科學(xué)路上，數(shù)學(xué)不是絆腳石

似乎很多人小時(shí)候都有一個(gè)“長(zhǎng)大要當(dāng)科學(xué)家”的理想。如今“長(zhǎng)大”已經(jīng)實(shí)現(xiàn)了，說好的“科學(xué)家”呢？也許你早就發(fā)現(xiàn)了，要想成為科學(xué)家并不是那么容易，要為科學(xué)奉獻(xiàn)一生需要多么濃厚的興趣和刻苦鉆研的精神??！什么？你是因?yàn)閿?shù)學(xué)不好才斷了當(dāng)科學(xué)家的念想？那還來得及，趕快重新投入科學(xué)的懷抱吧，誰告訴你數(shù)學(xué)不好就當(dāng)不了科學(xué)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(duì)于很多有志于成為科學(xué)家的年輕人來說，數(shù)學(xué)是個(gè)大難題。離開了高等數(shù)學(xué)，你怎么能在科學(xué)領(lǐng)域開展需要認(rèn)真思考的工作呢？不過，我有一個(gè)職業(yè)秘密要分享：當(dāng)今世界上很多非常成功的科學(xué)家在數(shù)學(xué)方面不過是半文盲罷了。

我在哈佛教授生物學(xué)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(xué)工作的可能性，他們擔(dān)心自己會(huì)因沒有出色的數(shù)學(xué)技能而失敗。這種錯(cuò)誤的臆斷使科學(xué)界痛失了無數(shù)亟需的人才。

在這方面我可是個(gè)權(quán)威，因?yàn)槲易约壕褪且粋€(gè)極端的例子。大學(xué)之前，我在條件相對(duì)較差的南部學(xué)校上學(xué)，在我去亞拉巴馬大學(xué)上大學(xué)一年級(jí)之前，我可沒學(xué)過代數(shù)。我到32歲才終于開始學(xué)習(xí)微積分，那時(shí)我已是哈佛大學(xué)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(diǎn)兒，其中有幾個(gè)還是我當(dāng)時(shí)正在教授的進(jìn)化生物學(xué)課上的學(xué)生。但我拋開了自尊，學(xué)會(huì)了微積分。

盡管我緊追猛趕，但我頂多也就是個(gè)C等生。不過令我安心的是，我發(fā)現(xiàn)出色的數(shù)學(xué)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(shí)間與母語人士交談，我的外語可能會(huì)變得很流利，但是因?yàn)槊τ趯?shí)地研究和實(shí)驗(yàn)室研究，我只進(jìn)步了一點(diǎn)點(diǎn)。

幸運(yùn)的是，對(duì)數(shù)學(xué)能力有極高要求的僅僅是少數(shù)幾個(gè)學(xué)科，如粒子物理學(xué)、天體物理學(xué)和信息論等。在科學(xué)的其他領(lǐng)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(xué)家那樣做白日夢(mèng)的時(shí)候。經(jīng)過升華與約束的幻想是所有創(chuàng)造性思維的源頭。牛頓做過夢(mèng)，達(dá)爾文做過夢(mèng)，你也做夢(mèng)。腦海中被喚起的那些圖像最初是模糊的，它們可能會(huì)變換形狀，漸漸顯形又漸漸消失。當(dāng)你把它們畫在紙上，形成圖形時(shí)，它們就變得更明確一些；當(dāng)你探尋并找到了真實(shí)的例證時(shí)，它們就開始有了生氣。

科學(xué)先驅(qū)們的發(fā)現(xiàn)極少是通過從純數(shù)學(xué)中提煉觀點(diǎn)而得來的。那些展現(xiàn)科學(xué)家研究黑板上一行一行方程式的老套照片其實(shí)大都是老師在解釋已有的發(fā)現(xiàn)。真正的科學(xué)進(jìn)步源自實(shí)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(shí)，源自獨(dú)自吃午飯時(shí)?！办`感突發(fā)時(shí)刻”的到來需要你努力工作并且專注其中。

