John von Neumann, the Mathematician

John von Neumann, 

the Mathematician 

DOMOKOS SZÁSZ 

Imagine a poll to choose the best-known mathematician of the twentieth century. No doubt the winner 

would be John von Neumann. Reasons are seen, for instance, in the title of the excellent biography [M] 

by Macrae: John von Neumann. The Scientific Genius who Pioneered the Modern Computer, Game 

Theory, Nuclear Deterrence, and Much More. Indeed, he was a fundamental figure not only in 

designing modern computers but also in defining their place in society and envisioning their potential. 

His minimax theorem, the first theorem of game theory, and later his equilibrium model of economy, 

essentially inaugurated the new science of mathematical economics. He played an important role in 

the development of the atomic bomb. However, behind all these, he was a brilliant mathematician. My 

goal here is to concentrate on his development and achievements as a mathematician and the evolution 

of his mathematical interests. 

The Years in Hungary (1903-1921) 

In 1802, the Hungarian mathematician János Bolyai was 

born in Kolozsvár (called Cluj today). János Bolyai was 

the first Hungarian mathematician, maybe the first Hungarian 

scholar, of world rank. His non-euclidean hyperbolic 

geometry, also discovered at the same time independently by 

the Russian mathematician Lobachevsky, is now everywhere 

recognized, but could not be appreciated in his time in 

Hungary. He never had a job as a mathematician. In the early 

nineteenth century, Hungary was in the hinterlands of world 

mathematics. 

A century later, John von Neumann was born as Neumann 

János on December 28, 1903, in Budapest. His family 

belonged to the cultural elite of the city. Not only were 

both of his parents’ families quite well-to-do, they understood 

how to use their wealth to live a rich and complete 

human life. But beyond all this, Hungary had changed 

dramatically during the 101 years since Bolyai’s birth. In 

1832, János Bolyai’s discovery was not understood by 

anyone in the country, except for his father. In contrast, 

János Neumann’s exceptional talent was discovered very 

early and was nurtured by top mathematicians on such a 

level that it is hard to imagine better circumstances for a 

prodigy in any other part of the world. 

Much has been written on the life of Neumann and his 

achievements. I have used extensively Norman Macrae’s 

exciting biography [M], which I recommend for the general 

reader. For the mathematician, I recommend the 

special issue 3 of Volume 64 (1958) of the Bulletin of the 

American Mathematical Society dedicated to von Neumann 

shortly after his death. In this issue, first of all, 

Stanisław Ulam, a lifelong friend and collaborator of 

Johnny, as he was called by his friends, gives a brief 

biography and concise mathematical overview (see also 

Ulam’s autobiography Adventures of a Mathematician). 

After this introductory paper, world experts in various 

fields place von Neumann’s achievements in a wider 

context. Excellent papers with a similar aim also 

appeared in the Hungarian periodical Matematikai Lapok, 

unfortunately in Hungarian only. I also mention the book 

of W. Aspray John von Neumann and the origins of 

Modern Computing [A]. I note that von Neumann’s works 

were edited by A. H. Taub [T]. A selection was edited by 

F. Bródy and T. Vámos [BV]. 

Neumann’s father, Neumann Miksa (Max Neumann), 

a doctor of law, prospered as a lawyer for a bank. ‘‘He 

was a debonair, fourth-generation-or-earlier, non-practicing 

Hungarian Jew, with a fine education in a Catholic 

Research supported by the Hungarian National Foundation for Scientific Research grants No. T 046187, K 71693, NK 63066, and TS 049835. This article is a version of 

the author’s speech at the conference in memory of John von Neumann held in Budapest, October 2003. 

Ó 2011 Springer Science+Business Media, LLC 

DOI 10.1007/s00283-011-9223-6

provincial high-school. He was well attuned to fin-de-siècle 

Austria-Hungary and indeed to being a lively intellectual in 

any age. His party piece was to compose two-line ditties 

about the latest vicissitudes of his personal or business life, 

and about national and international politics.’’ 1 Jacob Kann, 

Neumann’s grandfather through his mother Kann Margit 

(Margaret Kann), ‘‘was a distinctly prosperous man. With a 

partner, he had built up in Budapest a thriving business in 

agricultural equipment that grew even faster in 1880-1914 

than Hungary’s then soaring GNP.’’ The Kanns owned a 

rather big and nice house close to the Hungarian Parliament 

on the so-called Kiskörút, Small Boulevard. The 

Kanns and later the families of their four daughters lived 

there, while the ground floor was occupied by the firm 

Kann-Heller. When the families grew larger, they also 

rented the neighboring house. János’s mother Margit was 

‘‘a mother hen tending her children and protecting them. 

