# John von Neumann

John von Neumann (around 1940)

John von Neumann (born December 28, 1903 in Budapest , Austria-Hungary as János Lajos Neumann von Margitta ; † February 8, 1957 in Washington, DC , United States ) was a Hungarian-American mathematician . He made significant contributions to mathematical logic , functional analysis , quantum mechanics and game theory and is considered one of the fathers of computer science . He later published as Johann von Neumann ; Nowadays he is best known under his name, chosen in the USA, John von Neumann.

## life and work

### Origin and beginnings of the career

János Neumann came from a Jewish banking family. His father, the royal Hungarian councilor Max Neumann, was raised to the Hungarian nobility on July 1, 1913. Even as a child, John Neumann showed that above-average intelligence that later  amazed even Nobel Prize winners - for example Eugene Paul Wigner . As a six-year-old he could divide eight-digit numbers in his head at high speed. He had an extraordinary memory that enabled him, for example, to accurately reproduce the contents of a book page after a brief glance at it. Later he was able to memorize entire books such as Goethe's Faust and thus, for example, also shine through detailed historical knowledge. He attended the humanistic German-speaking Lutheran grammar school in Budapest , as did Eugene Paul Wigner at the same time, with the Abitur in 1921. The political situation in Hungary at that time was very uncertain because of the regime of the Soviet republic of Béla Kun , in which the von Neumanns were capitalists threatened by persecution, in 1919 the reactionary anti-Semitic regime of Miklós Horthy followed. Even as a high school student, von Neumann shone through his mathematical achievements and published his first mathematical article with his teacher Michael Fekete , which he conceived when he was not quite 18 years old. Following the wishes of his parents, however, he initially studied chemical engineering in Berlin from 1921 to 1923 and then at the ETH Zurich until his diploma in 1925 . At the same time he was enrolled at the University of Budapest, but only passed the exams there. However, his real interest was always in mathematics, to which he dedicated himself to a certain extent as a "hobby". He attended mathematics courses in Berlin and those of Hermann Weyl and George Pólya at the ETH Zurich and soon attracted attention. From 1928 to 1933 von Neumann was the (youngest) private lecturer at the University of Berlin and in the summer semester of 1929 at the University of Hamburg. Before that, he worked with David Hilbert in Göttingen in 1926/1927 .

Von Neumann in the course catalogs of the Friedrich-Wilhelms-Universität zu Berlin

The first three excerpts come from the summer semester 1928, the fourth excerpt from the winter semester 1928/29. Well-known colleagues mentioned here were Georg Feigl , Issai Schur , Erhard Schmidt , Leó Szilárd , Heinz Hopf , Adolf Hammerstein and Ludwig Bieberbach .

At the beginning of his career as a mathematician, von Neumann worked, among other things, on the development of axiomatic set theory , for which he found a new approach as a student (dissertation in Budapest 1926 with Leopold Fejér ), the Neumann-Bernays-Gödel set theory (NBG) , and with Hilbert's theory of proof . These topics were the current research area of ​​Hilbert's group in Göttingen, at that time one of the world centers of mathematics. His definition of ordinal numbers is now a standard: a new ordinal number is defined by the set of those already introduced. His preoccupation with mathematical logic ended when Gödel's incompleteness theorem became known , which dealt Hilbert's program a severe blow. Gödel was later a close friend and colleague of John von Neumann and Albert Einstein at Princeton.

### Work on quantum mechanics

Von Neumann was also the author of the first mathematically well-thought-out book on quantum mechanics , in which he dealt with the measurement process and the thermodynamics of quantum mechanics (see density matrix , introduced by him in 1927, Von Neumann entropy , Von Neumann equation ). The then "hot" topic of rapidly developing quantum mechanics was also the main reason why he turned to functional analysis and developed the theory of linear operators in Hilbert spaces , more precisely that of the unconstrained self-adjoint operators. The mathematicians in Göttingen objected to the new quantum mechanics that the canonical commutation relations could not be fulfilled with the linear restricted operators investigated up to then . Von Neumann clarified this and at the same time made numerous other contributions to this area. However, when later Werner Heisenberg was asked whether he was not grateful to von Neumann for this, he only asked the counter-question, where was the difference between limited and unlimited. Von Neumann's book on quantum mechanics enjoyed such a reputation that even his “proof” of the impossibility of hidden variable theories, which was correct but based on false assumptions, was not questioned for a long time. To von Neumann's chagrin, however, the physicists preferred the Principles of Quantum mechanics by Paul Dirac , which was published almost simultaneously , in which the mathematical problem addressed was circumvented by introducing distributions that were initially frowned upon by mathematicians before they were also used there in the late 1940s Triumphant advance ( Laurent Schwartz ).

