title: Moon landing

text: In contrast, crewed vehicles need additional fuel after a lunar landing to enable a return trip back to Earth for the crew. Leaving this massive amount of required Earth-return fuel in lunar orbit until it is used later in the mission is far more efficient than taking such fuel down to the lunar surface in a Moon landing and then hauling it all back into space yet again, working against lunar gravity both ways. Such considerations lead logically to a lunar orbit rendezvous mission profile for a crewed Moon landing.


title: Moon landing

text: Accordingly, beginning in mid-1966 both the U.S. and U.S.S.R. naturally progressed into missions featuring lunar orbit as a prerequisite to a crewed Moon landing. The primary goals of these initial uncrewed orbiters were extensive photographic mapping of the entire lunar surface for the selection of crewed landing sites and, for the Soviets, the checkout of radio communications gear that would be used in future soft landings.


title: Moon landing

text: An unexpected major discovery from initial lunar orbiters were vast volumes of dense materials beneath the surface of the Moon's maria. Such mass concentrations ("mascons") can send a crewed mission dangerously off course in the final minutes of a Moon landing when aiming for a relatively small landing zone that is smooth and safe. Mascons were also found over a longer period of time to greatly disturb the orbits of low-altitude satellites around the Moon, making their orbits unstable and forcing an inevitable crash on the lunar surface in the relatively short period of months to a few years.


title: Moon landing

text: Controlling the location of impact for spent lunar orbiters can have scientific value. For example, in 1999 the NASA Lunar Prospector orbiter was deliberately targeted to impact a permanently shadowed area of Shoemaker Crater near the lunar south pole. It was hoped that energy from the impact would vaporize suspected shadowed ice deposits in the crater and liberate a water vapor plume detectable from Earth. No such plume was observed. However, a small vial of ashes from the body of pioneer lunar scientist Eugene Shoemaker was delivered by the Lunar Prospector to the crater named in his honor – currently the only human remains on the Moon.


title: Moon landing

text: It is possible to aim a spacecraft from Earth so it will loop around the Moon and return to Earth without entering lunar orbit, following the so-called free return trajectory. Such circumlunar loop missions are simpler than lunar orbit missions because rockets for lunar orbit braking and Earth return are not required. However, a crewed circumlunar loop trip poses significant challenges beyond those found in a crewed low-Earth-orbit mission, offering valuable lessons in preparation for a crewed Moon landing. Foremost among these are mastering the demands of re-entering the Earth's atmosphere upon returning from the Moon.


title: Moon landing

text: Inhabited Earth-orbiting vehicles such as the Space Shuttle return to Earth from speeds of around . Due to the effects of gravity, a vehicle returning from the Moon hits Earth's atmosphere at a much higher speed of around . The g-loading on astronauts during the resulting deceleration can be at the limits of human endurance even during a nominal reentry. Slight variations in the vehicle flight path and reentry angle during a return from the Moon can easily result in fatal levels of deceleration force.


title: Moon landing

text: Achieving a crewed circumlunar loop flight prior to a crewed lunar landing became a primary goal of the Soviets with their Zond spacecraft program. The first three Zonds were robotic planetary probes; after that, the Zond name was transferred to a completely separate human spaceflight program. The initial focus of these later Zonds was extensive testing of required high-speed reentry techniques. This focus was not shared by the U.S., who chose instead to bypass the stepping stone of a crewed circumlunar loop mission and never developed a separate spacecraft for this purpose.


title: Moon landing

text: Initial crewed spaceflights in the early 1960s placed a single person in low Earth orbit during the Soviet Vostok and U.S. Mercury programs. A two-flight extension of the Vostok program known as Voskhod effectively used Vostok capsules with their ejection seats removed to achieve Soviet space firsts of multiple person crews in 1964 and spacewalks in early 1965. These capabilities were later demonstrated by the U.S. in ten Gemini low Earth orbit missions throughout 1965 and 1966, using a totally new second-generation spacecraft design that had little in common with the earlier Mercury. These Gemini missions went on to prove techniques for orbital rendezvous and docking crucial to a crewed lunar landing mission profile.


title: Moon landing

text: After the end of the Gemini program, the Soviet Union began flying their second-generation Zond crewed spacecraft in 1967 with the ultimate goal of looping a cosmonaut around the Moon and returning him or her immediately to Earth. The Zond spacecraft was launched with the simpler and already operational Proton launch rocket, unlike the parallel Soviet human Moon landing effort also underway at the time based on third-generation Soyuz spacecraft requiring development of the advanced N-1 booster. The Soviets thus believed they could achieve a crewed Zond circumlunar flight years before a U.S. human lunar landing and so score a propaganda victory. However, significant development problems delayed the Zond program and the success of the U.S. Apollo lunar landing program led to the eventual termination of the Zond effort.


title: Moon landing

text: Like Zond, Apollo flights were generally launched on a free return trajectory that would return them to Earth via a circumlunar loop if a service module malfunction failed to place them in lunar orbit. This option was implemented after an explosion aboard the Apollo 13 mission in 1970, which is the only crewed circumlunar loop mission flown to date.


title: Moon landing

text: Zond 5 was the first spacecraft to carry life from Earth to the vicinity of the Moon and return, initiating the final lap of the Space Race with its payload of tortoises, insects, plants, and bacteria. Despite the failure suffered in its final moments, the Zond 6 mission was reported by Soviet media as being a success as well. Although hailed worldwide as remarkable achievements, both these Zond missions flew off-nominal reentry trajectories resulting in deceleration forces that would have been fatal to humans.


title: Moon landing

text: As a result, the Soviets secretly planned to continue uncrewed Zond tests until their reliability to support human flight had been demonstrated. However, due to NASA's continuing problems with the lunar module, and because of CIA reports of a potential Soviet crewed circumlunar flight in late 1968, NASA fatefully changed the flight plan of Apollo 8 from an Earth-orbit lunar module test to a lunar orbit mission scheduled for late December 1968.


title: Moon landing

text: In early December 1968 the launch window to the Moon opened for the Soviet launch site in Baikonur, giving the USSR their final chance to beat the US to the Moon. Cosmonauts went on alert and asked to fly the Zond spacecraft then in final countdown at Baikonur on the first human trip to the Moon. Ultimately, however, the Soviet Politburo decided the risk of crew death was unacceptable given the combined poor performance to that point of Zond/Proton and so scrubbed the launch of a crewed Soviet lunar mission. Their decision proved to be a wise one, since this unnumbered Zond mission was destroyed in another uncrewed test when it was finally launched several weeks later.


title: Moon landing

text: By this time flights of the third generation U.S. Apollo spacecraft had begun. Far more capable than the Zond, the Apollo spacecraft had the necessary rocket power to slip into and out of lunar orbit and to make course adjustments required for a safe reentry during the return to Earth. The Apollo 8 mission carried out the first human trip to the Moon on 24 December 1968, certifying the Saturn V booster for crewed use and flying not a circumlunar loop but instead a full ten orbits around the Moon before returning safely to Earth. Apollo 10 then performed a full dress rehearsal of a crewed Moon landing in May 1969. This mission orbited within of the lunar surface, performing necessary low-altitude mapping of trajectory-altering mascons using a factory prototype lunar module too heavy to land. With the failure of the robotic Soviet sample return Moon landing attempt Luna 15 in July 1969, the stage was set for Apollo 11.


title: Moon landing

text: Plans for human Moon exploration began during the Eisenhower administration. In a series of mid-1950s articles in "Collier's" magazine, Wernher von Braun had popularized the idea of a crewed expedition to establish a lunar base. A human Moon landing posed several daunting technical challenges to the US and USSR. Besides guidance and weight management, atmospheric re-entry without ablative overheating was a major hurdle. After the Soviets launched Sputnik, von Braun promoted a plan for the US Army to establish a military lunar outpost by 1965.