She was a family woman, a good deal less rigorous than her 

husband, artistically inclined, a wafer-thin and later chainsmoking 

enthusiast for comporting oneself with what she 

called ‘elegance’ (which later became Neumann’s highest 

term of praise for neat mathematical calculations such as 

the ones that made the H-bomb possible).’’ 

Max and Margaret had three children, of which Johnny 

was the oldest. ‘‘Grandfather Jacob Kann had gone straight 

......................................................................... 

AUTHOR 

DOMOKOS SZÁSZ is a student of A. 

Prékopa, A. Rényi, B. V. Gnedenko, and Ya. 

G. Sinai. He founded the Budapest schools in 

mathematical statistical physics and dynamical 

systems. His current ambition is the rigorous 

derivation of laws of statistical physics 

(diffusion, heat conduction) from microscopic 

assumptions. His other enthusiasms are his 

family (three children, two grandchildren), 

culture, and skiing. 

Mathematical Institute 

Budapest University of Technology and 

Economics 

H-1111 Budapest 

Hungary 

from commercial high school into founding his business, 

but he proved demonic in his capacity for arithmetic 

manipulation. He could add in his head monstrous columns 

of numbers or multiply mentally two numbers in the 

thousands or even millions. The six-year-old Johnny would 

laboriously perform the computations with pencil and 

paper, and announce with glee that Grandfather had been 

absolutely on the mark. Later Johnny himself was known 

for his facility in mental computation, but he had long 

before persuaded himself that he could never match 

Jacob’s level of multiplication skill.’’ 

‘‘Nursemaids, governesses and preschool teachers were 

an integral part of upper-middle-class European households 

in those days, especially in countries where (as in 

Hungary) children in such families did not start school until 

age ten. As the Kann grandchildren and some of Johnny’s 

second cousins came along, the building on the Small 

Boulevard became an educational institution in its own 

right. There was an especially early emphasis on learning 

foreign languages. Father Max thought that youngsters who 

spoke only Hungarian would not merely fail in the Central 

Europe then darkening around them, they might not even 

survive.’’ So Johnny (or Jancsi in Hungarian) learned German, 

French, English, and Italian from various governesses 

from abroad. Johnny later told colleagues in Princeton that 

as a six-year-old he would converse with his father in 

classical Greek. This seems to have been a joke, but his 

knowledge of languages was real. It not only made his 

communication easier in Germany, Zürich, and in the US, 

but it may have imprinted at an early age axiomatic and 

abstract, algorithmic thinking into his brain. 

In such an environment, young Johnny was easily 

drinking in the knowledge surrounding him. One of his 

special interests was history: The family bought ‘‘an entire 

library, whose centerpiece was the Allgemeine Geschichte 

by Wilhelm Oncken. Johnny ploughed through all fortyfour 

volumes. Brother Michael was confounded by the fact 

that what Johnny read, Johnny remembered. Decades later 

friends were startled to discover that he remembered still.’’ 

Between 1914 and 1921, Johnny attended the Lutheran 

Gymnasium in Budapest. Here, too, he received a superb 

education. This was made possible by the fast economic 

and social progress after the compromise with Austria 2 and 

subsequent cultural and educational reforms. 3 

1 Here and below, many unattributed quotes are from [M]. 

2 Some data from the late nineteenth-century history of Hungary. 1867: Appeasement with Austria and the formation of the Austro-Hungarian Monarchy after the 

defeat of the 1848 revolution. 1873: Formation of the city Budapest from smaller cities such as Buda and Pest. In the years afterward, Hungary and Budapest 

experienced extremely fast economic, social, and intellectual progress. Society was quite open, with ‘‘a flood of Jewish immigration into Hungary in the 1880s and 

1890s as there was simultaneously to New York.’’ In 1896 a World Exposition was held in Budapest also commemorating the arrival of Hungarians to the Carpathians 

in 896. Hausmann-like reconstruction of the city contributed to its present character. For instance, the first subway of the European continent was inaugurated in 1896 

in Budapest. ‘‘In 1903 the Elisabeth Bridge over the Danube was the longest single span bridge in the world.’’ For the interested reader I strongly recommend the book 

of John Lukacs [Luk]. 