With Eugene Wigner , von Neumann published a series of papers in 1928/29 on the application of group theory in the atomic spectra. Here, too, the enthusiasm of the physicists was subdued, there was even talk of the “group plague”, which mathematicians tried to spread in quantum mechanics.

The Stone-von-Neumann-Theorem expresses the uniqueness of the canonical commutators of, for example, position and momentum operators in quantum mechanics and shows the equivalence of their two fundamental formulations by Schrödinger (wave function) and Heisenberg (matrices).

His work on quantum mechanics established his reputation in America - and, not least, with a view to changing to better-paying positions in the USA, he has dealt with it so intensively. In the fall of 1929 he was invited by Oswald Veblen to come to Princeton University in New Jersey and give lectures on it, and he switched between Princeton and Germany in the years that followed. From 1933 he worked at the newly founded, demanding Institute for Advanced Study in Princeton as a professor of mathematics. Some of his colleagues there were Albert Einstein and Hermann Weyl . Like them, von Neumann emigrated permanently to the USA after Hitler came to power .

### America, game theory and math

John von Neumann made outstanding contributions in many areas of mathematics . As early as 1928, an essay by the mathematician Émile Borel on minimax properties had led him to ideas that later resulted in one of his most original designs, game theory . In 1928 von Neumann proved the Min-Max theorem for the existence of an optimal strategy in " zero-sum games ". With the economist Oskar Morgenstern , he wrote the classic book The Theory of Games and Economic Behavior (3rd edition 1953) in 1944 , which also deals with the generalization to n-person games that is important for economics. He became the founder of game theory, which he applies less to classic games than to everyday conflict and decision-making situations with an imperfect knowledge of the opponent's intentions (as in poker). In economics, a seminar lecture from 1936 on the mathematical modeling of expanding economies is also frequently cited. In the second edition of The Theory of Games and Economic Behavior (1947), Morgenstern and von Neumann presented the Von-Neumann-Morgenstern expected utility and thus made significant contributions to utility theory .

In the 1930s, in a series of works with Francis Murray , von Neumann developed a theory of algebras of bounded operators in Hilbert spaces, which Jacques Dixmier later called Von Neumann algebras . These are now a current research area (for example Alain Connes , Vaughan F. R. Jones ), which - as von Neumann predicted - has applications in physics, albeit less in quantum mechanics than in quantum field theory and quantum statistics . Von Neumann and Murray proved a classification theorem for operator algebras as a direct sum of “factors” (with a trivial center) of type I, II, III, each with subdivisions.

Operator algebras were part of his search for a generalization of the quantum mechanical formalism, for he said in a letter to Garrett Birkhoff in 1935 that he would no longer believe in Hilbert dreams. Further attempts in this direction were the investigation of "lattice theory" ( theory of associations ), initially as an algebra of projection operators in the Hilbert space (in which Birkhoff was also involved), later interpreted as an extension of the logic to " quantum logic ", and continuous geometries, which in the end turned out to be no progress compared to operator algebras.

Another field of work in Princeton in the 1930s was the famous ergodic problem , which deals with the mathematical foundation of statistical mechanics in classical systems (equal distribution of orbits in phase space ). Von Neumann had already dealt with these questions from the quantum mechanical side in Germany. After Bernard Koopman had put the problem into operator form, von Neumann took it up and involuntarily engaged in a "duel" with the well-known American mathematician George David Birkhoff . As he later said, he would have preferred a collaboration.

### Manhattan Project and Government Advisor

Von Neumann worked on the Manhattan project in Los Alamos from 1943 . In previous years he was a sought-after consultant for the Army and Navy, for example for ballistics issues, shaped charges, operations research, fighting German magnetic mines or optimizing the effect of bombs with “oblique shock waves ”. One of his main areas of work was the theory of shock waves, which became relevant for supersonic flight in the 1950s and which he used, among other things, for the development of explosive lenses for the implosion mechanism of the plutonium bomb. His development of the first numerical method for solving hyperbolic partial differential equations , the Monte Carlo method with Stanislaw Ulam , the Von Neumann stability analysis and his pioneering achievements in computer architecture also belong in this context . Incidentally, with his expertise in the theory of shock waves during the Second World War, he also optimized British air mines over Germany. Von Neumann was also involved in the further development of the American nuclear bomb program through to the hydrogen bomb .

