入科學的懷抱吧，誰告訴你數學不好就當不了科學家的？

For many young people who aspire to be scientists， the great bugbear1） is mathematics. Without advanced math， how can you do serious work in the sciences？ Well， I have a professional secret to share： Many of the most successful scientists in the world today are mathematically no more than semiliterate.

During my decades of teaching biology at Harvard， I watched sadly as bright undergraduates turned away from the possibility of a scientific career， fearing that， without strong math skills， they would fail. This mistaken assumption has deprived science of an immeasurable amount of sorely needed talent.

I speak as an authority on this subject because I myself am an extreme case. Having spent my precollege years in relatively poor Southern schools， I didn’t take algebra until my freshman year at the University of Alabama. I finally got around to2） calculus as a 32-year-old tenured3） professor at Harvard， where I sat uncomfortably in classes with undergraduate students only a bit more than half my age. A couple of them were students in a course on evolutionary biology I was teaching. I swallowed my pride and learned calculus.

I was never more than a C student while catching up， but I was reassured by the discovery that superior mathematical ability is similar to fluency in foreign languages. I might have become fluent with more effort and sessions talking with the natives， but being swept up with field and laboratory research， I advanced only by a small amount.

Fortunately， exceptional mathematical fluency is required in only a few disciplines， such as particle physics， astrophysics and information theory. Far more important throughout the rest of science is the ability to form concepts， during which the researcher conjures4） images and processes by intuition.

Everyone sometimes daydreams like a scientist. Ramped up5） and disciplined， fantasies are the fountainhead of all creative thinking. Newton dreamed， Darwin dreamed， you dream. The images evoked are at first vague. They may shift in form and fade in and out. They grow a bit firmer when sketched as diagrams on pads of paper， and they take on life as real examples are sought and found.

Pioneers in science only rarely make discoveries by extracting ideas from pure mathematics. Most of the stereotypical photographs of scientists studying rows of equations on a blackboard are instructors explaining discoveries already made. Real progress comes in the field writing notes， at the office amid a litter of doodled paper， in the hallway struggling to explain something to a friend， or eating lunch alone. Eureka moments6） require hard work. And focus.

Ideas in science emerge most readily when some part of the world is studied for its own sake. They follow from thorough， well-organized knowledge of all that is known or can be imagined of real entities and processes within that fragment of existence. When something new is encountered， the follow-up steps usually require mathematical and statistical methods to move the analysis forward. If that step proves too technically difficult for the person who made the discovery， a mathematician or statistician can be added as a collaborator.

In the late 1970s， I sat down with the mathematical theorist George Oster to work out the principles of caste7） and the division of labor in the social insects. I supplied the details of what had been discovered in nature and the lab， and he used theorems8） and hypotheses from his tool kit to capture these phenomena. Without such information， Mr. Oster might have developed a general theory， but he would not have had any way to deduce which of the possible permutations9） actually exist on earth.

Over the years， I have co-written many papers with mathematicians and statisticians， so I can offer the following principle with confidence. Call it Wilson’s Principle No. 1： It is far easier for scientists to acquire needed collaboration from mathematicians and statisticians than it is for mathematicians and statisticians to find scientists able to make use of their equations.

This imbalance is especially the case in biology， where factors in a real-life phenomenon are often misunderstood or never noticed in the first place. The annals10） of theoretical biology are clogged with mathematical models that either can be safely ignored or， when tested， fail. Possibly no more than 10% have any lasting value. Only those linked solidly to knowledge of real living systems have much chance of being used.

If your level of mathematical competence is low， plan to raise it， but meanwhile， know that you can do outstanding scientific work with what you have. Think twice， though， about specializing in fields that require a close alternation of experiment and quantitative analysis. These include most of physics and chemistry， as well as a few specialties in molecular biology.

Newton invented calculus in order to give substance to his imagination. Darwin had little or no mathematical ability， but with the masses of information he had accumulated， he was able to conceive a process to which mathematics was later applied.

For aspiring scientists， a key first step is to find a subject that interests them deeply and focus on it. In doing so， they should keep in mind Wilson’s Principle No. 2： For every scientist， there exists a discipline for which his or her level of mathematical competence is enough to achieve excellence.

對于很多有志于成為科學家的年輕人來說，數學是個大難題。離開了高等數學，你怎么能在科學領域開展需要認真思考的工作呢？不過，我有一個職業秘密要分享：當今世界上很多非常成功的科學家在數學方面不過是半文盲罷了。

我在哈佛教授生物學的幾十年間，曾遺憾地看到一些聰明的本科生放棄了從事科學工作的可能性，他們擔心自己會因沒有出色的數學技能而失敗。這種錯誤的臆斷使科學界痛失了無數亟需的人才。

在這方面我可是個權威，因為我自己就是一個極端的例子。大學之前，我在條件相對較差的南部學校上學，在我去亞拉巴馬大學上大學一年級之前，我可沒學過代數。我到32歲才終于開始學習微積分，那時我已是哈佛大學的終身教授，不自在地與本科生坐在一起上課。那些本科生的年齡僅僅是我的一半多一點兒，其中有幾個還是我當時正在教授的進化生物學課上的學生。但我拋開了自尊，學會了微積分。