3 Some data from the nineteenth-century cultural progress of Hungary. End of nineteenth and first half of twentieth century: Development of national thought and 

nationalistic institutions, in particular, the foundation of the National Museum and the Hungarian Academy of Sciences. After the Appeasement of 1867, the educational 

system was transformed and modernized. (For some time the Minister of Education was Loránd Eötvös, the famous physicist, inventor of the torsion pendulum). The 

Gazette of Sciences, called today World of Nature, was founded in 1869, that is, in the same year as the magazine Nature in Britain! The Mathematical and Physical 

Society was founded in 1891. (Since 1947 there are separate societies for Mathematics and Physics, called the János Bolyai and the Loránd Eötvös Societies, 

respectively; in 1968 the John von Neumann Computer Society was founded.) In 1894 Dániel Arany founded the Középiskolai Matematikai Lapok, the oldest journal for 

high-school students; this journal still flourishes in an extended form including physics. In the same year 1894, the first Mathematical and Physical Competitions started 

(today the mathematical competitions are called József Kürschák Competitions). 

THE MATHEMATICAL INTELLIGENCER

As a result of these reforms the whole educational system 

was strong; for the talented (and let us also add, 

wealthy) pupils, there were élite high-schools. One of them 

was the Lutheran Gymnasium. It was quite liberal and had 

a number of Jewish pupils. Nobel Prize winners E. Wigner 

and later J. Harsányi also attended this high-school. As a 

matter of fact, Wigner studied a year above Neumann, and 

already at school a respect and a friendship were formed 

between the two. 

Johnny’s teacher of mathematics was the celebrated László 

Rátz. Between 1896 and 1914, he was the chief editor of the 

high-school journal Középiskolai Matematikai Lapok. ‘‘Wigner 

and others recall that Rátz’s recognition of Johnny’s 

mathematical talent was instant.’’ Rátz agreed with Max Neumann 

that Johnny would be supervised by professional 

mathematicians from the universities. Thus Professor J. Kurschak 

(Kürschák József) from Technical University arranged 

that Johnny would be tutored by the young G. Szego (Szeg}o 

Gábor) in 1915–1916. Later he was also taught by M. Fekete 

(Fekete Mihály) and Leopold Fejér (FejérLipót), and could 

also talk to A. Haar (Haar Alfréd) and Frédéric Riesz (Riesz 

Frigyes). All in all, the prodigy received education and 

supervision on the highest professional and intellectual level. 

In the years 1919–1921, when von Neumann graduated 

from the Lutheran Gymnasium, the mathematical competition 

for secondary schools did not take place because of 

the revolution in Hungary; but in 1918, Neumann was 

permitted to sit in as an unofficial participant, and would 

have won the first prize. The list of winners of the mathematical 

competitions before 1928 include among others L. 

Fejér, Th. von Kármán, D. Konig, A. Haar, M. Riesz, G. 

Szego, and E. Teller, whereas L. Szilárd won a second prize. 

It is commonplace that around and shortly after the turn 

of the nineteenth and twentieth centuries Hungary exported 

a tremendous amount of brains to the West besides von 

Neumann, but it may be worth listing some of these scientists: 

Dennis Gabor (Gábor Dénes; Nobel Prize 1971), 

John Harsányi (Harsányi János; Nobel Prize 1994), Georg 

von Hevesy (Hevesy György; Nobel Prize 1943), Theodore 

von Kármán (KármánTódor), Nicholas Kurti (Kürti Miklós), 

Cornelius Lanczos (Lánczos Kornél), Peter Lax (Lax Péter; 

Wolf Prize 1987), George Olah (Oláh György; Nobel Prize 

1994), Michael Polanyi (Polányi Mihály), George Pólya 

(Pólya György), Gabor Szego (Szeg}o Gábor), Albert Szent- 

Györgyi (Szent-Györgyi Albert; Nobel Prize 1937), Leo 

Szilárd (Szilárd Leó), Edward Teller (Teller Ede), Eugene 

Wigner (Wigner Jenö; Nobel Prize 1963), and Aurel Friedrich 

Wintner. One might wonder about the ‘‘von’’s figuring 

in several of these names. Macrae explains the case of 

Neumann: ‘‘In 1913 the forty-three-year-old Max was 

rewarded for his services to the government: he received 

the noble placename Margittai with hereditary nobility, so 

that in German he and his descendants could be called ‘von 

Neumann.’ Ennoblement was not an unusual award for 

prominent bankers and industrialists during those last years 

of the Austro-Hungarian Empire. Many of the 220 Hungarian 

Jewish families who were ennobled in 1900–1914 

(vs. just over half of that number in the whole century 

before) hastened to change their names. Ennoblement was 

a way through which one could seize the chance to call 

The house on Kiskörút today. (Photo by Domokos Szász) 

himself less Jewish. Max Neumann deliberately did not 

change his name.’’ 