Von Neumann was valued on the one hand because he freely passed on his ideas and helped colleagues (when visiting Los Alamos he was often surrounded by a cluster of scientists who wanted quick advice), on the other hand he was feared because he took up ideas quickly and developed his own at breathtaking speed Theories developed from it.

In addition to his mathematical achievements, von Neumann was also politically influential as a government advisor. Before the atomic bombs were dropped on Japan, he was a member of the Target Committee , which helped determine the exact targets of the bombs. He also calculated the optimal detonation height of the atomic bombs in order to achieve the greatest possible damage from the explosion on the ground. The idea of ending the East-West confrontation with the explosion of a hydrogen bomb over uninhabited Soviet territory, preventing the Soviet Union from developing its own bomb and permanently intimidating it is also allegedly associated with the name John von Neumann . Whether US President Eisenhower was actually urged to take such a step by von Neumann is controversial. But he was instrumental in getting the US military missile program off the ground.

### Computers and cybernetics

Von Neumann is considered one of the fathers of computer science. The Von Neumann architecture (also known as the Von Neumann computer ) was named after him, a computer in which data and program are binary-coded in the same memory . The program itself can thus be changed while the computing process is running and the specified sequence of the stored instructions can be deviated from by conditional jump commands. Loosely analogous to the human brain (as he writes in the report), it defines a computer architecture consisting of a control unit and an arithmetic unit as well as a storage unit. The commands are processed serially. He described this principle in 1945 in the First Draft of a Report on the EDVAC . The report was intended as a discussion report with the ENIAC group and initially remained unpublished, but quickly circulated in scientific circles. Almost all modern computers are based on von Neumann's idea.

Von Neumann's role as the sole inventor of the modern computer architecture named after him has been disputed and has long been the subject of disputes. Nowadays, the term “stored program computer” is therefore preferably used instead of “Von Neumann computer”. This applies in particular to the claims of the actual builders of the first tube computer ENIAC and its successor model EDVAC, John Presper Eckert and John William Mauchly from the Moore School of the University of Pennsylvania in Philadelphia, with whom von Neumann and Herman Goldstine initially worked closely. Von Neumann came across the computer developers at the Moore School, where Goldstine was a liaison officer in the US Army, through a chance encounter on a platform with the previously unknown mathematician Goldstine. As Goldstine reported, the liberal dissemination of the Edvac Report, which he himself pursued, ended the close relationship between himself and von Neumann with Eckert and Mauchly, who did not see their contribution in the Edvac Report (actually not intended for the public) and for essential parts of the Von Neumann computer asserted priority claims. For Eckert and Mauchly, patent considerations were in the foreground, which led them to leave the Moore School in 1946 to start their own company, and which later led to a decade-long dispute in court (they brought in patent attorneys as early as 1945). Von Neumann, however, initially saw a need for further research and development and advocated an open discussion and wide dissemination of the results. Parts of the concept were also developed independently by other computer pioneers - including Konrad Zuse in Germany. a. the idea of ​​separating memory and processor, which was already carried out in Zuse's still purely mechanical Z1 in 1938. Zuse's early computers, which were designed for special tasks, lacked the essential concept of conditional branching , although he knew it and used it in his plan calculus . At the time, Von Neumann strongly advocated the further development of calculating machines. The merits of von Neumann are based in particular on the mathematization and scientification of calculating machines.

Together with Norbert Wiener , at the end of the winter of 1943/44 in Princeton, he organized an interdisciplinary meeting with engineers, neuroscientists and mathematicians on commonalities between the brain and computers and thus the basics of cybernetics , which Wiener first described in detail in 1948.

Schematic representation of the Von Neumann architecture , 1947

From 1949 Von Neumann led his own computer project at the Institute for Advanced Study, the IAS computer , in which he was able to realize his ideas, including many programming concepts. Walk on it subroutines various methods for generating random numbers (including those with parameter passing a reference to a memory location, middle-square method and the rejection sampling ) and the Mergesort back. He contributed significantly to the use of binary codes in the computer systems and propagated the use of flowcharts , in which he also provided a kind of assertions that can be viewed as precursors for loop invariants in the Hoare calculus . Goldstine, whom he took over from the ENIAC group, became a close employee. He also let the reports from Princeton circulate freely from 1949 onwards, and computers based on these models were soon built all over the USA and England. The IAS calculator and the ENIAC, modified according to von Neumann's ideas, were used primarily for military calculations (ballistics). Von Neumann also used the Princeton computer for pioneering work in numerical weather forecasting, such as the first computer-aided 24-hour weather forecast.

