John von Neumann
John von Neumann (/vɒn ˈnɔɪmən/ von NOY-mən; Hungarian: Neumann János Lajos [ˈnɒjmɒn ˈjaːnoʃ ˈlɒjoʃ]; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist, engineer and polymath. He had perhaps the widest coverage of any mathematician of his time,[9] integrating pure and applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including cellular automata, the universal constructor and the digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA.
The native form of this personal name is Neumann János Lajos. This article uses Western name order when mentioning individuals.
John von Neumann
February 8, 1957 (aged 53)
Washington, D.C., U.S.
- Hungary
- United States
-
Marietta Kövesi(m. 1930; div. 1937)
- Bôcher Memorial Prize (1938)
- Navy Distinguished Civilian Service Award (1946)
- Medal for Merit (1946)
- Medal of Freedom (1956)
- Enrico Fermi Award (1956)
- Carl-Gustaf Rossby Research Medal (1957)
Az általános halmazelmélet axiomatikus felépítése (The axiomatic construction of general set theory) (1925)
During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lenses used in the implosion-type nuclear weapon.[10] Before and after the war, he consulted for many 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.[11] At the peak of his influence in the 1950s, he chaired a number of Defense Department committees including the Strategic Missile Evaluation Committee and the ICBM Scientific Advisory Committee. He was also a member of the influential Atomic Energy Commission in charge of all atomic energy development in the country. He played a key role alongside Bernard Schriever and Trevor Gardner in the design and development of the United States' first ICBM programs.[12] At that time he was considered the nation's foremost expert on nuclear weaponry and the leading defense scientist at the U.S. Department of Defense.
Von Neumann's contributions and intellectual ability drew praise from colleagues in physics, mathematics, and beyond. Accolades he received range from the Medal of Freedom to a crater on the Moon named in his honor.
Life and education[edit]
Family background[edit]
Von Neumann was born in Budapest, Kingdom of Hungary (then part of the Austro-Hungarian Empire),[13][14][15] on December 28, 1903, to a wealthy, non-observant Jewish family. His birth name was Neumann János Lajos. In Hungarian, the family name comes first, and his given names are equivalent to John Louis in English.[16]
He was the eldest of three brothers; his two younger siblings were Mihály (Michael) and Miklós (Nicholas).[17] His father Neumann Miksa (Max von Neumann) was a banker and held a doctorate in law. He had moved to Budapest from Pécs at the end of the 1880s.[18] Miksa's father and grandfather were born in Ond (now part of Szerencs), Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann);[19] her parents were Jakab Kann and Katalin Meisels of the Meisels family.[20] 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.[21]
On February 20, 1913, Emperor Franz Joseph elevated John's father to the Hungarian nobility for his service to the Austro-Hungarian Empire.[22] 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.[23]
Child prodigy[edit]
Von Neumann was a child prodigy who at six years old could divide two eight-digit numbers in his head[24][25] and converse in Ancient Greek.[26] He, 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.[27] By age eight, von Neumann was familiar with differential and integral calculus, and by twelve he had read Borel's La Théorie des Fonctions.[28] He was also interested in history, reading Wilhelm Oncken's 46-volume world history series Allgemeine Geschichte in Einzeldarstellungen (General History in Monographs).[29] One of the rooms in the apartment was converted into a library and reading room.[30]
Von Neumann entered the Lutheran Fasori Evangélikus Gimnázium in 1914.[31] Eugene Wigner was a year ahead of von Neumann at the school and soon became his friend.[32]
Although von Neumann's father insisted that he attend school at the grade level appropriate to his age, he agreed to hire private tutors to give von Neumann advanced instruction. At 15, he began to study advanced calculus under the analyst Gábor Szegő.[32] By 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.[33] At the conclusion of his education at the gymnasium, he applied for and won the Eötvös Prize, a national award for mathematics.[34]
University studies[edit]
According to his friend Theodore von Kármán, von Neumann's father wanted John to follow him into industry, and asked von Kármán to persuade his son not to take mathematics.[35] Von Neumann and his father decided that the best career path was chemical engineering. 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 ETH Zurich,[36] which he passed in September 1923.[37] Simultaneously von Neumann entered Pázmány Péter University in Budapest,[38] as a Ph.D. candidate in mathematics. For his thesis, he produced an axiomatization of Cantor's set theory.[39][40] He graduated as a chemical engineer from ETH Zurich in 1926, and simultaneously passed his final examinations summa cum laude for his Ph.D. in mathematics (with minors in experimental physics and chemistry).[41][42] He then went to the University of Göttingen on a grant from the Rockefeller Foundation to study mathematics under David Hilbert.[43] Hermann Weyl remembers how in the winter of 1926–1927 von Neumann, Emmy Noether, and he would walk through "the cold, wet, rain-wet streets of Göttingen" after class discussing hypercomplex number systems and their representations.[44]
Economics[edit]
Game theory[edit]
Von Neumann founded the field of game theory as a mathematical discipline.[260] 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.[261] Such strategies 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. The public interest in this work was such that The New York Times ran a front-page story.[262] 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.[260]
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 primary tools of mathematical economics ever since.[263]
Mathematical economics[edit]
Von Neumann raised the 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.[260] 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 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.[264][265]