The Mathematics of John von Neumann 

My aim here should certainly be quite modest, and I can go 

into a detailed description neither of his achievements nor 

of their influence. My goal is to follow some of the main 

and continuing interests of John von Neumann. The 

selection of these interests is subjective and reflects my 

knowledge and judgement. 

Despite the breadth of his interests, the picture is not as 

complicated as it looks at first glance. This is especially true 

for the years before the War. It seems he usually had one 

main interest, and thought about other problems with the 

left hand, so to speak. This does not mean that the other 

results are less important, only that even a genius is subject 

to laws of nature. If one wants to reach repeated breakthroughs 

in a problem or more than one (cf. Poincaré), 

then a necessary condition is full concentration on it for 

quite a period. In short, the genius is known both by the 

difficulty of the problems he can solve with full concentration, 

and by the difficulty of the problems he can solve 

with perhaps less persistent concentration. 

Axiomatic Set Theory 

In the first part of this paper I emphasized that this exceptional 

prodigy received an optimal launch in Hungary. In 

addition, one should note that the excellent mathematical 

support in Budapest ensured his start as a scholar and as a 

researcher. Apart from his first paper with Fekete, his big 

interest was the axiomatization of set theory. It is most likely 

that he heard about this problem from a tutor in Budapest. 

Julius König (König Gyula) (the father of the graph-theoretician 

Denes König (König Dénes)) was himself also working 

on set theory, and in particular on the continuum hypothesis, 

but he died in 1913, before Neumann entered high-school. 

Anyway, mathematicians in Budapest were definitely aware 

of this circle of problems (J. Kurschak, Neumann’s mentor, 

for instance, was a colleague of J. König). Macrae writes: 

‘‘The second of Johnny’s papers (about transfinite ordinals) 

had been prepared while he was still in high-school in 1921, 

although it was not published until 1923’’(see [vN23]). 

Ó 2011 Springer Science+Business Media, LLC

The young Neumann János. (Photo from http://www.ysfine. 

com.wigner/neum/ymath.jpeg) 

His papers on set theory and logic were published in 

1923 (1), 1925 (1), 1927 (1), 1928 (2), 1929 (1), and 1931 

(2). It is known that when he read about Gödel’s undecidability 

theorem in 1931, he dropped thinking seriously 

about this field. But there was another reason for that, too: 

starting at the latest from his scholarship in Göttingen in 

1926, his main interest had changed: it became centered 

around laying down the mathematical foundations of 

quantum mechanics and hence around functional analysis. 

Mathematical Foundations of Quantum Physics 

Hilbert certainly heard about the Hungarian Wunderkind 

quite early, and highly respected his results on set theory and 

on Hilbert’s proof theory. But in the mid-twenties, and very 

much in Göttingen, quantum mechanics was in the center of 

attention. Hilbert himself was strongly attracted to this disputed 

science. One should not forget that Göttingen was a 

center of quantum mechanics: Max Born (Nobel Prize 1954) 

and J. Franck (Nobel Prize 1925) were professors there 

between 1921 and 1933 and between 1920 and 1933, 

respectively, and W. Heisenberg (Nobel Prize 1932) and W. 

Pauli (Nobel Prize 1945) also spent some years there. In 1927, 

a long and important paper [HNvN27] on foundations of 

quantum mechanics of Neumann with Hilbert and L. Nordheim 

appeared. It also formulated a program that was later 

carried out in von Neumann’s book [vN32b]. 

The program became a success story in many ways. It 

meant a victory for the Hilbert-space approach. Moreover, 

it attracted the attention of mathematicians to the theory of 

operators and to functional analysis. Also, by translating the 

problems of quantum physics to the language of mathematics, 

it formulated intriguing questions, which either 

arose in physics or were generated by them by the usual 

process of mathematics. A substantial result of von Neumann 

was the spectral theory of unbounded operators, 

generalizing that given by Hilbert for bounded operators. 

Also, the language of operator theory helped to reconcile 

the complementary and—apparently contradictory— 

approaches of Heisenberg and of Schrödinger. 