In 1953 he also developed the theory of self-reproducing automata or self-replication , for which he gave a complicated example. Today much simpler ones emerge from the theory of cellular automata (for example John Horton Conway's Game of Life ). He is said to have tried out ideas for this while playing with a game of building blocks ( Tinkertoy ). Science fiction authors imagined the colonization of our galaxy with such automatons and coined the name Von Neumann probes for it . Von Neumann's cellular automata form an important basis for the research discipline Artificial Life and enable the simulation of biological organization, self-reproduction and the evolution of complexity.

### Appreciation and end

Numerous anecdotes have circulated about von Neumann (Halmos collected some in the article cited in the literature). For example, someone tried to test it with the following riddle: “The end points of a route move towards each other with speed , a runner whizzes back and forth between the two end points at one speed . What distance does he cover? ”There is a simple and a somewhat more complicated solution method (summation of the partial distances). Von Neumann gave the answer at lightning speed and, when asked, stated that he had added up the series - so he had chosen the complicated route, which, however, did not mean that he needed more time. ${\ displaystyle s}$${\ displaystyle v}$${\ displaystyle w> v}$

Because of his ability to quickly break down complex issues into simple questions and often to find a solution straight away, as well as his strictly factual attitude, which avoids any unnecessary disputes, von Neumann was happy to be hired as a technical consultant; like IBM , Standard Oil or the RAND Corporation . His name is therefore a household name in a wide variety of areas. In 1952 he published the Von Neumann law , which describes the change in the size of cells of two-dimensional foam over time . For Standard Oil he helped develop methods to better exploit oil deposits. His death prevented a planned larger collaboration with IBM. For the RAND Corporation he applied game theory to strategic thinking games, as did other mathematicians such as John Nash and John Milnor . In an unpublished paper in 1953, he also set out the principles of the semiconductor laser.

John von Neumann was a fun-loving and sociable person (nickname "Good Time Johnny"); he was married twice - to Marietta Kövesi and Klára Dán - and had a daughter ( Marina ). His Princeton home was the focus of academic circles at the legendary Princeton parties. Von Neumann also loved fast cars like Cadillac or Studebaker, but his driving style was feared because he quickly got bored with quiet traffic and then fell into absent-mindedness. Even in the middle of a party, he could suddenly say goodbye to think through a math problem. Some of his alcohol consumption was only faked, as one guest's child was surprised to find out. Another aspect of the "entertainer" von Neumann was his inexhaustible reservoir of often slippery jokes and his fondness for limericks .

Von Neumann died in the Walter Reed Military Hospital in Washington after suffering from excruciating cancer, possibly caused by his participation in nuclear tests . A soldier kept watch in front of the room so that when he was delirious - the cancer ultimately also attacked his brain - he would not divulge any state secrets. While still on his deathbed, he wrote his book “The Calculator and the Brain”, in which he investigated the special features of the “computer” in the human head.

Most recently he professed to be a Catholic again (the family had converted in 1929/30) and at the end of his life he had an intensive exchange of ideas with a priest. He is buried in Princeton next to his mother, his second wife Klari (who drowned in the sea in 1963, probably suicide) and Karl Dan, Klari's father, who committed suicide in 1939 after moving from Hungary to the USA. It is located in Princeton Cemetery in Mercer County.

## Honors and memberships

After Neumann is IEEE John von Neumann Medal of the IEEE , the John von Neumann Theory Prize in Operations Research , the John von Neumann Lecture of SIAM and the Von Neumann lunar crater named. The institutes for computer science and mathematics at the Humboldt University in Berlin are located in the Johann von Neumann House.

## Quotes

Von Neumann in a discussion with Jacob Bronowski in 1943 while studying bomb craters on aerial photographs:

“No, no, you don't see it correctly. Your visualizing mind cannot see this properly. You have to think abstractly. What happens is that the first differential quotient disappears identically and therefore what becomes visible is the trace of the second differential quotient. "

Bronowski reports that on this advice he rethought the discussed problem and found it confirmed from Neumann's point of view late at night - when he informed him of this the next morning, von Neumann only asked him to ask him for one of these the next time Only to disturb von Neumann early in the morning if he was wrong and not if he was right.

Von Neumann described the problem of overfitting mathematical models very clearly using the example of an elephant:

"With four parameters I can fit an elephant, and with five I can make him wiggle his trunk."

