Mathematics during World War I

SUMMARY: World War I saw an increased emphasis on applied mathematics but ultimately disrupted mathematics research.

Although mathematicians were not as heavily involved with the conduct of World War I as they would be with World War II, the four years of conflict impacted the field of mathematics in two main ways: they severed international ties among researchers, thus slowing collaborative research efforts; and the war provided the circumstances for applied mathematics to develop more fully through military research. Many mathematicians contributed their knowledge and abilities to the war effort. At the same time, others published papers unrelated to the military, worked to encourage reconciliation among mathematicians of warring nations, or strove to end the war outright. World War I, which was fought from 1914 to 1918, was precipitated by the assassination of Archduke Franz Ferdinand of Austria. After the initial declaration of war on Serbia by Austria, countries with various political alliances joined the fighting, with the result that more than 30 countries on five continents were ultimately named as combatants. The massive scope of this first truly global war led U.S. President Woodrow Wilson to refer to it as the “war to end all wars.”

Mathematics Applied to Military Research

Some mathematicians turned their attention to more practical and applied uses of the field. World War I saw extensive use of both trench warfare, which the United States had already experienced somewhat during the U.S. Civil War; and potent chemical weapons, like mustard gas. In the United States and in Europe, mathematicians researched ballistics and aeronautics as the warring countries sought advantages in firepower on land and began to realize the potential of air power. Mathematician John Littlewood performed research on ballistics and improved tables for the British Royal Garrison Artillery. In the United States, important figures such as Gilbert Bliss, Oswald Veblen, Norbert Wiener, and Forest Ray Moulton worked at the U.S. Army’s Aberdeen Proving Ground, Maryland, in ordnance and improvement in ballistics calculations. The American Mathematical Society published, in 1919, a list of over 175 mathematicians working in some capacity to support the war effort. The National Advisory Committee on Aeronautics also began construction of the Langley Laboratory in 1917, although research did not fully get underway until a few years later.

Similarly, Europeans conducted research with the aim of improving military operations. The British mathematician Frederick William Lanchester devised a formula to calculate the likely outcome of a battle between opponents of different strengths. He also published a series of articles on the military potential of aeronautics, which in 1916 were collected into a book. At Göttingen, Germany, Felix Klein and others instituted the Aerodynamic Proving Ground in 1917. In Italy, Mauro Picone investigated new methods for calculating ballistics tables, and Vito Volterra proposed using helium in airships.

As was the case in many wars dating back into antiquity, codes and cyphers played an important role. For example, “trench codes” consisting of three number or letter groups were used for rapid communications of tactical situations but they were fairly easily cracked and were quickly supplanted by more complex structures. The Germans widely employed the ADFGVX cypher, so named because only those six letters were used in coded messages. They had been chosen to minimize operator error because when those letters are sent by Morse code, they sound very different from one other. The code was a fractionating transposition cipher using a modified Polybius square, named for second-century B.C.E. historian Polybius of Megalopolis, with a single columnar transposition.

The cypher keys were typically changed every few days and the code was broken in only a few isolated cases during the war. A general solution was found in the 1930s by William Friedman, who is often referred to as the “father of modern U.S. cryptography.” The Germans also used some double transposition cyphers, which applied the same transposition key horizontally and vertically to the same matrix. In addition, they proved to be skillful in deciphering the codes of others, and the U.S. Army began to experiment with using Native American languages as military code. Several Choctaw soldiers served in the U.S. Army in Europe during World War I and are credited with helping to win some major battles.

The goal of war-related mathematicians was to improve the efficiency of military action. In the United States, this goal also applied to the home front. Allyn A. Young, the president of the American Statistical Association, proposed in a December 1917 address that a central statistical office or commission be established to aid the coordination of various boards and agencies then gathering statistics related to the war.

A greater division between mathematics research and teaching concerns also occurred around the time of World War I, as evidenced by the founding and branching off of the Mathematician Association of America in 1915 and the National Council of Teachers of Mathematics in 1920 from the more research-focused American Mathematical Society.

Non-Military Research During the War Years

Although much mathematical work from 1914 to 1918 related to improving military capability, there were many other notable advances that did not have immediate effects on war power. For instance, Albert Einstein published his general theory of relativity in 1915. David Hilbert also published field equations about that time. While a prisoner of war in Russia, the Polish mathematician Waclaw Sierpinski published a paper on his fractal triangle. Together with Godfrey Harold Hardy, after arriving at Cambridge University on Hardy’s invitation, Srinivasa Iyengar Ramanujan published a series of papers on number theory during the war.

Efforts for Peace and Reconciliation

At the same time, some mathematicians focused not on improving the conduct of war or other research, but instead on ending the conflict and reconciling with their colleagues in the peace that would follow. Perhaps the most famous case is that of the British mathematician Bertrand Russell, who soon after the turn of the century had identified a paradox that challenged assumptions of set theory and in the years immediately before the war had co-authored Principia Mathematica with Alfred North Whitehead. Repulsed by the battlefield slaughters and the general support of his countrymen for the war, Russell became an increasingly active pacifist, eventually taking part in public demonstrations and spending six months in prison for his antiwar writings.

Less dramatically, but still forcefully, the German David Hilbert made a point of recognizing the accomplishment of colleagues in enemy countries. The Dutch geometer Luitzen Egbertus Jan “L. E. J.” Brouwer worked after the war to bring German mathematicians back into recognition. Gosta Mittag-Leffler, a Swedish mathematician, deliberately published English, French, and German papers in his journal Acta Mathematica. After the war, he and Godfrey Harold Hardy worked to encourage reconciliation with German researchers.

Approaching the cause of peace from another angle, the Quaker mathematician Lewis Fry Richardson, who had served in an ambulance unit in France during the war, worked to understand the causes of wars so as to better prevent them. A limited printing of his first paper on the subject, “The Mathematical Psychology of War,” appeared in 1919. In later decades, as World War II loomed, Richardson would return to the subject.

Laying the Mathematical Groundwork for World War II

An adage about World War II, the greatest calamity to befall humankind, was that its seeds were sown in World War I. This is accurate in almost all contexts and certainly in regards to mathematical advancements. Many of the technological innovations of the First World War came to maturation in World War II, during which they would unleash unprecedented levels of destruction. In terms of military systems, and with perhaps the exception of submarines, weapons such as airplanes and tanks started in World War I as oddities only to become effective weapons in their own right by the end of the hostilities. Their effectiveness would finally render obsolete such historical assets such as horses, a mainstay of warfare since time immemorial. These weapons would continue to advance during the interwar years and would become national strategic assets as the Second World War began in 1939.

Some technologies not utilized during World War I saw their underlying principles become widespread shortly afterwards. These, too, would see great employment during World War II. An example was Radio Detection and Ranging (Radar). Radar, based on the mathematical principle of how radio waves change direction, shape, and velocity upon impact with other objects, became a weapon of war. Wave changes became measurable by machines, and the trajectory of aircraft and ships were identifiable such that they were attacked and destroyed. Rocketry began to make significant advances in this era, and by the next war, it evolved into ballistic missiles. A scant half-century later, rockets would propel humankind to the moon. Finally, the physical properties of the atom, harnessed by government programs of unprecedented scale, became channeled into weapons of mass destruction.

Conclusion

The death of possible future contributors to the field of mathematics during World War I as a whole was, of course, an incalculable loss. By disrupting the continuity of research and discovery, the war also delayed advances in areas of mathematics such as topology and set theory. At the same time, however, the possible applied uses of mathematics began to receive more attention and appreciation. In addition, national governments became more aware of the military value of mathematicians—a value that they would exploit much more thoroughly and effectively in World War II.

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