Von Neumann’s papers on the mathematical foundations 

of quantum mechanics and functional analysis appeared in 

1927 (4), 1928 (5), 1929 (6), 1931 (2), 1932 (2), 1934 (2), 

1935 (3), and 1936 (1). I add that physicists appreciate, in 

particular, his theories of hidden variables (more exactly, his 

proof of their nonexistence), of quantum logic, and of the 

measuring process (cf. Geszti’s article in [BV]). From the 

physical and also from gnoseological point of view I stress 

his disproof of the existence of hidden variables. The laws 

of quantum physics are by their very nature stochastic, 

basically contrary to the deterministic Laplacian view of the 

universe. Several scholars, including Albert Einstein (his 

saying ‘‘God does not throw dice’’ became famous), did not 

accept a nondeterministic universe. There was a belief that 

probabilistic laws are always superficial and that behind 

them there must be some hidden variables, by the use of 

which the world becomes deterministic. This belief was 

irreversibly repudiated by von Neumann. 4 

Von Neumann writes in 1947 in an article entitled The 

Mathematician [vN47], ‘‘It is undeniable that some of the 

best inspirations of mathematics—in those parts of it which 

are as pure mathematics as one can imagine—have come 

from the natural sciences.’’ It is certainly undeniable that 

having shown ‘‘his lion’s claws’’ in set theory, Neumann got 

to the right place: To elaborate the mathematical foundations 

of quantum mechanics was the best imaginable 

challenge for the young genius. The completion of this task 

led to a victorious period for functional analysis, the theory 

of unbounded operators. Von Neumann kept up this 

interest until his death. Peter Lax remembers, ‘‘I recall how 

pleased and excited Neumann was in 1953 when he 

learned of Kato’s proof of the self-adjointness of the 

Schrödinger operator for the helium atom.’’ [Lax]. 

Von Neumann, being a true mathematician, was also 

aware of the other fundamental motivation of a mathematician. 

In the same 1947 article [27] he starts, ‘‘I think it is a 

relatively good approximation to truth ... that mathematical 

ideas originate in empirics, although the genealogy is 

sometimes long and obscure.’’ Then he continues, ‘‘But, 

once they are so conceived, the subject begins to live a 

peculiar life of his own and is better compared to a creative 

one, governed by almost entirely aesthetical motivations, 

than to anything else and, in particular, to an empirical science.’’ 

I can not resist adding one more idea of von 

Neumann about his favourite criterion of elegance: ‘‘One 

expects from a mathematical theorem or from a mathematical 

theory not only to describe and to classify in a simple 

and elegant way numerous and a priori special cases. One 

also expects ‘elegance’ in its ‘architectural’ structural 

makeup. Ease in stating the problem, great difficulty in 

getting hold of it and in all attempts at approaching it, then 

again some very surprising twist by which the approach 

becomes easy, etc. Also, if the deductions are lengthy or 

4 

I learnt from Peter Lax that von Neumann made a subtle error about hidden variables. It was fixed up by S. Kochen and E. P. Specker in J. of Math.& Mech. 17 (1967), 

59-87. Editor’s note: This issue remains contentious, and will be revisited in a future issue of The Mathematical Intelligencer. 

THE MATHEMATICAL INTELLIGENCER

complicated, there should be some simple general principle 

involved, which ‘explains’ the complications and detours, 

reduces the apparent arbitrariness to a few simple guiding 

motivations, etc.’’ 

von Neumann Algebras 

In von Neumann’s hands, operator theory started to live an 

independent life in 1929. In that year he published a long 

paper [vN29] inMath. Ann., whose first half was his initial 

work on algebras of operators. This was a completely new 

branch of mathematics, whose study he continued in a couple 

of papers published in 1936 (3), 1937 (1), 1940 (1), and in 1943 

(2) (these include his classic four-part series of works with F. J. 

Murray). Most likely von Neumann had no external motivation, 

for instance from physics, at least he does not say so in his 

first paper. He, partially with Murray, obtained his celebrated 

classification, leaving the classification of type III algebras 

open (this is what Alain Connes completed later). The creation 

of the theory of von Neumann algebras shows that his unrivalled 

knowledge, speed of thinking, and intuition led him to 

a brilliant discovery, whose real value only became evident 

three decades after its invention. One important idea was that 

the dimension of an algebra (or of a space) is strongly related 

to the invariance group acting on the object. The theory of von 

Neumann algebras started to flourish around the 1960s with 

many deep results, and, in particular, with the Tomita-Takesaki 

theory also revealing profound links of von Neumann 

algebras to physics. Not much later their relation to field theories 

was also discovered. Subsequently, this theory became 

deeper, and absolutely new connections and applicability 

were also discovered. In fact, at least four Fields Medals have 

been awarded for radically new results on or in connection 

with von Neumann algebras: to A. Connes in 1982, to V. F. R. 

