數學家解得開那維葉‧史托克斯（Navier-Stokes)方程式的奧秘?氣流的平順與狂暴之關鍵

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SCIENCE P102

Will Malthus Continue to Be Wring?

馬爾薩斯的理論會持續被推翻嗎？

In 1798, a 32-year-old curate at a small parish church in Albury, England, published a sobering pamphlet entitled An Essay on the Principle of Population. As a grim rebuttal of the utopian philosophers of his day, Thomas Malthus argued that human populations will always tend to grow and, eventually, they will always be checked—either by foresight, such as birth control, or as a result of famine, war, or disease. Those speculations have inspired many a dire warning from environmentalists.

　　一七九八年，在英國亞伯利（Albury）一位三十二歲的教堂牧師，發表了一本內容嚴謹的小冊子，取名叫「人口學原理（An Eassy on the Principle of Population）」。他是托瑪斯‧馬爾薩斯（Thomas Malthus）。他對當時蔚為風潮的烏托邦理論提出嚴厲反駁，並且指出人類總人口恐會不斷增長，而且最必定會受到下列因素的控制──像是出生率的預測，以及饑荒、戰爭或疾病。這些預言使得環境學家紛紛提出悲觀而可怕的警告。

Since Malthus’s time, world population has risen six fold to more than 6 billion. Yet happily, apocalyptic collapses have mostly been prevented by the advent of cheap energy, the rise of science and technology, and the green revolution. Most demographers predict that by 2100, global population will level off at about 10 billion.

　　由馬爾薩斯的時代至今，人類總人口已經成長了六倍、總人次已經超過六十億。幸好，這毀滅性的末日卻因便宜的能源、科技的進步以及綠色產業革命等等的到來而使人類躲過一劫。多數人口學家都預估，西元2100年以前，世界人口將超過一百億。

The urgent question is whether current standards of living can be sustained while improving the plight of those in need. Consumption of resources—not just food but also water, fossil fuels, timber, and other essentials—has grown enormously in the developed world. In addition, humans have compounded the direct threats to those resources in many ways, including by changing climate (see p. 100), polluting land and water, and spreading invasive species.

　　當前最急迫的問題是，在將資源分配給那些急切需要的人後，當今的生活水平如何維持。資源的消耗量──不只是食物，還有水、化石燃料、林木以及其他必需品，在今天快速發展的當代社會中，都逐漸上升。此外，人類在許多方面也加重了對這些能源的直接威脅，像是改變氣候（參閱第100頁)，汙染土地與水質，還有散布侵略性的品種。

How can humans live sustainably on the planet and do so in a way that manages to preserve some biodiversity? Tackling that question involves a broad range of research for natural and social scientists. It’s abundantly clear, for example, that humans are degrading many ecosystems and hindering their ability to provide clean water and other “goods and services” (Science, 1 April, p. 41). But exactly how bad is the situation? Researchers need better information on the status and trends of wetlands, forests, and other areas. To set priorities, they’d also like a better understanding of what makes ecosystems more resistant or vulnerable and whether stressed ecosystems, such as marine fisheries, have a threshold at which they won’t recover.

人類究竟如何靠著植物而保持一定水準以上的生活，並保持生物多樣性？解決這個問題必須牽涉到許多自然科學研究以及社會學家的理論。舉例而言，人類正在降低許多生態系統的能量，並且妨礙這些生態系生產乾淨的水以及其他「物質與資源」（科學雜誌，同年四月一號第四十一期）。但現在狀態究竟有多糟？研究人員需要有更多關於濕地、森林與其他地方的相關資訊和和發展趨勢。為了做出優先選擇的順序，研究人員也需要知道更多關於如何使生態系統有抵抗力的方式或使之脆弱的原因，還有某些強大環境壓力下的物種（像是深海魚）在什麼情況下會完全滅絕。

Agronomists face the task of feeding 4 billion more mouths. Yields may be maxing out in the developed world, but much can still be done in the developing world, particularly sub-Saharan Africa, which desperately needs more nitrogen. Although agricultural biotechnology clearly has potential to boost yields and lessen the environmental impact of farming, it has its own risks, and winning over skeptics has proven difficult.

農藝學家面對的問題是要餵飽四十億人口，甚至更多。在已開發國家中，產量要極大化是沒有問題的，但當中也有許多可以在開發中地區執行，尤其是急需更多氦氣的撒哈拉沙漠以南的非洲。雖然農業生化技術可以將產能潛力提高，並減少農產對環境的負面危害，但還是有其風險的，而且要說服、壓制懷疑論者的言論，已被證實為一個非常嚴峻的考驗。

There’s no shortage of work for social scientists either. Perverse subsidies that encourage overuse of resources—tax loopholes for luxury Hummers and other inefficient vehicles, for example—remain a chronic problem. A new area of activity is the attempt to place values on ecosystems’ services, so that the price of clear-cut lumber, for instance, covers the loss of a forest’s ability to provide clean water. Incorporating those “externalities” into pricing is a daunting challenge that demands much more knowledge of ecosystems. In addition, economic decisions often consider only net present value and discount the future value of resources—soil erosion, slash-and-burn agriculture, and the mining of groundwater for cities and farming are prime examples. All this complicates the process of transforming industries so that they provide jobs, goods, and services while damaging the environment less.