title: Moon landing

text: After the early Soviet successes, especially Yuri Gagarin's flight, US President John F. Kennedy looked for a project that would capture the public imagination. He asked Vice President Lyndon Johnson to make recommendations on a scientific endeavor that would prove US world leadership. The proposals included non-space options such as massive irrigation projects to benefit the Third World. The Soviets, at the time, had more powerful rockets than the US, which gave them an advantage in some kinds of space mission.


title: Moon landing

text: Advances in US nuclear weapon technology had led to smaller, lighter warheads; the Soviets' were much heavier, and the powerful R-7 rocket was developed to carry them. More modest missions such as flying around the Moon, or a space lab in lunar orbit (both were proposed by Kennedy to von Braun), offered too much advantage to the Soviets; "landing", however, would capture the world's imagination.


title: Moon landing

text: Johnson had championed the US human spaceflight program ever since Sputnik, sponsoring legislation to create NASA while he was still a senator. When Kennedy asked him in 1961 to research the best achievement to counter the Soviets' lead, Johnson responded that the US had an even chance of beating them to a crewed lunar landing, but not for anything less. Kennedy seized on Apollo as the ideal focus for efforts in space. He ensured continuing funding, shielding space spending from the 1963 tax cut, but diverting money from other NASA scientific projects. These diversions dismayed NASA's leader, James E. Webb, who perceived the need for NASA's support from the scientific community.


title: Moon landing

text: The Moon landing required development of the large Saturn V launch vehicle, which achieved a perfect record: zero catastrophic failures or launch vehicle-caused mission failures in thirteen launches.


title: Moon landing

text: For the program to succeed, its proponents would have to defeat criticism from politicians both on the left (more money for social programs) and on the right (more money for the military). By emphasizing the scientific payoff and playing on fears of Soviet space dominance, Kennedy and Johnson managed to swing public opinion: by 1965, 58 percent of Americans favored Apollo, up from 33 percent two years earlier. After Johnson became President in 1963, his continuing defense of the program allowed it to succeed in 1969, as Kennedy had planned.


title: Moon landing

text: Soviet leader Nikita Khrushchev said in October 1963 the USSR was "not at present planning flight by cosmonauts to the Moon," while insisting that the Soviets had not dropped out of the race. Only after another year did the USSR fully commit itself to a Moon-landing attempt, which ultimately failed.


title: Moon landing

text: At the same time, Kennedy had suggested various joint programs, including a possible Moon landing by Soviet and U.S. astronauts and the development of better weather-monitoring satellites, eventually resulting in the Apollo-Soyuz mission. Khrushchev, sensing an attempt by Kennedy to steal Russian space technology, rejected the idea at first: if the USSR went to the Moon, it would go alone. Though Khrushchev was eventually warming up to the idea, but the realization of a joint Moon landing was choked by Kennedy's assassination.


title: Moon landing

text: Sergey Korolev, the Soviet space program's chief designer, had started promoting his Soyuz craft and the N1 launcher rocket that would have the capability of carrying out a human Moon landing. Khrushchev directed Korolev's design bureau to arrange further space firsts by modifying the existing Vostok technology, while a second team started building a completely new launcher and craft, the Proton booster and the Zond, for a human cislunar flight in 1966. In 1964 the new Soviet leadership gave Korolev the backing for a Moon landing effort and brought all crewed projects under his direction.


title: Moon landing

text: With Korolev's death and the failure of the first Soyuz flight in 1967, coordination of the Soviet Moon landing program quickly unraveled. The Soviets built a landing craft and selected cosmonauts for a mission that would have placed Alexei Leonov on the Moon's surface, but with the successive launch failures of the N1 booster in 1969, plans for a crewed landing suffered first delay and then cancellation.


title: Moon landing

text: A program of automated return vehicles was begun, in the hope of being the first to return lunar rocks. This had several failures. It eventually succeeded with Luna 16 in 1970. But this had little impact, because the Apollo 11 and Apollo 12 lunar landings and rock returns had already taken place by then.


title: Moon landing

text: In total, twenty-four U.S. astronauts have traveled to the Moon. Three have made the trip twice, and twelve have walked on its surface. Apollo 8 was a lunar-orbit-only mission, Apollo 10 included undocking and Descent Orbit Insertion (DOI), followed by LM staging to CSM redocking, while Apollo 13, originally scheduled as a landing, ended up as a lunar fly-by, by means of free return trajectory; thus, none of these missions made landings. Apollo 7 and Apollo 9 were Earth-orbit-only missions. Apart from the inherent dangers of crewed Moon expeditions as seen with Apollo 13, one reason for their cessation according to astronaut Alan Bean is the cost it imposes in government subsidies.


title: Moon landing

text: President Richard Nixon had speechwriter William Safire prepare a condolence speech for delivery in case Armstrong and Aldrin became marooned on the Moon's surface and could not be rescued.


title: Moon landing

text: On 16 August 2006, the Associated Press reported that NASA is missing the original Slow-scan television tapes (which were made before the scan conversion for conventional TV) of the Apollo 11 Moon walk. Some news outlets have mistakenly reported the SSTV tapes found in Western Australia, but those tapes were only recordings of data from the Apollo 11 Early Apollo Surface Experiments Package. The tapes were found in 2008 and sold at auction in 2019 for the 50th anniversary of the landing.


title: Moon landing

text: Scientists believe the six American flags planted by astronauts have been bleached white because of more than 40 years of exposure to solar radiation. Using LROC images, five of the six American flags are still standing and casting shadows at all of the sites, except Apollo 11. Astronaut Buzz Aldrin reported that the flag was blown over by the exhaust from the ascent engine during liftoff of Apollo 11.


title: Moon landing

text: Launched on 24 January 1990, 11:46 UTC. At the end of its mission, the Japanese lunar orbiter Hiten was commanded to crash into the lunar surface and did so on 10 April 1993 at 18:03:25.7 UT (11 April 03:03:25.7 JST).


title: Moon landing

text: Lunar Prospector was launched on 7 January 1998. The mission ended on 31 July 1999, when the orbiter was deliberately crashed into a crater near the lunar south pole after the presence of water ice was successfully detected.


title: Moon landing

text: Launched 27 September 2003, 23:14 UTC from the Guiana Space Centre in Kourou, French Guiana. At the end of its mission, the ESA lunar orbiter SMART-1 performed a controlled crash into the Moon, at about 2 km/s. The time of the crash was 3 September 2006, at 5:42 UTC.


title: Moon landing

text: The impactor, the Moon Impact Probe, an instrument on Chandrayaan-1 mission, impacted near Shackleton crater at the south pole of the lunar surface at 14 November 2008, 20:31 IST. Chandrayaan-1 was launched on 22 October 2008, 00:52 UTC.


title: Moon landing

text: The Chinese lunar orbiter Chang'e 1, executed a controlled crash onto the surface of the Moon on 1 March 2009, 20:44 GMT, after a 16-month mission. Chang'e 1 was launched on 24 October 2007, 10:05 UTC.


title: Moon landing

text: SELENE or "Kaguya" after successfully orbiting the Moon for a year and eight months, the main orbiter was instructed to impact on the lunar surface near the crater Gill at 18:25 UTC on 10 June 2009. SELENE or "Kaguya" was launched on 14 September 2007.


title: Moon landing

text: The LCROSS data collecting shepherding spacecraft was launched together with the Lunar Reconnaissance Orbiter (LRO) on 18 June 2009 on board an Atlas V rocket with a Centaur upper stage. On 9 October 2009, at 11:31 UTC, the Centaur upper stage impacted the lunar surface, releasing the kinetic energy equivalent of detonating approximately 2 tons of TNT (8.86 GJ). Six minutes later at 11:37 UTC, the LCROSS shepherding spacecraft also impacted the surface.