Jones and to E. Witten in 1990, and to M. Kontsevich in 1998. It 

is intriguing that Jones’s discovery of the famous Jones polynomials, 

a topological concept related to knots/braids, was 

motivated by ideas from von Neumann algebras. The easier 

and more natural topological construction was afterward 

guessed by Witten and rigorously executed by Kontsevich. 

See [AI] for lectures of the first three medalists and by H. Araki, 

J. S. Birman, and L. Fadeev for their laudations. See also 

http://math.berkeley.edu/*vfr/. 

Despite the fact that the revolutionary progress of the 

theory of von Neumann algebras only occurred in the 

1960s, von Neumann was conscious of their potential 

importance. In 1954, answering a questionnaire of the 

National Academy of Sciences, he selected as his most 

important scientific contributions 1) his work on mathematical 

foundations of quantum theory, 2) his theory of 

operator algebras, and 3) his work in ergodic theory. 

‘‘Side’’ Interests, Ergodic Theory, and Game Theory 

Among Others 

Ergodic Theory. The appearance of ergodic theory in the 

aforementioned list is somewhat surprising. To be sure, von 

Neumann was well aware of the importance of the ergodic 

theorem for the foundations of statistical mechanics. (The 

ergodic hypothesis grew from ideas of Maxwell, Boltzmann, 

Kelvin, Poincaré, Ehrenfest, etc.) As a matter of fact, von 

Neumann found and proved the first-ever ergodic theorem: 

the L2 version in 1931 (which only appeared in 1932) 

[vN32a]. Then George Birkhoff established the individual 

ergodic theorem in 1932. Yet despite his numerous papers 

on ergodic theory (in 1927 (1), 1932 (3), 1941 (1), 1942 (1), 

and 1945 (1)), it seems to me that it was not one of his most 

persistent interests. Furthermore, he may have felt it as less 

than a triumph. When he heard about Birkhoff’s result 

‘‘Johnny expressed pleasure rather than resentment, 

although he did kick himself for not spotting the next steps 

from his calculations that Birkhoff saw.’’ 

Progress in applying the notion of ergodicity to statistical 

mechanics, that is, showing that interesting mechanical systems 

are ergodic, has been rather slow. Only in 1970 could 

Sinai [S] find the first true mechanical system, that of two hard 

discs on the 2-torus, whose ergodicity he could prove; he 

conjectured that the situation is similar for any number of 

balls in arbitrary dimension. In 1999 Simányi and I showed 

that typical hard-ball systems of N balls of masses m1; ...; mN 

and radii r moving on the m-torus are hyperbolic; in 2006 

Simányi established ergodicity in general under an additional 

hypothesis: the Chernov-Sinai Ansatz. 

Other Interests. Von Neumann’s intelligence, culture, 

speed, motivation, and communication abilities were at 

play in many fields from the very start. He touched upon 

algebra, theory of functions of real variables, measure theory, 

topology, continuous groups, lattice theory, continuous 

geometry, almost periodic functions, representation of 

groups, quantum logic, etc. I suspect in a sense these were 

done ‘‘with his left hand.’’ Nevertheless, for his two-part 

memoir ([vN34] and [BvN35]) on almost periodic functions 

and groups, he received the prestigious Bôcher Prize in 1938. 

He would have liked to construct invariant measures for 

groups, a goal actually completed by A. Haar in 1932. Von 

Neumann returned several times to this question, and the 

idea of the construction was also exploited in the construction 

of the dimension in his theory of operator algebras. 

Game Theory and Mathematical Economics. Two related 

left-hand topics deserve special mention: game theory 

and mathematical economics. It is well known that von 

Neumann’s two game-theory papers in 1928 (the more 

detailed one is [vN28]) contained the formulation and the 

proof of the minimax theorem (a model of symmetric, twoperson 

games had already been suggested by Émile Borel 

in 1921, but he had doubts about the validity of a minimax 

theorem). Also, in 1937 von Neumann published a model 

of general economic equilibrium [vN37], known today by 

his name. These two sources were basic for his big joint 

enterprise with O. Morgenstern, the fundamental monograph 

[vNM44] Theory of Games and Economic Behavior. 

These topics were also the objects of his papers in 1950 (1), 

1953 (3), 1954 (1), and 1956 (1) (and two further unfinished 

manuscripts that appeared in 1963). Judging from the fact 

that in 1994 three Nobel laureates in economics were 

named for achievements in game theory (J. Harsányi, J. 

Nash, and Reinhart Selten), it is not too bold to surmise that 

had Neumann lived until this Nobel Prize was founded in 

1969, he would have been the first to receive it. 


