2011年10月20日 星期四

New ways with old numbers

New ways with old numbers

Academics are always being asked to demonstrate the “impact” of their research. (Is it like being hit by a rogue cyclist? Or is it more like a pile-driver, or even an asteroid strike?) But while it is not unreasonable to ask whether a particular piece of academic research is useful, the difficulties in answering the question are extraordinary.

學者總是被要求證明其研究的“影響”。 (是像被一輛不規矩的自行車撞了一下?還是更像一台打樁機、甚至是一次小行星撞擊?)儘管詢問某項學術研究是否有用並非不合情理,但要回答這一問題,難度非常大。

The quality of a piece of research is subjective, and using measures such as the number of peer-reviewed articles published simply outsources the subjective judgment to somebody else. But there is a deeper problem: in a complex world, it is impossible for anyone to judge what the significance of a research breakthrough might eventually be.


Nowhere is this more true than in the field of mathematics. The most famous example is the development of imaginary numbers. The very name conveys the supposed uselessness of the concept. Square the imaginary unit, i, and you get minus one. Baffling.


Imaginary numbers were regarded with great scepticism after they were developed in Bologna in the 16th century as the logical solution to an abstract problem. Eventually, however, they turned out to be essential for, among other applications, electrical engineering – hardly something that could have been imagined by their creators.


So are imaginary numbers typical of the unexpected bounties of pure mathematics – or an unrepresentative poster child? Two recent commentators have tried to expand the number of examples. Professor ​​Caroline Series of the University of Warwick devoted a recent presidential lecture at the British Science Festival to this topic, focusing on the applications of non-Euclidian geometry.

那麼,虛數在理論數學的意外收穫中到底有沒有代表性呢?最近,兩位評論人士試圖增加樣本數量。華威大學(University of Warwick)的卡羅琳•希爾麗絲(Caroline Series)教授最近在英國科學節(British Science Festival)上的校長演講就關注於這一主題,主要討論了非歐幾里得幾何學的應用。

When Euclid originally laid down the axioms of his geometric system 23 centuries ago, one of them seemed less than obvious. For 2,000 years mathematicians tried to derive the “fifth postulate” – equivalent to the claim that the internal angles of a triangle add up to 180 degrees – from more basic building blocks, and failed. Eventually it transpired that the axiom was optional. Consistent systems of geometry were possible in which the internal angles of triangles summed to more than 180 degrees, or even to fewer.

2300年前,當歐幾里得(Euclidian)最初提出構成他幾何體系的那些公理時,其中一個公理似乎一點也不顯眼。 2000年來,數學家們試圖用更基礎的公設推導出“第五公設”(fifth postulate)——其內容相當於一個三角形的內角之和等於180度——但都沒有成功。最終,事實表明,這一公理並非放之四海皆準。有可能存在一些一致的幾何體系,其中三角形的內角之和可能超過180度,甚至可能小於180度。

Surfaces on which the sum of angles in a triangle is less than 180 degrees look like leaves of kale. Prof Series points out that the development of kale-like geometric systems, called hyperbolic geometry, initially seemed a curiosity but made possible Einstein's theory of special relativity. Now, says Prof Series, hyperbolic geometry promises to advance our understanding of the way complex networks such as the internet behave and grow.


Peter Rowlett, a maths educator and historian, recently gathered further examples together in the journal Nature. The “sphere packing problem” – beginning with the conje​​cture that grocers have found the most efficient way to stack oranges – has been an open area of​​ research for four centuries, but in the 1970s a solution for eight-dimensional “spheres” was used to design efficient modems. This meant that internet access no longer required specialised cables.

數學教育家、歷史學家彼得•羅萊特(Peter Rowlett)最近在《自然》(Nature)雜誌上收集了更多的例子。 400年來,“球體填充問題”一直是一個懸而未決的研究領域。該問題源自於一種猜想,即水果店老闆發現瞭如何最有效地擺放橙子的方法,但在上世紀70年代,用於解決八維“空間”的一種方法被用於設計高效調製解調器這意味著,互聯網接入不再需要專門電纜。

Quaternions, which extend imaginary numbers into a further dimension, began to be developed by William Hamilton in Dublin in 1843. They were eclipsed by matrix algebra, before being rediscovered as indispensable for generating 3D computer graphics efficiently. Rowlett's contributors offered several other examples.

1843年,威廉•漢密爾頓(William Hamilton)在都柏林發明了四元數,將虛數擴展到四維空間。這一理論在很長一段時間內都因矩陣代數的存在而不受重視,直到被重新發現,成為高效製作3D電腦圖像不可或缺的理論。羅萊特的投稿人還提供了其它幾個例子。

Cost-benefit analysis has its place. But the benefits of academic research can pop up in such unexpected ways, sometimes immediately and sometimes after centuries. We should not set too much store by any bureaucrat's analysis of “academic impact”.


Tim Harford's latest book is 'Adapt:​​ Why Success Always Starts with Failure' (Little, Brown)

本文作者的新書《適者生存:為何失敗是成功之母》(Adapt:​​ Why Success Always Starts with Failure)由利特爾-布朗公司(Little, Brown)出版