title: Moon landing

text: The GRAIL mission consisted of two small spacecraft: GRAIL A ("Ebb"), and GRAIL B ("Flow"). They were launched on 10 September 2011 on board a Delta II rocket. GRAIL A separated from the rocket about nine minutes after launch, and GRAIL B followed about eight minutes later. The first probe entered orbit on 31 December 2011 and the second followed on 1 January 2012. The two spacecraft impacted the Lunar surface on 17 December 2012.


title: Moon landing

text: LADEE was launched on 7 September 2013. The mission ended on 18 April 2014, when the spacecraft's controllers intentionally crashed LADEE into the far side of the Moon, which, later, was determined to be near the eastern rim of Sundman V crater.


title: Moon landing

text: The Manfred Memorial Moon Mission was launched on the 23 October 2014. It conducted a lunar flyby and operated for 19 days which was four times longer than expected. The Manfred Memorial Moon Mission remained attached to the upper stage of its launch vehicle (CZ-3C/E). The spacecraft along with its upper stage impacted the Moon on 4 March 2022.


title: Moon landing

text: On 14 December 2013 at 13:12 UTC Chang'e 3 soft-landed a rover on the Moon. This was China's first soft landing on another celestial body and world's first lunar soft landing since Luna 24 on 22 August 1976. The mission was launched on 1 December 2013. After successful landing, the lander release the Yutu rover, which moved 114 meters before being immobilized due to system malfunction. But the rover was still operational until July 2016.


title: Moon landing

text: On 3 January 2019 at 2:26 UTC Chang'e 4 became the first spacecraft to land on the far side of the Moon. Chang'e 4 was originally designed as the backup of Chang'e 3. It was later adjusted as a mission to the far side of the Moon after the success of Chang'e 3. After making a successful landing within Von Kármán crater, the Chang'e 4 lander deployed the 140kg Yutu-2 rover and began human's very first close exploration of the far side of the Moon. Because the Moon blocks the communications between far side and the earth, a relay satellite, Queqiao, was launched to the Earth–Moon L2 Lagrangian point a few months prior to the landing to enable communications.


title: Moon landing

text: Yutu-2, the second lunar rover from China, was equipped with panoramic camera, lunar penetrating radar, visible and near-infrared Imaging spectrometer and advanced small analyzer for neutrals. As of July 2022, it has survived more than 1000 days on the lunar surface and is still driving with cumulative travel distance of over 1200 meters.


title: Moon landing

text: On 22 February 2019 at 01:45 UTC, SpaceX launched the "Beresheet" lunar lander, developed by Israel's SpaceIL organization. Launched from Cape Canaveral, Florida on a Falcon 9 booster, with the lander being one of three payloads on the rocket. Beresheet arrived near the Moon using a slow but fuel-efficient trajectory. Taking six weeks and several increasingly large orbits around the Earth, it first achieved a large elliptical orbit around Earth with an apogee near . At that point, with a short deceleration burn, it was caught by the Moon's gravity in a highly elliptical lunar orbit, an orbit which was circularized and reduced in diameter over a week's time, before attempting a landing on the Moon's surface on 11 April 2019. The mission was the first Israeli, and the first privately funded, lunar landing attempt. SpaceIL was originally conceived in 2011 as a venture to pursue the Google Lunar X Prize. On 11 April 2019 "Beresheet" crashed on the surface of the Moon, as a result of a main engine failure in the final descent. The "Beresheet" lunar lander's target landing destination was within Mare Serenitatis, a vast volcanic basin on the Moon's northern near side. Despite the failure, the mission represents the closest a private entity has come to a soft lunar landing.


title: Moon landing

text: ISRO, the Indian National Space agency, launched Chandrayaan 2 on 22 July 2019. It has 3 major modules: Orbiter, Lander and Rover. Each of these modules has scientific instruments from scientific research institutes in India and the US. The spacecraft was launched by the GSLV Mk III. On 7 September 2019 at 1:50 IST Chandryaan 2's "Vikram" lander started the soft landing sequence. Contact was lost on above the lunar surface after the rough braking phase, and was not regained. From the images of the Lunar Reconnaissance Orbiter and chandrayaan orbiter it was found that the "Vikram" lander had crashed on the Moon and was destroyed.


title: Moon landing

text: On 6 December 2020 at 21:42 UTC Chang'e 5 landed and collected the first lunar soil samples in over 40 years, and then returned the samples to Earth. The 8.2t stack consisting of lander, ascender, orbiter and returner was launched to lunar orbit by a Long March 5 rocket on November 24. The lander-ascender combination was separated with the orbiter and returner before landing near Mons Rümker in Oceanus Procellarum. The ascender was later launched back to lunar orbit, carrying samples collected by the lander, and completed the first-ever robotic rendezvous and docking in lunar orbit. The sample container was then transferred to the returner, which successfully landed on Inner Mongolia on 16 December 2020, completing China's first extraterrestrial sample return mission.


title: Moon landing

text: Progress in space exploration has recently broadened the phrase "moon landing" to include other moons in the Solar System as well. The "Huygens" probe of the "Cassini–Huygens" mission to Saturn performed a successful moon landing on Titan in 2005. Similarly, the Soviet probe Phobos 2 came within of performing a landing on Mars' moon Phobos in 1989 before radio contact with that lander was suddenly lost. A similar Russian sample return mission called Fobos-Grunt ("grunt" means "soil" in Russian) launched in November 2011, but stalled in low Earth orbit. There is widespread interest in performing a future landing on Jupiter's moon Europa to drill down and explore the possible liquid water ocean beneath its icy surface.


title: Moon landing

text: After the failure of the "Vikram" lander of Chandrayaan-2, the Indian Space Research Organisation (ISRO) plans to re-attempt a soft landing with a third lunar exploration mission, Chandrayaan-3. It is scheduled to launch in the third quarter of 2022.


title: Moon landing

text: The Lunar Polar Exploration Mission is a robotic space mission concept by ISRO and Japan's space agency JAXA that would send a lunar rover and lander to explore south pole region of the Moon in 2024. JAXA is likely to provide launch service using the future H3 rocket, along with responsibility for the rover. ISRO would be responsible for the lander.


title: Moon landing

text: On 11 December 2017, US President Trump signed Space Policy Directive 1, which directed NASA to return to the Moon with a crewed mission, for "long-term exploration and use" and missions to other planets. On 26 March 2019, Vice President Mike Pence formally announced that the mission will include the first female lunar astronaut. The Artemis program has the goal of returning to the Moon with new launch systems.


title: Moon landing

text: Many conspiracists hold that the Apollo Moon landings were a hoax; however, empirical evidence is readily available to show that human Moon landings did occur. Anyone on Earth with an appropriate laser and telescope system can bounce laser beams off three retroreflector arrays left on the Moon by Apollo 11, 14 and 15, verifying deployment of the Lunar Laser Ranging Experiment at historically documented Apollo Moon landing sites and so proving equipment constructed on Earth was successfully transported to the surface of the Moon. In addition, in August 2009 NASA's Lunar Reconnaissance Orbiter began to send back high resolution photos of the Apollo landing sites. These photos show the large descent stages of the six Apollo Lunar Modules which were left behind, the tracks of the three Lunar Roving Vehicles, and the paths left by the twelve astronauts as they walked in the lunar dust. In 2016, then-U.S. president Barack Obama acknowledged that the Moon landing was not a hoax and publicly thanked the members of the television show "Mythbusters" for publicly proving as such in season 6 episode 2.


title: John von Neumann

text: John von Neumann (; , ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest coverage of any mathematician of his time and was said to have been "the last representative of the great mathematicians who were equally at home in both pure and applied mathematics". He integrated pure and applied sciences.