Von Neumann’s Mean Ergodic Theorem 

Domokos Szász 

Ludwig Boltzmann’s ergodic hypothesis, formulated in 

the 1870s, says: for large systems of interacting particles 

in equilibrium, time averages of observables are equal to 

their ensemble averages (i.e., averages w.r.t. the timeinvariant 

measure) (cf. D. Szász, Boltzmann’s Ergodic 

Hypothesis, a Conjecture for Centuries? Hard Ball Systems 

and the Lorentz Gas, Springer Encyclopaedia of 

Mathematical Sciences, vol. 101, 2000, pp. 421-446.). For 

Hamiltonian systems the natural invariant measure is the 

Liouvillian one. In 1931, Koopman (B. O. Koopman, 

Hamiltonian systems and transformations in Hilbert 

space, Proc. Nat. Acad. Sci. vol. 17 (1931) pp. 315-318.) 

made a fundamental observation: If T is a transformation 

of a measurable space X keeping the measure l invariant, 

then the operator U defined by (Uf)(x) := f(Tx) is 

unitary on L2. This functional-analytic translation of the 

ergodic problem led von Neumann in the same year to 

finding his famous 

MEAN ERGODIC THEOREM 

T HEOREM 1 (John von Neumann, Proof of the quasiergodic 

hypothesis, Proc. Nat. Acad. Sci. vol. 18 (1932) 

pp. 70-82.) 

For f 2 L2ðlÞ the L2-limit 1 

lim 

n!1 n f þ f ðTxÞþ þf ðT n 1 xÞ ¼ FðxÞ ð1Þ 

exists. The T-invariant function F equals the L2-projec tion of the function f to the subspace of T-invariant 

functions, and moreover, R Fdl ¼ R fdl. 

Note that for ergodic systems, that is, for those where 

all invariant functions are constant almost everywhere, 

the last claim of the theorem is just the equality of 

(asymptotic) time and space averages. 

Shortly after von Neumann discussed this result with 

G. D. Birkhoff, the latter established his individual 

ergodic theorem (G. D. Birkhoff, Proof of the ergodic 

theorem, Proc. Nat. Acad. Sci. vol. 17 (1931) pp. 656- 

660.) stating that in (1) the convergence also holds 

almost everywhere. (For a more detailed history of the 

first ergodic theorems see G. D. Birkhoff and B. O. 

Koopman, Recent contributions to the ergodic theory, 

Proc. Nat. Acad. Sci. vol. 18 (1932) pp. 279-282.) Von 

Neumann was also aware of the L2 form of his theorem: 

if U is any unitary operator in a Hilbert space H, then the 

averages 1 

n f þ Uf þ þU n 1 ð f Þ converge for every 

f 2 H. 

Moving to the US 

Moving to the US in 1930–1931 apparently did not make a 

big change in von Neumann’s mathematics. For a time he 

did not face challenges such as encountering the revolution 

of quantum physics in Göttingen. It is known that he had 

THE MATHEMATICAL INTELLIGENCER 

Von Neumann’s Minimax Theorem 

András Simonovits 

Von Neumann made a number of outstanding contributions 

to game theory, most notably the minimax theorem of game 

theory, a linear input-output model of an expanding economy, 

and the introduction of the von Neumann- 

Morgenstern utility function. Here I present the first of these. 

There are two players, 1 and 2, each has a finite number 

of pure strategies: s1; ...; sm and t1; ...; tn. If player 1 

chooses strategy si and player 2 strategy tj, then the first 

one gets payoff uij and the second one -uij. To hide their 

true intentions, each randomizes his behavior by choosing 

strategies si and tj independently with probabilities pi and qj, respectively. For random strategies p = (pi) and 

q = (qj), the payoffs are the expected values u(p, 

q) = P i=1 

m Pn j=1 piqjuij and -u(p, q), respectively. An 

equilibrium pair is defined as (p, q) such that 

minq maxp u(p, q) = maxp minq u(p, q). 

T HEOREM 1 (von Neumann 1928) For any twoplayer, 

zero-sum matrix game, there exists at least one 

equilibrium pair of strategies. 

R EMARKS 

1. Although another mathematical genius, Émile Borel, 

studied such games before von Neumann, he was 

uncertain if the minimax theorem holds or not. 

2. This theorem was generalized by Nash (1951) (Nobel 

Prize in Economics, 1994) for any finite number of 

players with arbitrary payoff matrices (J. Nash, ‘‘Non- 

Cooperative Games,’’ Annals of Mathematics 54, 

(1951) pp. 289–295.). 