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.[266][267] 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.[268] 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.[269] The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding of Nobel prizes in 1972 to Kenneth Arrow, in 1983 to Gérard Debreu, and in 1994 to John Nash who used fixed point theorems to establish equilibria for non-cooperative games and for bargaining problems in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureates Tjalling Koopmans, Leonid Kantorovich, Wassily Leontief, Paul Samuelson, Robert Dorfman, Robert Solow, and Leonid Hurwicz.[270]
Von Neumann's interest in the topic began while he was lecturing at Berlin in 1928 and 1929. He spent his summers in Budapest, as did the economist Nicholas Kaldor; Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann 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 his paper.[271]
Linear programming[edit]
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.[272]
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.[273]
Personality[edit]
Work habits[edit]
Herman Goldstine commented on von Neumann's ability to intuit hidden errors and remember old material perfectly.[361][362] When he had difficulties he would not labor on; instead, he would go home and sleep on it and come back later with a solution.[363] This style, 'taking the path of least resistance', sometimes meant that he could go off on tangents. It also meant that if the difficulty was great from the very beginning, he would simply switch to another problem, not trying to find weak spots from which he could break through.[364] At times he could be ignorant of the standard mathematical literature, finding it easier to rederive basic information he needed rather than chase references.[365]
After World War II began, he became extremely busy with both academic and military commitments. His habit of not writing up talks or publishing results worsened.[366] He did not find it easy to discuss a topic formally in writing unless it was already mature in his mind; if it was not, he would, in his own words, "develop the worst traits of pedantism and inefficiency".[367]
Mathematical range[edit]
The mathematician Jean Dieudonné said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions".[160] According to Dieudonné, his specific genius was in analysis and "combinatorics", with combinatorics being understood in a very wide sense that described his ability to organize and axiomize complex works that previously seemed to have little connection with mathematics. His style in analysis followed the German school, based on foundations in linear algebra and general topology. While von Neumann had an encyclopedic background, his range in pure mathematics was not as wide as Poincaré, Hilbert or even Weyl: von Neumann never did significant work in number theory, algebraic topology, algebraic geometry or differential geometry. However, in applied mathematics his work equalled that of Gauss, Cauchy or Poincaré.[116]
According to Wigner, "Nobody knows all science, not even von Neumann did. But as for mathematics, he contributed to every part of it except number theory and topology. That is, I think, something unique."[368] Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory; he recalled an incident where von Neumann failed to recognize the topological definition of a torus.[369] Von Neumann admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to Hermann Weyl, who he thought was deeper and broader.[363]
In his biography of von Neumann, Salomon Bochner wrote that much of von Neumann's works in pure mathematics involved finite and infinite dimensional vector spaces, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as topology, differential geometry and harmonic integrals, algebraic geometry and other such fields. Von Neumann rarely worked in these fields and, as Bochner saw it, had little affinity for them.[129]
In one of von Neumann's last articles, he lamented that pure mathematicians could no longer attain deep knowledge of even a fraction of the field.[370] In the early 1940s, Ulam had concocted for him a doctoral-style examination to find weaknesses in his knowledge; von Neumann was unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. They concluded that doctoral exams might have "little permanent meaning". However, when Weyl turned down an offer to write a history of mathematics of the 20th century, arguing that no one person could do it, Ulam thought von Neumann could have aspired to do so.[371]
Preferred problem-solving techniques[edit]
Ulam remarked that most mathematicians could master one technique that they then used repeatedly, whereas von Neumann had mastered three:
Legacy[edit]
Accolades[edit]
Nobel Laureate Hans Bethe said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man".[29] Edward Teller observed "von Neumann would carry on a conversation with my 3-year-old son, and the two of them would talk as equals, and I sometimes wondered if he used the same principle when he talked to the rest of us."[396] Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics".[366] Eugene Wigner said, "He understood mathematical problems not only in their initial aspect, but in their full complexity."[397] Claude Shannon called him "the smartest person I've ever met", a common opinion.[398] Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius."[399]
"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei.[400] Peter Lax commented that von Neumann would have won a Nobel Prize in Economics had he lived longer, and that "if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too."[401] Rota writes that "he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first large computer" and consequently that "No other mathematician in this century has had as deep and lasting an influence on the course of civilization."[402] He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century.[403]
Neurophysiologist Leon Harmon described him in a similar manner, calling him the only "true genius" he had ever met: "von Neumann's mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless."[404] While consulting for non-academic projects von Neumann's combination of outstanding scientific ability and practicality gave him a high credibility with military officers, engineers, and industrialists that no other scientist could match. In nuclear missilery he was considered "the clearly dominant advisory figure" according to Herbert York.[405] Economist Nicholas Kaldor said he was "unquestionably the nearest thing to a genius I have ever encountered."[268] Likewise, Paul Samuelson wrote, "We economists are grateful for von Neumann's genius. It is not for us to calculate whether he was a Gauss, or a Poincaré, or a Hilbert. He was the incomparable Johnny von Neumann. He darted briefly into our domain and it has never been the same since."[406]