title: John von Neumann

text: Von Neumann made major contributions to many fields, including mathematics (foundations of mathematics, measure theory, functional analysis, ergodic theory, group theory, lattice theory, representation theory, operator algebras, matrix theory, geometry, and numerical analysis), physics (quantum mechanics, hydrodynamics, ballistics, nuclear physics and quantum statistical mechanics), economics (game theory and general equilibrium theory), computing (Von Neumann architecture, linear programming, numerical meteorology, scientific computing, self-replicating machines, stochastic computing), and statistics. He was a pioneer of the application of operator theory to quantum mechanics in the development of functional analysis, and a key figure in the development of game theory and the concepts of cellular automata, the universal constructor and the digital computer.


title: John von Neumann

text: Von Neumann published over 150 papers in his life: about 60 in pure mathematics, 60 in applied mathematics, 20 in physics, and the remainder on special mathematical subjects or non-mathematical ones. His last work, an unfinished manuscript written while he was dying in hospital, was later published in book form as "The Computer and the Brain".


title: John von Neumann

text: His analysis of the structure of self-replication preceded the discovery of the structure of DNA. In a shortlist of facts about his life he submitted to the National Academy of Sciences, he wrote, "The part of my work I consider most essential is that on quantum mechanics, which developed in Göttingen in 1926, and subsequently in Berlin in 1927–1929. Also, my work on various forms of operator theory, Berlin 1930 and Princeton 1935–1939; on the ergodic theorem, Princeton, 1931–1932."


title: John von Neumann

text: During World War II, von Neumann worked on the Manhattan Project with theoretical physicist Edward Teller, mathematician Stanislaw Ulam and others, problem-solving key steps in the nuclear physics involved in thermonuclear reactions and the hydrogen bomb. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon and coined the term "kiloton" (of TNT) as a measure of the explosive force generated. During this time and after the war, he consulted for a vast number of organizations including the Office of Scientific Research and Development, the Army's Ballistic Research Laboratory, the Armed Forces Special Weapons Project and the Oak Ridge National Laboratory. At the peak of his influence in the 1950s he was the chair for a number of critical Defense Department committees including the Nuclear Weapons Panel of the Air Force Scientific Advisory Board and the ICBM Scientific Advisory Committee as well as a member of the influential Atomic Energy Commission. He played a key role alongside Bernard Schriever and Trevor Gardner in contributing to the design and development of the United States' first ICBM programs. During this time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the Pentagon. As a Hungarian émigré, concerned that the Soviets would achieve nuclear superiority, he designed and promoted the policy of mutually assured destruction to limit the arms race.


title: John von Neumann

text: In honor of his achievements and contributions to the modern world, he was named in 1999 the "Financial Times" Person of the Century, as a representative of the century's characteristic ideal that the power of the mind could shape the physical world, and of the "intellectual brilliance and human savagery" that defined the 20th century.


title: John von Neumann

text: Von Neumann was born on December 28, 1903, to a wealthy, acculturated and non-observant Jewish family. His Hungarian birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.


title: John von Neumann

text: Von Neumann was born in Budapest, Kingdom of Hungary, which was then part of the Austro-Hungarian Empire. He was the eldest of three brothers; his two younger siblings were Mihály (English: Michael von Neumann; 1907–1989) and Miklós (Nicholas von Neumann, 1911–2011). His father, Neumann Miksa (Max von Neumann, 1873–1928) was a banker, who held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s. Miksa's father and grandfather were both born in Ond (now part of the town of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (English: Margaret Kann); her parents were Jakab Kann and Katalin Meisels of the Meisels family. Three generations of the Kann family lived in spacious apartments above the Kann-Heller offices in Budapest; von Neumann's family occupied an 18-room apartment on the top floor.


title: John von Neumann

text: On February 20, 1913, Emperor Franz Joseph elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire. The Neumann family thus acquired the hereditary appellation "Margittai", meaning "of Margitta" (today Marghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosen coat of arms depicting three marguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.


title: John von Neumann

text: Von Neumann was a child prodigy. When he was six years old, he could divide two eight-digit numbers in his head and could converse in Ancient Greek. When the six-year-old von Neumann caught his mother staring aimlessly, he asked her, "What are you calculating?"


title: John von Neumann

text: When they were young, von Neumann, his brothers and his cousins were instructed by governesses. Von Neumann's father believed that knowledge of languages other than their native Hungarian was essential, so the children were tutored in English, French, German and Italian. By the age of eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read and understood Borel's Théorie des Fonctions. But he was also particularly interested in history. He read his way through Wilhelm Oncken's 46-volume world history series ("General History in Monographs"). A copy was contained in a private library Max purchased. One of the rooms in the apartment was converted into a library and reading room, with bookshelves from ceiling to floor.


title: John von Neumann

text: Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914. Eugene Wigner was a year ahead of von Neumann at the Lutheran School and soon became his friend. This was one of the best schools in Budapest and was part of a brilliant education system designed for the elite. Under the Hungarian system, children received all their education at the one gymnasium. The Hungarian school system produced a generation noted for intellectual achievement, which included Theodore von Kármán (born 1881), George de Hevesy (born 1885), Michael Polanyi (born 1891), Leó Szilárd (born 1898), Dennis Gabor (born 1900), Eugene Wigner (born 1902), Edward Teller (born 1908), and Paul Erdős (born 1913). Collectively, they were sometimes known as "The Martians".


title: John von Neumann

text: Although von Neumann's father insisted von Neumann attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction in those areas in which he had displayed an aptitude. At the age of 15, he began to study advanced calculus under the renowned analyst Gábor Szegő. On their first meeting, Szegő was so astounded with the boy's mathematical talent that he was brought to tears. Some of von Neumann's instant solutions to the problems that Szegő posed in calculus are sketched out on his father's stationery and are still on display at the von Neumann archive in Budapest. By the age of 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition of ordinal numbers, which superseded Georg Cantor's definition. At the conclusion of his education at the gymnasium, von Neumann sat for and won the Eötvös Prize, a national prize for mathematics.


title: John von Neumann

text: According to his friend Theodore von Kármán, von Neumann's father wanted John to follow him into industry and thereby invest his time in a more financially useful endeavor than mathematics. In fact, his father asked von Kármán to persuade his son not to take mathematics as his major. Von Neumann and his father decided that the best career path was to become a chemical engineer. This was not something that von Neumann had much knowledge of, so it was arranged for him to take a two-year, non-degree course in chemistry at the University of Berlin, after which he sat for the entrance exam to the prestigious ETH Zurich, which he passed in September 1923. At the same time, von Neumann also entered Pázmány Péter University in Budapest, as a Ph.D. candidate in mathematics. For his thesis, he chose to produce an axiomatization of Cantor's set theory. He graduated as a chemical engineer from ETH Zurich in 1926 (although Wigner says that von Neumann was never very attached to the subject of chemistry), and passed his final examinations for his Ph.D. in mathematics simultaneously with his chemical engineering degree, of which Wigner wrote, "Evidently a Ph.D. thesis and examination did not constitute an appreciable effort." He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.


title: John von Neumann

text: Von Neumann's habilitation was completed on December 13, 1927, and he began to give lectures as a "Privatdozent" at the University of Berlin in 1928. He was the youngest person ever elected "Privatdozent" in the university's history in any subject. By the end of 1927, von Neumann had published 12 major papers in mathematics, and by the end of 1929, 32, a rate of nearly one major paper per month. In 1929, he briefly became a "Privatdozent" at the University of Hamburg, where the prospects of becoming a tenured professor were better, but in October of that year a better offer presented itself when he was invited to Princeton University as a visiting lecturer in mathematical physics.