"Four parameters are enough for me to fit an elephant, and at five he can still wiggle his trunk."

- John von Neumann, quoted from Freeman Dyson , quoted from Enrico Fermi : Nature

Even if a rough sketch of an elephant is actually possible with the help of four complex numbers, the statement is aimed at critically questioning excessive adjustments of a model to existing data.

## Fonts

• Collected works , 6 volumes. Pergamon Press, from 1961
• Brody, Vamos (Ed.): The von Neumann compendium . World Scientific (reprint of important articles by Neumanns)
• The computer and the brain (Silliman Lectures). Yale University Press, 2000 (German The Calculator and the Brain , 1958)
• The mathematician . In: Heywood (Ed.): The works of the mind . 1948. Reprinted in: Kasner, Newman (Ed.): The world of mathematics , Vol. 4
• Mathematical foundations of quantum mechanics . 2nd Edition. Springer Verlag, 1996, ISBN 978-3-540-59207-5 (first 1932)
• Theory of games and economic behavior , together with Oskar Morgenstern. Princeton Univ. Press, 1944, Theory of Games and Economic Behavior. (PDF; 31.6 MB). German translation: Game theory and economic behavior , ISBN 3-7908-0134-8 .

Some essays and books online:

Some of Neumann's works that were created in Los Alamos (for example on shock waves, detonation waves) are available online from the Federation of American Scientists .

Some other work, for example on continuous geometries, operator rings, or ergodic theory, is available online at the National Academy of Sciences .

chronologically

## Documentaries

• John von Neumann. The thinker of the computer age. Documentary, France, 2014, 56:44 min., Script and director: Philippe Calderon, production: arte France, BFC Productions, first broadcast: 4th August 2015 on arte, synopsis by ARD , online video by Internet Archive .
• The Struggle for Freedom: Six Friends and Their Mission - From Budapest to Manhattan. Documentary, Germany, 2013, 88:42 min., Book: Thomas Ammann and Judith Lentze, director: Thomas Ammann, production: Prounen Film, Mythberg Films, Agenda Media, MDR , arte , first broadcast: December 17th, 2013 by arte, summary from ARD .

