Dangerous Liaison 

From mathematics to war

Thursday, Feb 19 2004
Illustration by P. Jay Fidler

As Bush’s proposed military budget — $401.7 billion, not counting Iraq and Afghanistan — sickeningly reminds us, numbers have always figured large in the annals of war. Behind the dollars lies another story about the increasing dependence of the modern military on numerical and mathematical techniques. In a new book simply titled Mathematics and War, two Danish writer-editors, Bernhelm Booss-Bavnbek and Jens Hoyrup, argue that although not widely recognized by the public, “mathematical thinking, mathematical methods and mathematics-based technology has become an integral and even essential part of modern warfare.”

While the role of physics in the service of weaponry has been analyzed extensively, little has been written about the military use of its ethereal cousin. Hence the importance of this seminal volume, which grew out of an International Meeting on Mathematics and War held in Karlskrona, Sweden, in 2002. Though most participants were Scandinavian, much of their analysis focused on developments in Britain and the U.S. during and after the Second World War, for as many of the authors here stress, WWII ushered in a partnership between science and the military unprecedented in history.

The results of this increasingly dangerous liaison are illustrated in a table that details improvements in bomber precision (see below). The CEP, or Circular Error Probable, is a common measure (given in meters) of the average bombing proficiency range. Technically, it is the radius of a disc around a target within which 50 percent of shots hit. From WWII to the Kosovo conflict we witness an almost hundredfold shrinkage in the CEP — and inversely, a corresponding hike in bombing efficiency. The final column shows the number of bombs required to “destroy” (i.e. hit directly) a 20-by-30-meter target, a truly staggering upgrade which, the authors note, “depends essentially on the application of mathematics.”

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War CEP(m) Number of bombs WWII 1,100 9,140 Korea 330 823 Vietnam 130 128 Gulf 70 38 Kosovo 13 2

Until the 20th century, the use of math in the service of war had little to do with the development of weapons. For hundreds of years, military officers were almost the only people outside universities with general mathematical training, but this skill was deployed toward tasks like navigation, building fortifications, logistical planning and accounting. Bookkeeping has long been an engine of mathematical advance, and one of the most historically important of all mathematical inventions, the “place value system,” may well have been the bastard child of accountancy and war.

In place value systems, numbers are written with thousands, hundreds, tens, and so on, in consecutive positions — as in 372 or 4,967. Seemingly obvious to us, this Babylonian innovation marked a turning point in the history of mathematics. In 2074 B.C.E., the Babylonian king Shulgi implemented a military reform of the Sumerian empire, followed by an administrative reform in which large swaths of the population were forced into quasi-servile labor crews. Overseers ruthlessly clocked productivity in units of 1/60th of a day, or 12-minute chunks, and in an effort to standardize accounting practices, a base-60 place value system was introduced throughout the realm.

While the mathematical fuse was slow to ignite, it has burned brightly in recent years. Russian mathematician A.N. Kolmogorov virtually invented control theory to help stabilize guided missiles. American mathematician Norbert Wiener used theoretical research in statistics and information theory to improve the performance of anti-aircraft fire-control predictors. The effective use of radar and sonar both depend on mathematically aided signal processing, which is also essential in GPS systems so vital for military reconnaissance. And in 1940, Otto Frisch and Rudolf Peierls formulated the basic equations for the construction of a uranium bomb and famously determined that the critical mass was small enough to be militarily feasible.

If, as the authors note, “Hitler preached German invincibility by presenting the Wehrmacht as ‘Fast as greyhounds, tough as German Lederhosen, hard as Krupp steel,’ mathematics presents modern warfare as ‘fast by avionics, precise by GPS, safe by optimized operations planning.’” It is just this vision of war — swift, accurate and safe (at least for our side) — that the Pentagon has been shilling.


NOT ALL mathematical innovation is geared toward things that fire and fly. One of the most important uses of math in war is behind the scenes in what is known as Operations Research, which military planners rely on to help coordinate their vast resources. When tens of thousands of troops are deployed and hundreds of thousands of meals and vehicles and pieces of equipment have to be sent to them, the logistical nightmares hardly need elaborating. (The enormous difficulty of this task was highlighted earlier this month when an Army report detailed how dismally logistical operations are being handled in Iraq — radios, truck parts, even howitzer parts have often been dangerously low at the front.) Operations Research was initially invented by the British to help get the best use out of radar resources, but it quickly blossomed into a field of its own. In U.S. hands it soon gave rise to several new branches of mathematics, including linear programming, which formalizes the problem of how to ensure in a hugely complex system that things get where they need to be when they are supposed to be there.

The mathematization of Operations Research dates back to the Office of Scientific Research and Development (OSRD), set up by Vannevar Bush in 1941 to coordinate U.S. research on behalf of the war: Though initially left out in the cold, the mathematical community lobbied hard to be included, and in 1942 the OSRD established its Applied Mathematics Panel. In an illuminating essay, Tinne Hoff Kjeldsen, a professor in the history of mathematics at Roskilde University in Denmark, describes how many U.S. mathematicians desperately wanted to be part of the war effort, in part because they understood that postwar funding for scientific fields would depend on how much they were perceived to have contributed to the presumed victory.

By far the most important technological offspring of World War II was the computer, a device whose very conception was spawned in the brains of mathematicians originally involved in military projects like computing ballistic tables and cracking German encryption codes. Its exponential rise in power, abetted at every step by mathematical acumen, continues to transform the practice of war. If, as OSRD honcho James Conant famously remarked, WWII was the physicist’s war, World War III (god forbid) will surely be the computer scientist’s war.

Computers have become critical to almost every aspect of armed conflict — from guiding missiles and encrypting data to processing spy-satellite images and crunching numbers in those logistics equations. So central has the microchip become that another essay in this volume describes how, before the Gulf War, discussions were held to consider the possibility of exploding a high-altitude nuke over Iraq for the purpose of disabling enemy computers. Atmospheric atomic tests in the Pacific had shown that the electromagnetic pulse (EMP) generated would destroy most electronic equipment across a wide area. “Unfortunately,” the authors remark, “the EMP affects the equipment of friendly forces as well as that of an adversary.” “Directed energy weapons” that will apparently deliver a targeted pulse are currently being developed.

Increasingly, the military is also using computers for mathematical modeling. Beginning with flight simulators, modeling technology has given rise to amazingly complex simulated battle games. “Q,” the world’s second-fastest supercomputer, at Los Alamos National Laboratory, was built specifically to model nuclear tests — actual tests now being prohibited by the nonproliferation treaty. Booss-Bavnbek and Hoyrup tell us that when weapons designers wanted to understand the effects of fragmentation bombs on human bodies “but humanitarian concerns prohibited testing on pigs,” they turned instead to mathematical simulation.

So here’s my idea (listen up, W.): Instead of actually having the next war, let’s hand the job over to the mathematicians and computer scientists and let them simulate the entire thing. It’ll be faster, safer, and a hell of a lot cheaper.

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