盡管我緊追猛趕，但我頂多也就是個C等生。不過令我安心的是，我發現出色的數學能力類似于流利的外語水平。如果我付出更多努力，花更多時間與母語人士交談，我的外語可能會變得很流利，但是因為忙于實地研究和實驗室研究，我只進步了一點點。

幸運的是，對數學能力有極高要求的僅僅是少數幾個學科，如粒子物理學、天體物理學和信息論等。在科學的其他領域，更重要的是形成概念的能力，在此過程中，研究者利用直覺來想象出圖像和過程。

人人都有像科學家那樣做白日夢的時候。經過升華與約束的幻想是所有創造性思維的源頭。牛頓做過夢，達爾文做過夢，你也做夢。腦海中被喚起的那些圖像最初是模糊的，它們可能會變換形狀，漸漸顯形又漸漸消失。當你把它們畫在紙上，形成圖形時，它們就變得更明確一些；當你探尋并找到了真實的例證時，它們就開始有了生氣。

科學先驅們的發現極少是通過從純數學中提煉觀點而得來的。那些展現科學家研究黑板上一行一行方程式的老套照片其實大都是老師在解釋已有的發現。真正的科學進步源自實地考察所做的筆記中，源自到處堆著涂鴉紙張的辦公室里，源自在走廊里努力向朋友解釋某事時，源自獨自吃午飯時?！办`感突發時刻”的到來需要你努力工作并且專注其中。

科學領域的觀點最容易出現在為了世上某物本身而進行研究時。當人們對現存事物中的真正實體和過程的所有已知情況或可想象情況有了詳盡和條理清晰的了解后，科學觀點才會誕生。當某種新發現出現時，后續的步驟往往需要用數學和統計學方法來推進分析。如果做出發現的人覺得這一步驟的技術難度太大，那可以增加一位數學家或統計學家作為其合作者。

在20世紀70年代末，我與數學理論家喬治·奧斯特一起研究社會性昆蟲中的等級原則和勞動分工。我提供了自然界中和實驗室內已經發現的細節，他則使用其“工具包”內的定理和假設來描述這些現象。如果沒有我提供的那些信息，奧斯特先生或許可以提出一個籠統的理論，但他將無法推斷出哪些可能的排列是地球上真正存在的。

多年來，我與數學家和統計學家合寫過很多論文，所以我可以自信地給出以下定律，姑且稱之為“威爾遜第一定律”：比起讓數學家和統計學家找到能運用其方程式的科學家，讓科學家從數學家和統計學家處得到其所需的合作要容易得多。

這種不平衡在生物學領域尤為顯著，因為在這個領域，真實生活中某個現象的某些因素往往被誤解，或者一開始就根本沒被注意到。理論生物學的歷史記載中充斥著要么可以完全忽略、要么經過驗證是錯誤的數學模型，有長久價值的模型可能頂多只占10%。只有那些與真實生命系統的知識緊密相連的模型才有較大可能得到運用。

如果你的數學能力較低，那就做個計劃提升一下。但同時你也要知道，運用現有的數學能力你同樣可以完成杰出的科學工作。但是，如果你想專攻需要不斷交替進行實驗和定量分析的領域時，那就要三思了。這些領域包括物理學和化學的大多數專業，還有分子生物學方面的幾個專業。

牛頓發明了微積分，以便為他的想象賦予實質內容。達爾文幾乎或者說根本沒有數學能力，但他卻能憑借自己積累的大量信息構想出一個過程，數學被應用于此過程是后來的事了。

對于有抱負的科學家來說，關鍵的第一步是找到一個非常感興趣的學科，并專攻該學科。在這樣做時，他們應當牢記“威爾遜第二定律”：對于每一位科學家來說，都有一個學科是其數學能力足以使之取得杰出成就的。

1.bugbear [?b?ɡ?be?（r）] n. 棘手的問題，難題；恐懼（或煩惱）的原因

2.get around to：抽出時間做（或考慮）某事

3.tenured [?tenj?（r）d] adj. 〈主美〉享有終身職位的

4.conjure [?k?nd??（r）] vt. 想象；提出

5.ramp up：增加，提高

6.Eureka moment：“尤里卡”（Eureka）原是古希臘語，意思是“好啊！有辦法啦！”古希臘學者阿基米德有一次在浴盆里洗澡，突然來了靈感，發現了他久未解決的計算浮力問題的辦法，于是驚喜地叫了一聲“尤里卡”?！坝壤锟〞r刻”因此用來形容靈感突現、豁然開朗的時刻。

7.caste [kɑ?st] n. [昆]級（社會性昆蟲中成熟個體如兵、工等的不同型）

8.theorem [?θ??r?m] n. [數]定理

9.permutation [?p??（r）mj??te??（?）n] n. [數]排列，置換

10.annals [??n（?）lz] n. [復]歷史記載