title: John von Neumann

text: On New Year's Day 1930, von Neumann married Marietta Kövesi, who had studied economics at Budapest University. Von Neumann and Marietta had one child, a daughter, Marina, born in 1935. As of 2021 Marina is a distinguished professor emerita of business administration and public policy at the University of Michigan. The couple divorced on November 2, 1937. On November 17, 1938, von Neumann married Klara Dan, whom he had met during his last trips back to Budapest before the outbreak of World War II.


title: John von Neumann

text: In 1930, before marrying Marietta, von Neumann was baptized into the Catholic Church. Von Neumann's father, Max, had died in 1929. None of the family had converted to Christianity while Max was alive, but all did afterward.


title: John von Neumann

text: In 1933 Von Neumann was offered and accepted a life tenure professorship at the Institute for Advanced Study in New Jersey, when that institution's plan to appoint Hermann Weyl appeared to have failed. Von Neumann remained a mathematics professor there until his death, although he had announced his intention to resign and become a professor at large at the University of California, Los Angeles. His mother, brothers and in-laws followed von Neumann to the United States in 1939. Von Neumann anglicized his first name to John, keeping the German-aristocratic surname von Neumann. His brothers changed theirs to "Neumann" and "Vonneumann". Von Neumann became a naturalized citizen of the United States in 1937, and immediately tried to become a lieutenant in the United States Army's Officers Reserve Corps. He passed the exams easily but was rejected because of his age. His prewar analysis of how France would stand up to Germany is often quoted: "Oh, France won't matter."


title: John von Neumann

text: Klara and John von Neumann were socially active within the local academic community. His white clapboard house at 26 Westcott Road was one of Princeton's largest private residences. He always wore formal suits. He once wore a three-piece pinstripe while riding down the Grand Canyon astride a mule. Hilbert is reported to have asked, "Pray, who is the candidate's tailor?" at von Neumann's 1926 doctoral exam, as he had never seen such beautiful evening clothes.


title: John von Neumann

text: Von Neumann held a lifelong passion for ancient history and was renowned for his historical knowledge. A professor of Byzantine history at Princeton once said that von Neumann had greater expertise in Byzantine history than he did. He knew by heart much of the material in Gibbon's Decline and Fall and after dinner liked to engage in various historical discussions. Ulam noted that one time while driving south to a meeting of the American Mathematical Society, von Neumann would describe even the minutest details of the battles of the Civil War that occurred in the places they drove by. This kind of travel where he could be in a car and talk for hours on topics ranging from mathematics to literature without interruption was something he enjoyed very much.


title: John von Neumann

text: Von Neumann liked to eat and drink. His wife, Klara, said that he could count everything except calories. He enjoyed Yiddish and "off-color" humor (especially limericks). He was a non-smoker. In Princeton, he received complaints for regularly playing extremely loud German march music on his phonograph, which distracted those in neighboring offices, including Albert Einstein, from their work. Von Neumann did some of his best work in noisy, chaotic environments, and once admonished his wife for preparing a quiet study for him to work in. He never used it, preferring the couple's living room with its television playing loudly. Despite being a notoriously bad driver, he enjoyed driving—frequently while reading a book—occasioning numerous arrests as well as accidents. When Cuthbert Hurd hired him as a consultant to IBM, Hurd often quietly paid the fines for his traffic tickets.


title: John von Neumann

text: Von Neumann's closest friend in the United States was mathematician Stanislaw Ulam. A later friend of Ulam's, Gian-Carlo Rota, wrote, "They would spend hours on end gossiping and giggling, swapping Jewish jokes, and drifting in and out of mathematical talk." When von Neumann was dying in the hospital, every time Ulam visited, he came prepared with a new collection of jokes to cheer him up. Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up. Ulam noted that von Neumann's way of thinking might not be visual, but more aural.


title: John von Neumann

text: In February 1951 for the New York Times he had his brain waves scanned while at rest and while thinking (along with Albert Einstein and Norbert Wiener). "They generally showed differences from the average".


title: John von Neumann

text: In 1955, von Neumann was diagnosed with what was either bone, pancreatic or prostate cancer after he was examined by physicians following a fall, they discovered a mass growing near his collarbone. The cancer was possibly caused by his radiation exposure during his time in Los Alamos National Laboratory. He was not able to accept the proximity of his own demise, and the shadow of impending death instilled great fear in him. He invited a Catholic priest, Father Anselm Strittmatter, O.S.B., to visit him for consultation. Von Neumann reportedly said, "So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end," referring to Pascal's wager. He had earlier confided to his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't." Father Strittmatter administered the last rites to him. Some of von Neumann's friends, such as Abraham Pais and Oskar Morgenstern, said they had always believed him to be "completely agnostic". Of this deathbed conversion, Morgenstern told Heims, "He was of course completely agnostic all his life, and then he suddenly turned Catholic—it doesn't agree with anything whatsoever in his attitude, outlook and thinking when he was healthy." Father Strittmatter recalled that even after his conversion, von Neumann did not receive much peace or comfort from it, as he still remained terrified of death.


title: John von Neumann

text: On his deathbed he entertained his brother by reciting by heart and word-for-word the first few lines of each page of Goethe's "Faust". For example, it is recorded that one day his brother Mike read "Faust" to him, and when Mike paused to turn the pages, Von Neumann recited from memory the first few lines of the following page. On his deathbed, his mental capabilities became a fraction of what they were before, causing him much anguish. At times Von Neumann even forgot the lines that his brother recited from "Faust". Meanwhile, Clay Blair remarked that Von Neumann did not give up research until his death: "It was characteristic of the impatient, witty and incalculably brilliant John von Neumann that although he went on working for others until he could do no more, his own treatise on the workings of the brain—the work he thought would be his crowning achievement in his own name—was left unfinished." He died on February 8, 1957, at the Walter Reed Army Medical Center in Washington, D.C., under military security lest he reveal military secrets while heavily medicated. He was buried at Princeton Cemetery of Nassau Presbyterian Church in Princeton, Mercer County, New Jersey.


title: John von Neumann

text: Ulam reflected on his death in his autobiography, originally intended to be a book on von Neumann, saying that he died so prematurely, "seeing the promised land but hardly entering it". His published work on automata and the brain contained only the barest sketches of what he planned to think about, and although he had a great fascination with them, many of the significant discoveries and advancements in molecular biology and computing were made only after he died before he could make any further contributions to them. On his deathbed he was still unsure of whether he had done enough important work in his life. Although he never lived to see it, he had also accepted an appointment as professor at large at UCLA should he have recovered from his cancer.


title: John von Neumann

text: The axiomatization of mathematics, on the model of Euclid's "Elements", had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and in geometry, thanks to Hilbert's axioms. But at the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell's paradox (on the set of all sets that do not belong to themselves). The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the "axiom of foundation" and the notion of "class."


title: John von Neumann

text: The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the principles of Zermelo and Fraenkel. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced a method of demonstration called the "method of inner models", which became an essential instrument in set theory.


title: John von Neumann

text: The second approach to the problem of sets belonging to themselves took as its base the notion of class, and defines a set as a class that belongs to other classes, while a "proper class" is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is a "proper class", not a set.


title: John von Neumann

text: Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of the ordinal and cardinal numbers as well as the first strict formulation of principles of definitions by the transfinite induction".


title: John von Neumann

text: Building on the work of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But in a 1929 paper, von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators. The group of area-preserving transformations contains such subgroups, and this opens the possibility of performing paradoxical decompositions using these subgroups. The class of groups von Neumann isolated in his work on Banach–Tarski decompositions was very important in many areas of mathematics, including von Neumann's own later work in measure theory (see below).


title: John von Neumann

text: With the aforementioned contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of its consistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding stronger axioms that could be used to prove a broader class of theorems.


title: John von Neumann

text: By 1925 he was involving himself in discussions with others in Göttingen on whether elementary arithmetic followed from Peano axioms. Building on the work of Ackermann, von Neumann began attempting to prove (using the finistic methods of Hilbert's school) the consistency of first-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions on induction). He continued looking for a more general proof of the consistency of classical mathematics using methods from proof theory.