Commons : János Lajos Neumann  - Album with pictures, videos and audio files

## Individual evidence

1. See Ulf Hashagens article about the habilitation in Berlin (p. 265). It was completed on December 13, 1927.
2. In the winter semester 1928/29 Neumann von Margitta is mentioned as in the summer semester 1928 also at the mathematical colloquium and when discussing recent work on quantum theory with Leó Szilárd . Further lecturers in the discussion of more recent work on quantum theory were Hartmut Kallman and Fritz London in the winter semester of 1928/29 .
3. ^ John (Janos) von Neumann in the Mathematics Genealogy Project (English)
4. The operators for measured quantities used in quantum mechanics are linear (superposition principle for solutions to the linear Schrödinger equation, for example) and self-adjoint, since the eigenvalues, the possible measured values, are then real.
5. The anecdote comes from Kurt Friedrichs, cf. Peter Lax Mathematics and Physics , Bulletin American Mathematical Society, Vol. 45, 2008, pp. 135-152.
6. First probably used by Paul Ehrenfest in a letter to Wolfgang Pauli in September 1928, see Martina Schneider, Between two disciplines. BL van der Waerden and the development of quantum mechanics, Springer 2011, p. 63.
7. Von Neumann was like Edward Teller and a number of other theoretical physicists after the end of the war in Los Alamos during his visits (they worked on the hydrogen bomb) a member of a poker round. Stanislaw Ulam Adventures of a Mathematician , Scribners 1976, p. 169.
8. In the Menger Colloquium, translated as A model of general equilibrium . In: Review of Economic Studies , Vol. 13, 1945, 1, also in Brody, Vamos (Ed.): The von Neumann compendium . Among other things, the use of inequalities instead of just equations as with Walras was new, cf. McRae, pp. 217ff.
9. Poundstone “Prisoners dilemma”, p. 4 quotes an obituary in Life Magazine 1957, in which von Neumann even spoke out in 1950 for a preventive nuclear war against the Soviet Union, as did other personalities at the same time, such as the pacifist Bertrand, who was transformed by contemporary history Russell.
10. ^ Friedrich L. Bauer : Historical Notes on Computer Science . Springer Verlag, 2009, p. 139.
11. Nicholas Metropolis , J. Worlton: A trilogy on errors in the history of computing . In: IEEE Annals of the history of computing , Volume 2., 1980, pp. 49-55, take the view that the concept of stored program was developed by Eckert and Mauchly before the involvement of von Neumann. See also Friedrich L. Bauer : Historical Notes on Computer Science . Springer Verlag, 2009, chapter Who invented the von Neumann calculator? Reprinted from Informatik Spektrum , Volume 21, 1998, p. 84. Also Joel Shurkin: Engines of the Mind. The history of the computer . Norton, 1984, sees the contributions by Eckert and Mauchly as central to Edvac, and von Neumann's important role only begins with his own IAS computer, Goldstine: The Computer from Pascal to von Neumann . 1993, pp. 186f. speaks against it for a central role of Neumanns, who after Goldstine was already involved in the discussions in the Moore School at the beginning of August 1944.
12. ^ Goldstine: The Computer from Pascal to von Neumann . 1993, p. 182.
13. ^ Goldstine: The Computer from Pascal to von Neumann . Princeton University Press, 1993, p. 229.
14. ^ Bauer: Historical notes on computer science . P. 138.
15. ^ Raúl Rojas : The architecture of Konrad Zuse's early computing machines . In: Rojas, Hashagen: The first computers . MIT Press, 2000. According to Rojas, the logical structure of the Z1 was very similar to the later Z3 relay computer and both could be used as a universal calculating machine, even if that was not practical.
16. ^ Raúl Rojas : Zuse and Turing. The wire of Mephistopheles. In: Telepolis , December 21, 2011.
17. Thomas Rid : Machine Dawn . A Brief History of Cybernetics . Propylaen, Berlin 2016, ISBN 978-3-549-07469-5 (492 pages, American English: Rise of the Machines. A Cybernetic History . New York 2016. Translated by Michael Adrian, first edition: WW Norton & Company).
18. ^ Norbert Wiener: Cybernetics. Regulation and communication in living beings and in machines . With the addition of 1961 learning and self-reproducing machines. Second, revised and expanded edition. Econ-Verlag, Düsseldorf 1963 (287 pages, American English: Cybernetics or Control and Communication in the Animal and the Machine . 1948. Translated by EH Serr, E. Henze, first edition: MIT-Press).
19. ^ John von Neumann: Theory of Self-reproducing Automata . published posthumously. Ed .: Arthur W. Burks . University of Illinois Press, 1967, ISBN 978-0-252-72733-7 (English, 388 pages).
20. ^ Poundstone: Prisoner's Dilemma . P. 24
21. ^ Howard Eves: Return to Mathematical Circles , PWS-Kent Publishing, 1988, p. 140.
22. Stephen Dunwell from IBM reports about it in his Oral History Interview 1989.  ( Page no longer available , search in web archivesInfo: The link was automatically marked as broken. Please check the link according to the instructions and then remove this notice. (PDF) Babbage Institute. After Dunwell, his role as a consultant at IBM was very limited, but IBM was aware that as the father of the modern computer, they owed him a lot. This was also due to von Neumann's general attitude - he was of the opinion, according to Dunwell, that the problem with computers is not low memory space, but rather unimaginative programmers.
23. ^ Russell Dupuis: The Diode Laser - the first 30 days 40 years ago . In: Optics and Photonics News , Vol. 15, 2004, p. 30, The Diode Laser — the First Thirty Days Forty Years Ago ( Memento from June 19, 2010 in the Internet Archive )
24. According to other statements, he also used to sing loudly in the car with appropriate steering movements. He wrecked a car almost every year. Poundstone Prisoner's Dilemma , p. 25.
25. ^ McRae, John von Neumann, Birkhäuser, p. 328
26. ^ Find a grave , John von Neumann
27. ^ Members of the American Academy. Listed by election year, 1900-1949 ( PDF ). Retrieved October 8, 2015
28. McRae: Von Neumann , p. 186, after Jacob Bronowski: The Ascent of man , BBC book, 1973 and in his BBC television series of the same name, episode 13. McRae says “differential coefficient”, obviously a translation error.
29. ^ Freeman Dyson: A meeting with Enrico Fermi . In: Nature . 427, No. 297, 2004.
30. ^ Review of Neumann's Selected Letters by George Dyson: Review. In: Notices AMS , June / July 2007.
31. ^ Review of George Dyson, Turing Cathedral by Brian Blank: Review. In: Notices AMS , August 2014. Like Regis' book, much about the history of the IAS. He evaluates Klara von Neumann's unpublished memoirs.