Mathematical analysis of the Attack on Pearl Harbor

• SUMMARY: Mathematicians were involved in both the planning of and the response to Pearl Harbor. 

The attack on Pearl Harbor, a major engagement of World War II and the impetus for the United States’ entry into the war, took place early Sunday morning, December 7, 1941, on the island of Oahu, Hawaii. The Japanese Navy, commanded by Admiral Isoroku Yamamoto, planned and executed the surprise attack against the U.S. naval base and nearby army airfields. As a result, the United States declared war on Japan. In his address to Congress, President Franklin D. Roosevelt famously proclaimed December 7 “a date which will live in infamy.” 

Both leading up to and as a result of the attack on Pearl Harbor, mathematicians in Japan and the United States mobilized for the war. For instance, after Pearl Harbor, the American Mathematical Society and the Mathematical Association of America converted their War Preparedness Committee to a War Policy Committee to increase research on “mathematical problems for military or naval science, or rearmament” and to strengthen mathematics education in order to prepare undergraduate students for military service. The attack has also been surrounded by speculation as to how the United States could have been caught off guard so easily. The naval base had been designed as nearly impenetrable to surprise attack because of the geography and geometry of the island. However, new technologies made the attack possible: the aircraft carrier could bring low-flying aircraft within attack range, and the Japanese development of shallow-running torpedoes could skim the surface of the harbor’s relatively shallow water. One of the largest controversies involves U.S. efforts to decode Japanese communications that may have given forewarning of the attack. 

Logistical Computations

An important, but often unacknowledged, aspect of warfare consisted of logistics. These were the processes where every object that a fighting force would be consuming in the course of a battle needed to be identified, sourced, scheduled for delivery, and then transported to a location where it could be distributed to the forces that needed it. The same processes of re-supply needed to be undertaken to sustain the fighting force as it executed the battle plan. Both the Japanese and Americans conducted these complicated mathematical computations well in advance of the first Japanese ship departing port and heading to Pearl Harbor. Japanese logisticians would have computed the amount of fuel every participating Japanese ship needed to complete the round trip from Japan to the Hawaiian Islands. These computations would have also included the amount of food its naval forces would be consuming. An estimation had to be made how much aviation fuel would be needed for the aircraft, how many bombs were projected to be dropped, anti-aircraft shells needed to ward off attacks, medical supplies to care for a projection of wounded, etc. The American Navy, Marines, and Army Air Forces assigned the mission of defending Pearl Harbor would have had to complete the similar calculations. In a high-stakes operation such as the Pearl Harbor attack, the failure of either side to have completed these mathematical computations would have been criminally incompetent and an invitation to disaster. Due to technologies at the time, these types of computations were done without the use of computers. Today, twenty-first century logisticians employ sophisticated software systems and applications to manage these complex processes. In December 1941, Japanese and American logisticians had to compute all their estimations without the benefit of automation.

Japanese Mathematicians

Leading up to Pearl Harbor, the number of Japanese graduate students increased and several studied in Germany. Mathematicians applied lattice theory and logic to the design of circuits. In the 1930s, both the United States and Japan successfully built a cyclotron, an early particle accelerator. Mathematics was also important in electrical engineering and airplane design. With a focus on aerodynamics and science and technology policy, the Japanese Technology Board was founded in 1941. A statistical institute contributed to war production. Japanese cryptologists also created many variations of military codes that were in use prior to Pearl Harbor, such as Kaigun Ango—Sho D, later referred to as “JN-25B” by cryptanalysts in the United States. Before committing to the attack on Pearl Harbor, the Japanese Navy conducted feasibility studies that included calculations and considerations of their current military resources; the need for a longer, circuitous route outside the customary naval traffic lanes to avoid detection by both military and civilian ships; the probability of encountering severe winter storms and critical data obtained from spies in Hawaii, such as the patterns of military activity at Pearl Harbor. They concluded that the attack was possible, if dangerous, and they originally intended to specifically target U.S. aircraft carriers to optimize the long-term effects of the attack. Experimentation and simulated training attacks yielded a satisfactory plan only a few weeks before the event. 

U.S. Mathematicians

In the United States, mathematicians conducted ballistics research at Aberdeen Proving Ground. Max Munk used the calculus of variations in airfoil design at the National Advisory Committee for Aeronautics, a precursor to the National Aeronautics and Space Administration (NASA). Technology such as radar, developed by scientists and mathematicians including Christian Doppler and Luis Alvarez, served military uses, though it was still in its infancy at the time of Pearl Harbor. Responsibility for compiling codes for military use and using cryptology to decipher codes shifted from Military Intelligence to the Army Signal Corps in 1929. William Friedman was the chief civilian cryptologist at the Signal Intelligence Service. The U.S. Army at that time realized the importance of mathematics in deciphering, and the first three civilian cryptanalysts hired by the U.S. Army were mathematics teachers. 

Forewarning

Many wonder how the United States could not have known of the impending Japanese attack, which had been planned and practiced months in advance. The new radar installation on Opana Point did, in fact, detect the incoming Japanese attack planes, but they were ultimately mistaken for a group of U.S. planes that were due to arrive from the mainland that morning. A U.S. destroyer also spotted a Japanese submarine attempting to enter the harbor, which it reported, but the information was not acted upon immediately. Both would have given at least short-term warnings to the ships and personnel. However, much of the accountability is assigned to the U.S. and Japanese intelligence and counter-intelligence efforts. Correspondence declassified many years after the war suggests that the United States could at least partially understand the codes needed to monitor Japanese naval movements on the eve of Pearl Harbor. While U.S. and British cryptanalysts had successfully broken some Japanese codes, such as the MAGIC code, the United States was not able to determine from those messages that the attack was about to happen. The broken codes were the ones used primarily for diplomatic messages sent by the Japanese Foreign Office and military strategy was rarely shared with the Japanese Foreign Office. The U.S. Navy had three cryptanalysis centers devoted to breaking Japanese naval codes. Prior to the attack, American cryptanalysts had been using traffic analysis to follow Japanese naval movements. Traffic analysis is the process of looking for patterns in communications to infer if an attack is about to occur. According to the National Security Agency, the Japanese, aware that their communications were being monitored, issued “dummy traffic to mislead the eavesdroppers into thinking that some of the ships sailing through the North Pacific were still in home waters.” Additionally, as Japanese forces were preparing for the attack, radio traffic was limited, greatly reducing the ability of American intelligence to determine a pattern. These efforts to stymie cryptologists were effective in keeping the impending attack a secret from the United States. Mathematicians and historians continue to analyze whether signal intelligence techniques could have revealed Japan’s intentions. 

In the years following the 9/11 terror attacks on New York City and Washington DC, the investigating congressional commission wrote that the United States' intelligence agencies had suffered a "failure of imagination" in their inability to predict such a catastrophic event. This is likely what contributed to the inadequate focus the American military and intelligence organizations gave in 1941 to the possibility of a Japanese attack. Americans did not believe a Japanese attack would occur at Pearl Harbor because it was inconvenient and would have necessitated too many countermeasures. In retrospect, these same countermeasures would have been miniscule compared to the scope of military actions that had to be undertaken in the decades after. The same failure of imagination likely befell Israeli military and intelligence officials in their inability to forecast the October 7, 2023, Hamas attack. The Hamas offensive, achieving the same type of strategic surprise as did the Japanese in 1941 and Al Qaeda in 2001, breached the Israeli wall complex that separated Gaza from Israeli settlers. Over 1,200 Israeli citizens, mostly civilians were killed and 251 taken hostage.

Bibliography

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