title: John von Neumann

text: A strongly negative answer to whether it was definitive arrived in September 1930 at the historic Second Conference on the Epistemology of the Exact Sciences of Königsberg, in which Kurt Gödel announced his first theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.


title: John von Neumann

text: Less than a month later, von Neumann, who had participated in the Conference, communicated to Gödel an interesting consequence of his theorem: that the usual axiomatic systems are unable to demonstrate their own consistency. Gödel had already discovered this consequence, now known as his second incompleteness theorem, and sent von Neumann a preprint of his article containing both theorems. Von Neumann acknowledged Gödel's priority in his next letter. He never thought much of "the American system of claiming personal priority for everything." However von Neumann's method of proof differed from Gödel's, as his used polynomials to explain consistency. With this discovery, von Neumann ceased work in mathematical logic and foundations of mathematics and instead spent time on problems connected with applications.


title: John von Neumann

text: In a series of papers published in 1932, von Neumann made foundational contributions to ergodic theory, a branch of mathematics that involves the states of dynamical systems with an invariant measure. Of the 1932 papers on ergodic theory, Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality". By then von Neumann had already written his articles on operator theory, and the application of this work was instrumental in his mean ergodic theorem.


title: John von Neumann

text: The theorem is about arbitrary one-parameter unitary groups formula_1 and states that for every vector formula_2 in the Hilbert space, formula_3 exists in the sense of the metric defined by the Hilbert norm and is a vector formula_4 which is such that formula_5 for all formula_6. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating to Boltzmann's ergodic hypothesis. He also pointed out that ergodicity had not yet been achieved and isolated this for future work.


title: John von Neumann

text: Later in the year he published another long and influential paper that began the systematic study of ergodicity. In this paper he gave and proved a decomposition theorem showing that the ergodic measure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction with Paul Halmos have significant applications in other areas of mathematics.


title: John von Neumann

text: In measure theory, the "problem of measure" for an -dimensional Euclidean space may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets of ?" The work of Felix Hausdorff and Stefan Banach had implied that the problem of measure has a positive solution if or and a negative solution (because of the Banach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character" - the existence of a measure could be determined by looking at the properties of the transformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that the Euclidean group is a solvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space." Around 1942 he told Dorothy Maharam how to prove that every complete σ-finite measure space has a multiplicative lifting, however he did not publish this proof and she later came up with a new one.


title: John von Neumann

text: In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions. A major contribution von Neumann made to measure theory was the result of a paper written to answer a question of Haar regarding whether there existed an algebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions". He proved this in the positive, and in later papers with Stone discussed various generalizations and algebraic aspects of this problem. He also proved by new methods the existence of disintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked for compact groups. He had to create entirely new techniques to apply this to locally compact groups. He also gave a new, ingenious proof for the Radon–Nikodym theorem. His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.


title: John von Neumann

text: Using his previous work on measure theory von Neumann made several contributions to the theory of topological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extended Bohr's theory of almost periodic functions to arbitrary groups. He continued this work with another paper in conjunction with Bochner that improved the theory of almost periodicity to include functions that took on elements of linear spaces as values rather than numbers. In 1938, he was awarded the Bôcher Memorial Prize for his work in analysis in relation to these papers.


title: John von Neumann

text: In a 1933 paper, he used the newly discovered Haar measure in the solution of Hilbert's fifth problem for the case of compact groups. The basic idea behind this was discovered several years earlier when von Neumann published a paper on the analytic properties of groups of linear transformations and found that closed subgroups of a general linear group are Lie groups. This was later extended by Cartan to arbitrary Lie groups in the form of the closed-subgroup theorem.


title: John von Neumann

text: Von Neumann was the first person to axiomatically define an abstract Hilbert space whereas it was previously defined as the Lp space. He defined it as a complex vector space with a Hermitian scalar product, with the corresponding norm being both separable and complete. In the same papers he also defined several other abstract inequalities such as the Cauchy–Schwarz inequality that were previously only defined for Euclidean spaces. He continued with the development of the spectral theory of operators in Hilbert space in 3 seminal papers between 1929 and 1932. This work cumulated in his Mathematical Foundations of Quantum Mechanics which among two other books by Stone and Banach in the same year were the first monographs on Hilbert space theory. Previous work by others showed that a theory of weak topologies could not be obtained by using sequences, and von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defining locally convex spaces and topological vector spaces for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas from Hausdorff from Euclidean to Hilbert spaces) such as boundness and total boundness are still used today. For twenty years von Neumann was considered the 'undisputed master' of this area. These developments were primarily prompted by needs in quantum mechanics where von Neumann realized the need to extend the spectral theory of Hermitian operators from the bounded to the unbounded case. Other major achievements in these papers include a complete elucidation of spectral theory for normal operators, the first abstract presentation of the trace of a positive operator, a generalisation of Riesz's presentation of Hilbert's spectral theorems at the time, and the discovery of Hermitian operators in a Hilbert space, as distinct from self-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. In addition he wrote a paper detailing how the usage of infinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general of operator algebras.


title: John von Neumann

text: His later work on rings of operators lead to him revisiting his earlier work on spectral theory and providing a new way of working through the geometric content of the spectral theory by the use of direct integrals of Hilbert spaces. Like in his work on measure theory he proved several theorems that he did not find time to publish. Nachman Aronszajn and K. T. Smith were told by him that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on the invariant subspace problem.


title: John von Neumann

text: With I. J. Schoenberg he wrote several items investigating translation invariant Hilbertian metrics on the real number line which resulted in their complete classification. Their motivation lie in various questions related to embedding metric spaces into Hilbert spaces.


title: John von Neumann

text: With Pascual Jordan he wrote a short paper giving the first derivation of a given norm from an inner product by means of the parallelogram identity. His trace inequality is a key result of matrix theory used in matrix approximation problems. He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric absolute norms). This paper leads naturally to the study of symmetric operator ideals and is the beginning point for modern studies of symmetric operator spaces.


title: John von Neumann

text: Later with Robert Schatten he initiated the study of nuclear operators on Hilbert spaces, tensor products of Banach spaces, introduced and studied trace class operators, their ideals, and their duality with compact operators, and preduality with bounded operators. The generalization of this topic to the study of nuclear operators on Banach spaces was among the first achievements of Alexander Grothendieck. Previously in 1937 von Neumann published several results in this area, for example giving 1-parameter scale of different cross norms on formula_7 and proving several other results on what are now known as Schatten–von Neumann ideals.


title: John von Neumann

text: Von Neumann founded the study of rings of operators, through the von Neumann algebras (originally called W*-algebras). While his original ideas for rings of operators existed already in 1930, he did not begin studying them in depth until he met F. J. Murray several years later. A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. The von Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to the bicommutant. After elucidating the study of the commutative algebra case, von Neumann embarked in 1936, with the partial collaboration of Murray, on the noncommutative case, the general study of factors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century". The nearly 500 pages that the papers span collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification of factors. In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors yet he did not find time to publish this result until 1949. Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out. Another important result on polar decomposition was published in 1932. His work here lead on to the next major topic.


title: John von Neumann

text: Between 1935 and 1937, von Neumann worked on lattice theory, the theory of partially ordered sets in which every two elements have a greatest lower bound and a least upper bound. Garrett Birkhoff described his work, "John von Neumann's brilliant mind blazed over lattice theory like a meteor". Von Neumann combined traditional projective geometry with modern algebra (linear algebra, ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of general modules over rings. His work laid the foundations for modern work in projective geometry.