社會科學家的工作量也未有減少。反常且不合情理的補貼使得資源過度使用量持續增長──例如對高級跑車的或是其他沒效率的交通載具課稅的稅法漏洞，使得問題一而再再而三地持續。有一個可以努力的方向是，將生態系統產物的附加價值提高，舉例而言，把伐工精細的木材的售價提高，即可彌補森林產生乾淨空氣的能力。然而，將這些「外部性」操作反應在售價上，是個令人氣餒的挑戰，因為這需要更多對於生態系統的知識。此外，商業性的決策通常只考慮當下的銷售淨值，而忽略了資源的未來價值，例如土壤侵蝕。刀耕式的原始農業，為了城市需求與農業產量而抽取地下水，這些都是最好的例子。在減少對環境危害的同時，這些產業轉型的複雜過程將帶來工作機會、生活用品、日常服務。

Researchers must also grapple with the changing demographics of housing and how it will impact human well-being: In the next 35 to 50 years, the number of people living in cities will double. Much of the growth will likely happen in the developing world in cities that currently have 30,000 to 3 million residents. Coping with that huge urban influx will require everything from energyefficient ways to make concrete to simple ways to purify drinking water.

研究人員還必須與正在變遷的人口地理學中，房屋興建及其如何影響人類的幸福等問題搏鬥：在接下來的三十五到五十年，居住在城市裡的人口將加倍成長。人口成長主要發生在開發中地區的城市，尤其那些目前人口在三萬到三百萬的城市。要處理這種城市群聚的問題，需要具備高效率使用能源的技巧，並在城市中興建一個匝渠來簡化淨水過程。

And in an age of global television and relentless advertising, what will happen to patterns of consumption? The world clearly can’t support 10 billion people living like Americans do today. Whether science—both the natural and social sciences—and technology can crank up efficiency and solve the problems we’ve created is perhaps the most critical question the world faces. Mustering the political will to make hard choices is, however, likely to be an even bigger challenge.

同時，在一個全球電視傳播以及廣告永不停息的時代，我們也不禁擔心消費的形態將會如何演變？地球很明顯無法負擔一百億人以當今美國人的生活型態。不論自然科學、社會科學以及科技，都可以增進資源使用的效率與解決問題，這個問題是由人類製造出來，卻也是當今人類面對最嚴肅的考驗。但要結合政治理念來作為決定的關鍵，其實是一個更大的挑戰。

問題討論

 Will mathematicians unleash the power of the Navier-Stokes equations?

 數學家解得開那維葉‧史托克斯（Navier-Stokes)方程式的奧秘嗎？

 First written down in the 1840s, the equations hold the keys to understanding both smooth and turbulent flow. To harness them, though, theorists must find out exactly when they work and under what conditions they break down.

 寫於1840年代，此方程式是理解氣流的平順與狂暴之關鍵。為了解釋此方程式，理論家必須找出什麼時候此方程式適用，以及什麼情況下它們會失靈。

 Does Poincaré’s test identify spheres in our-dimensional space?

 龐加萊（Poincaré）的實驗是驗證了四度空間的領域呢？

 You can tie a string around a doughnut, but it will slide right off a sphere. The mathematical principle behind that observation can reliably spot every spherelike object in 3D space. Henri Poincaré conjectured that it should also work in the next dimension up, but no one has proved it yet.

 你可以順著甜甜圈綁一圈線，但這條線最後還是會滑出該環形。經過觀察後的數學理論可以準確地點出在三度空間中的每個環型物。昂希‧龐加萊（Henri Poincaré）因此推測，在下一度空間中也適用。但至今沒人能證明它。

 Do mathematically interesting zero-value solutions of the Riemann zeta function all have the

 form a + bi?

 饒富數學趣味的里曼（Riemann）「zeta 作用」的零值（zero-value）解決方案，都可以構成a + bi方程式嗎？

 Don’t sweat the details. Since the mid-19th century, the “Riemann hypothesis” has been the monster catfish in mathematicians’ pond. If true, it will give them a wealth of information about the distribution of prime numbers and other longstanding mysteries.

 不要竭盡心思去探究細節。自從十九世紀中葉以後，「里曼假說」成了數學界的魔鬼鯰魚。如果此假說為真，這將為解開許多關於質數的分佈以及其他長久以來的迷思，提供豐富的資訊。

 Does the Standard Model of particle physics rest on solid mathematical foundations?

 在粒子物理學中的「標準模式」是否建立在紮實的數學基礎上？

 For almost 50 years, the model has rested on “quantum Yang-Mills theory,” which links the behavior of particles to structures found in geometry. The theory is breathtakingly elegant and useful—but no one has proved that it’s sound.

 大約五十年來，這個模式是建立在「量子楊‧米爾斯理論」(quantum Yang-Mills theory)基礎上。這和幾何學基礎的粒子活動有關。這種理論極度簡潔而且實用，但卻沒有人能夠徹底證實它。

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