科學(xué)領(lǐng)域的觀點(diǎn)最容易出現(xiàn)在為了世上某物本身而進(jìn)行研究時(shí)。當(dāng)人們對(duì)現(xiàn)存事物中的真正實(shí)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(xué)觀點(diǎn)才會(huì)誕生。當(dāng)某種新發(fā)現(xiàn)出現(xiàn)時(shí)，后續(xù)的步驟往往需要用數(shù)學(xué)和統(tǒng)計(jì)學(xué)方法來推進(jìn)分析。如果做出發(fā)現(xiàn)的人覺得這一步驟的技術(shù)難度太大，那可以增加一位數(shù)學(xué)家或統(tǒng)計(jì)學(xué)家作為其合作者。

在20世紀(jì)70年代末，我與數(shù)學(xué)理論家喬治·奧斯特一起研究社會(huì)性昆蟲中的等級(jí)原則和勞動(dòng)分工。我提供了自然界中和實(shí)驗(yàn)室內(nèi)已經(jīng)發(fā)現(xiàn)的細(xì)節(jié)，他則使用其“工具包”內(nèi)的定理和假設(shè)來描述這些現(xiàn)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(gè)籠統(tǒng)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(shù)學(xué)家和統(tǒng)計(jì)學(xué)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(shù)學(xué)家和統(tǒng)計(jì)學(xué)家找到能運(yùn)用其方程式的科學(xué)家，讓科學(xué)家從數(shù)學(xué)家和統(tǒng)計(jì)學(xué)家處得到其所需的合作要容易得多。

這種不平衡在生物學(xué)領(lǐng)域尤為顯著，因?yàn)樵谶@個(gè)領(lǐng)域，真實(shí)生活中某個(gè)現(xiàn)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(xué)的歷史記載中充斥著要么可以完全忽略、要么經(jīng)過驗(yàn)證是錯(cuò)誤的數(shù)學(xué)模型，有長(zhǎng)久價(jià)值的模型可能頂多只占10%。只有那些與真實(shí)生命系統(tǒng)的知識(shí)緊密相連的模型才有較大可能得到運(yùn)用。

如果你的數(shù)學(xué)能力較低，那就做個(gè)計(jì)劃提升一下。但同時(shí)你也要知道，運(yùn)用現(xiàn)有的數(shù)學(xué)能力你同樣可以完成杰出的科學(xué)工作。但是，如果你想專攻需要不斷交替進(jìn)行實(shí)驗(yàn)和定量分析的領(lǐng)域時(shí)，那就要三思了。這些領(lǐng)域包括物理學(xué)和化學(xué)的大多數(shù)專業(yè)，還有分子生物學(xué)方面的幾個(gè)專業(yè)。

牛頓發(fā)明了微積分，以便為他的想象賦予實(shí)質(zhì)內(nèi)容。達(dá)爾文幾乎或者說根本沒有數(shù)學(xué)能力，但他卻能憑借自己積累的大量信息構(gòu)想出一個(gè)過程，數(shù)學(xué)被應(yīng)用于此過程是后來的事了。

對(duì)于有抱負(fù)的科學(xué)家來說，關(guān)鍵的第一步是找到一個(gè)非常感興趣的學(xué)科，并專攻該學(xué)科。在這樣做時(shí)，他們應(yīng)當(dāng)牢記“威爾遜第二定律”：對(duì)于每一位科學(xué)家來說，都有一個(gè)學(xué)科是其數(shù)學(xué)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(shí)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??！有辦法啦！”古希臘學(xué)者阿基米德有一次在浴盆里洗澡，突然來了靈感，發(fā)現(xiàn)了他久未解決的計(jì)算浮力問題的辦法，于是驚喜地叫了一聲“尤里卡”?！坝壤锟〞r(shí)刻”因此用來形容靈感突現(xiàn)、豁然開朗的時(shí)刻。

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

8.theorem [?θ??r?m] n. [數(shù)]定理

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

10.annals [??n（?）lz] n. [復(fù)]歷史記載