title: John von Neumann

text: His biggest contribution was founding the field of continuous geometry. It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complex projective geometry, where instead of the dimension of a subspace being in a discrete set formula_8 it can be an element of the unit interval formula_9. Earlier, Menger and Birkhoff had axiomatized complex projective geometry in terms of the properties of its lattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened those axioms to describe a broader class of lattices, the continuous geometries.


title: John von Neumann

text: While the dimensions of the subspaces of projective geometries are a discrete set (the non-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval formula_9. Von Neumann was motivated by his discovery of von Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was the projections of the hyperfinite type II factor.


title: John von Neumann

text: In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of formula_11 (continuous-dimensional projective geometry over an arbitrary division ring formula_12) in abstract language of lattice theory. Von Neumann provided an abstract exploration of dimension in completed complemented modular topological lattices (properties that arise in the lattices of subspaces of inner product spaces): "Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity."


title: John von Neumann

text: For any integer formula_13 every formula_14-dimensional abstract projective geometry is isomorphic to the subspace-lattice of an formula_14-dimensional vector space formula_16 over a (unique) corresponding division ring formula_17. This is known as the Veblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continous dimensional case. This coordinatization theorem is a deep and important result that stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques.


title: John von Neumann

text: The theorem as described by Birkhoff: "[I]n the general case, von Neumann proved the following basic representation theorem. Any complemented modular lattice having a "basis" of pairwise perspective elements, is isomorphic with the lattice of all principal right-ideals of a suitable regular ring . This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe."


title: John von Neumann

text: This work required the creation of regular rings. A von Neumann regular ring is a ring where for every formula_18, an element formula_19 exists such that formula_20. These rings came from and have connections to his work on von Neumann algebras, as well as AW*-algebras and various kinds of C*-algebras.


title: John von Neumann

text: Many smaller technical results were proven during the creation and proof of the above theorems, particularly regarding distributivity (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory of metric lattices.


title: John von Neumann

text: Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication and afterwards became busy with war work and his interests moved to computers. He finished his article by saying, "One wonders what would have been the effect on lattice theory, if von Neumann's intense two-year preoccupation with lattice theory had continued for twenty years!"


title: John von Neumann

text: Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.


title: John von Neumann

text: Subsequently, Denis Sargan and Alok Bhargava extended the results for testing if the errors on a regression model follow a Gaussian random walk ("i.e.", possess a unit root) against the alternative that they are a stationary first order autoregression.


title: John von Neumann

text: In his early years von Neumann published several papers relating to set-theoretical real analysis and number theory. In a paper from 1925, he proved that for any dense sequence of points in formula_9, there existed a rearrangement of those points that is uniformly distributed. In 1926 his sole publication was on Prüfer's theory of ideal algebraic numbers where von Neumann found a new way of constructing them, extending Prüfer's theory to the field of all algebraic numbers, and their relation to p-adic numbers. In 1928 he wrote a couple papers on set-theoretic analysis. The first dealt with partitioning an interval into countably many congruent subsets. It solved a problem of Hugo Steinhaus asking whether an interval is formula_22-divisible. Von Neumann proved that indeed that all intervals, half-open, open, or closed are formula_22-divisible by translations (i.e. that these intervals can be decomposed into formula_22 subsets that are congruent by translation). His next paper dealt with giving a constructive proof without the axiom of choice that formula_25 algebraically independent reals exist. He proved that formula_26 are algebraically independent for formula_27. Consequently, there exists a perfect algebraically independent set of reals the size of the continuum. Other minor results from his early career include a proof of a maximum principle for the gradient of a minimizing function in the field of calculus of variations, specifically proving the following theorem: Let formula_28 be a Lipschitz function with constant formula_29, and formula_30 an open and bounded set in formula_31. If formula_32 is a minimum for formula_17 in formula_34, then formula_35 and a small simplification of Hermann Minkowski's theorem for linear forms in geometric number theory.


title: John von Neumann

text: Later in his career together with Pascual Jordan and Eugene Wigner he wrote a foundational paper classifying all finite-dimensional formally real Jordan algebras and discovering the Albert algebras. A couple years later in 1936 he wrote another paper by himself in an attempt to further the program of replacing the axioms of his previous Hilbert space program with those of Jordan algebras. In this paper he investigated the infinite-dimensional case and planned to write at least one further paper on the topic however this paper never came to fruition.


title: John von Neumann

text: Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his widely influential 1932 work "Mathematical Foundations of Quantum Mechanics". After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system.


title: John von Neumann

text: The "physics" of quantum mechanics was thereby reduced to the "mathematics" of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the "non-commutativity" of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?"


title: John von Neumann

text: Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1935, Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid. Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966. In 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories. The validity of Bub's argument is, in turn, disputed. In any case, Gleason's theorem of 1957 fills the gaps in von Neumann's approach.


title: John von Neumann

text: Von Neumann's proof inaugurated a line of research that ultimately led, through Bell's theorem and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a "notion of reality" substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity.


title: John von Neumann

text: In a chapter of "The Mathematical Foundations of Quantum Mechanics", von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. Although this view was accepted by Eugene Wigner, the Von Neumann–Wigner interpretation never gained acceptance among the majority of physicists. The Von Neumann–Wigner interpretation has been summarized as follows:


title: John von Neumann

text: The rules of quantum mechanics are correct but there is only one system which may be treated with quantum mechanics, namely the entire material world. There exist external observers which cannot be treated within quantum mechanics, namely human (and perhaps animal) "minds", which perform measurements on the brain causing wave function collapse.


title: John von Neumann

text: Though theories of quantum mechanics continue to evolve, there is a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches that can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.


title: John von Neumann

text: Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of Hilbert's sixth problem, noted mathematical physicist A. S. Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. In the publication of his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems. Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed (operator theory, von Neumann algebras, continuous geometries, etc.), he did not find a satisfactory mathematical framework for quantum theory as a whole (including quantum field theory).


title: John von Neumann

text: Von Neumann entropy is extensively used in different forms (conditional entropy, relative entropy, etc.) in the framework of quantum information theory. Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix formula_36, it is given by formula_37 Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and conditional quantum entropy.


title: John von Neumann

text: Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy. The von Neumann entropy is the cornerstone in the development of quantum information theory, while the Shannon entropy applies to classical information theory. This is considered a historical anomaly, as Shannon entropy might have been expected to be discovered before Von Neumann entropy, given the latter's more widespread application to quantum information theory. But Von Neumann discovered von Neumann entropy first, and applied it to questions of statistical physics. Decades later, Shannon developed an information-theoretic formula for use in classical information theory, and asked von Neumann what to call it. Von Neumann said to call it Shannon entropy, as it was a special case of von Neumann entropy.


title: John von Neumann

text: The formalism of density operators and matrices was introduced by von Neumann in 1927 and independently, but less systematically by Lev Landau and Felix Bloch in 1927 and 1946 respectively. The density matrix is an alternative way to represent the state of a quantum system, which could otherwise be represented using the wavefunction. The density matrix allows the solution of certain time-dependent problems in quantum mechanics.


title: John von Neumann

text: The von Neumann measurement scheme, the ancestor of quantum decoherence theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories.


title: John von Neumann

text: Von Neumann first proposed a quantum logic in his 1932 treatise "Mathematical Foundations of Quantum Mechanics", where he noted that projections on a Hilbert space can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated, in a famous paper of 1936 by von Neumann and Garrett Birkhoff, the first work ever to introduce quantum logics, wherein von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly ("e.g.", horizontally and vertically), and therefore, "a fortiori", it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added "between" the other two, the photons will indeed pass through. This experimental fact is translatable into logic as the "non-commutativity" of conjunction formula_38. It was also demonstrated that the laws of distribution of classical logic, formula_39 and formula_40, are not valid for quantum theory.


title: John von Neumann

text: The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is in turn attributable to the fact that it is frequently the case in quantum mechanics that a pair of alternatives are semantically determinate, while each of its members is necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (spin angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., "x" and "y") results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the "x" direction is positive." By the principle of indeterminacy, the value of the spin in the direction "y" will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of "y" is positive" nor the proposition "the spin in the direction of "y" is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of "y" is positive or the spin in the direction of "y" is negative" must be true for ɸ.


title: John von Neumann

text: In the case of distribution, it is therefore possible to have a situation in which "formula_41", while formula_42. As Hilary Putnam writes, von Neumann replaced classical logic with a logic constructed in orthomodular lattices (isomorphic to the lattices of subspaces of the Hilbert space of a given physical system).


title: John von Neumann

text: Nevertheless, he was never satifisied with his work on quantum logic. He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at the Washington Philosophical Society in 1945 he found that he could not, especially given that he was busy with war work at the time. He just could not make himself write something he did not fully understand to his satisfaction. During his address at the 1954 International Congress of Mathematicians he gave this issue as one of the unsolved problems that future mathematicians could work on.


title: John von Neumann

text: Von Neumann's contributions to fluid dynamics included his discovery of the classic flow solution to blast waves, and the co-discovery (independently of Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives. During the 1930s, von Neumann became an authority on the mathematics of shaped charges.


title: John von Neumann

text: Later with Robert D. Richtmyer, von Neumann developed an algorithm defining "artificial viscosity" that improved the understanding of shock waves. When computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of "artificial viscosity" smoothed the shock transition without sacrificing basic physics.


title: John von Neumann

text: Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During WW2, he arrived one day at the office of R.H. Kent, the Director of the US Army's Ballistic Research Laboratory, with a computer program he had created for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann then gave a seminar on his computer program to an audience which included his friend Theodore von Kármán. After von Neumann had finished, von Kármán said "Well, Johnny, that's very interesting. Of course you realize Lagrange also used digital models to simulate continuum mechanics." It was evident from von Neumann's face, that he had been unaware of Lagrange's Mécanique analytique.


title: John von Neumann

text: While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions to it. His pioneering papers with Subrahmanyan Chandrasekhar on the statistics of a fluctuating gravitational field generated by randomly distributed stars were considered a "tour de force". In this paper they developed a theory of two-body relaxation and used the Holtsmark distribution to model the dynamics of stellar systems. He wrote several other unpublished manuscripts on topics in stellar structure, some of which were included in Chandresekhar's other works. In some earlier work led by Oswald Veblen von Neumann helped develop basic ideas involving spinors that would lead to Roger Penrose's twistor theory. Much of this was done in seminars conducted at the IAS during the 1930s. From this work he wrote a paper with A. H. Taub and Veblen extending the Dirac equation to projective relativity, maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity in the 1930s. Additionally in the same time period he made several proposals to colleagues for dealing with the problems in the newly created quantum theory of fields and for quantizing spacetime, however both his colleagues and he himself did not consider the ideas fruitful and he did not work on them further. Nevertheless, he maintained at least some interest in these ideas as he had as late as 1940 written a manuscript on the Dirac equation in De Sitter space.


title: John von Neumann

text: Von Neumann founded the field of game theory as a mathematical discipline. He proved his minimax theorem in 1928. It establishes that in zero-sum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss.


title: John von Neumann

text: Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 "Theory of Games and Economic Behavior", written with Oskar Morgenstern. Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book. The public interest in this work was such that "The New York Times" ran a front-page story. In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.


title: John von Neumann

text: Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been the primary tools of mathematical economics ever since.


title: John von Neumann

text: Von Neumann raised the intellectual and mathematical level of economics in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. Von Neumann's model of an expanding economy considered the matrix pencil " A − λB" with nonnegative matrices A and B; von Neumann sought probability vectors "p" and "q" and a positive number "λ" that would solve the complementarity equation


title: John von Neumann

text: along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector "p" represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution "λ" represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate.


title: John von Neumann

text: Von Neumann's results have been viewed as a special case of linear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists with interests in computational economics. This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself.


title: John von Neumann

text: Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in German that was eventually translated into English. His interest in economics that led to that paper began while he was lecturing at Berlin in 1928 and 1929. He spent his summers back home in Budapest, as did the economist Nicholas Kaldor, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann found some faults in the book and corrected them–for example, replacing equations by inequalities. He noticed that Walras's General Equilibrium Theory and Walras's law, which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced the paper.


title: John von Neumann

text: Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming.


title: John von Neumann

text: Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming.


title: John von Neumann

text: Von Neumann was a founding figure in computing. Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. Von Neumann wrote the 23 pages long sorting program for the EDVAC in ink. On the first page, traces of the phrase "TOP SECRET", which was written in pencil and later erased, can still be seen. He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s.


title: John von Neumann

text: Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanisław Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers.


title: John von Neumann

text: Von Neumann's algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators. Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.


title: John von Neumann

text: While consulting for the Moore School of Electrical Engineering at the University of Pennsylvania on the EDVAC project, von Neumann wrote an incomplete "First Draft of a Report on the EDVAC". The paper, whose premature distribution nullified the patent claims of EDVAC designers J. Presper Eckert and John Mauchly, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space. This architecture is the basis of most modern computer designs, unlike the earliest computers that were "programmed" using a separate memory device such as a paper tape or plugboard. Although the single-memory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the architecture was based on the work of Eckert and Mauchly, inventors of the ENIAC computer at the University of Pennsylvania.


title: John von Neumann

text: Von Neumann consulted for the Army's Ballistic Research Laboratory, most notably on the ENIAC project, as a member of its Scientific Advisory Committee.


title: John von Neumann

text: The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. Complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved.


title: John von Neumann

text: The next computer that von Neumann designed was the IAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. Von Neumann recommended that the IBM 701, nicknamed "the defense computer", include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704.


title: John von Neumann

text: However, the theory could not be implemented until advances in computing of the 1960s. Around 1950 he was also among the first people to talk about the time complexity of computations, which eventually evolved into the field of computational complexity theory.


title: John von Neumann

text: Herman Goldstine once described how he felt that even in comparison to all his technical achievements in computer science, it was the fact that he was held in such high esteem, had such a reputation, that the digital computer was accepted so quickly and worked on by others. As an example, he talked about Tom Watson, Jr.'s meetings with von Neumann at the Institute for Advanced Study, whom he had come to see after having heard of von Neumann's work and wanting to know what was happening for himself personally. IBM, which Watson Jr. later became CEO and president of, would play an enormous role in the forthcoming computer industry. The second example was that once von Neumann was elected Commissioner of the Atomic Energy Commission, he would exert great influence over the commission's laboratories to promote the use of computers and to spur competition between IBM and Sperry-Rand, which would result in the Stretch and LARC computers that lead to further developments in the field. Goldstine also notes how von Neumann's expository style when speaking about technical subjects, particularly to non-technical audiences, was very attractive. This view was held not just by him but by many other mathematicians and scientists of the time too.


title: John von Neumann

text: Von Neumann's rigorous mathematical analysis of the structure of self-replication (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA.


title: John von Neumann

text: Von Neumann created the field of cellular automata without the aid of computers, constructing the first self-replicating automata with pencil and graph paper.


title: John von Neumann

text: The detailed proposal for a physical non-biological self-replicating system was first put forward in lectures von Neumann delivered in 1948 and 1949, when he first only proposed a kinematic self-reproducing automaton. While qualitatively sound, von Neumann was evidently dissatisfied with this model of a self-replicator due to the difficulty of analyzing it with mathematical rigor. He went on to instead develop a more abstract model self-replicator based on his original concept of cellular automata.


title: John von Neumann

text: Subsequently, the concept of the Von Neumann universal constructor based on the von Neumann cellular automaton was fleshed out in his posthumously published lectures "Theory of Self Reproducing Automata".