E XAMPLE. Let us consider the following game. Players 

1 and 2 play Head (H) or Tail (T) in the following 

way. Each puts a coin (say a silver dollar) on a table 

simultaneously and independently. If the result is either 

HH or TT, then 1 receives the coin of 2, otherwise 2 

receives the coin of 1. There is a single equilibrium: 

p i* = 1/2 = q j*, which can be achieved by simply 

throwing fair coins. 

http://www.econ.core.hu/english/inst/simonov.html 

trouble getting used to the higher publication-centeredness 

of the American mathematical community, and the lower 

level of informal communication as compared with his 

European experiences. The famous parties of the von 

Neumann family in Princeton, offering beyond alcoholic 

drinks both mathematical and intellectual communication, 

partly solved this problem. 

Events of his private life may have influenced his work. 

In the years around his divorce and second marriage, his 

productivity was less than his average (in 1938 he published 

just one work, and in 1939 none), but this is not 

essential. The proximity and then the beginning of the war,

Johnny during his years in the United States. (Photo by Alan 

Richards, Fuld Hall in Institute for Advanced Study, Princeton) 

however, brought a dramatic change in von Neumann’s 

mathematical interests and perhaps in his style. His hatred 

for Nazi Germany and his engagement in promoting victory 

over it had certainly pushed him toward essentially new 

questions. These were connected to new fields and new 

problems whose solution he hoped would contribute to the 

big goal. At the same time, these problems offered new 

mathematical challenges. 

War Effort: Ballistics and Shocks 

Ballistics. From the technological point of view, the war 

after the First World War was its continuation; the important 

work on improving firing tables of shells did not cease. 

(During the Second World War Kolmogorov also participated 

in similar work; see B. Booss-Bavnbek and 

J. Høyrup: ‘‘Mathematics and War: an Invitation to Revisit,’’ 

Math. Intelligencer 25 (2003), no. 3, 12-25). Oswald Veblen, 

who became the first professor of the Princeton Institute for 

Advanced Study in 1932, and was responsible for inviting 

von Neumann to Princeton and later to the Institute, had 

been the commanding officer of this research between 

1917 and 1919 in the US Army Ordnance Office at Aberdeen 

Proving Grounds, Maryland. (Among others, the great 

mathematicians J. W. Alexander and H. C. M. Morse also 

worked there under his guidance.) In 1937, von Neumann, 

who from his knowledge of history and from experience 

had already expected a war in Europe, decided to join the 

work in Aberdeen. Details of his progressive involvement 

are again very nicely described in Macrae’s book, and I 

restrain myself from repetitions. The essential thing is 

that—because of the decreasing density of air with altitude—the 

equations governing the trajectories are 

nonlinear and cannot be solved exactly and even have new 

types of solutions. Later, especially during his 1943 visit to 

England, von Neumann also became interested in the 

dynamics of magnetic mines. His first publications about 

shock waves were i) an informal progress report to the 

National Defense Committee, which dates back to 1941, 

and ii) its detailed version [vN43]. As usual in von Neumann’s 

mathematics, this very first work already gave a 

deep and broad overview, proved interesting new results, 

and provided a program, too. This report was then followed 

by an analogous report on detonation waves in 

1942, and by further works in 1943 (3), 1945 (2), 1947 (1), 

1948 (1), 1950 (1), 1951 (1), and 1955 (1). (A concise and 

nice description of von Neumann’s approach and results is 

given by Fritz in [BV].) 

Meteorology and Hydrodynamics. One of von Neumann’s 

favorite subjects was meteorology. He was quite optimistic 

that with progress in understanding the nature of aero- and 

hydrodynamic equations and of their solutions, and with 

the development of computation, a practically satisfactory 

weather forecast would become possible. His hopes were 

only partially realized. He was certainly aware of the chaotic 

nature of the solutions of hydrodynamic equations, but he 

probably did not see sufficiently clearly the limitations caused 

by chaos (one should not forget that E. Lorentz’s revolutionary 

work [Lor] appeared only in 1963, and its import was 

not appreciated for quite a while after that). 

The study of trajectories of shells, his study of the 

dynamics of magnetic mines in Britain in 1943, the work on 

hydrodynamics equations, and later his participation from 

1943 in the Manhattan project, all of these projects brought 

home to him the fundamental importance of computations, 

and consequently that of the development of computers. 

His Memorandum written in March 1945 to O. Veblen on 

the use of variational methods in hydrodynamics [vN45b] 

starts with the sentence ‘‘Numerical calculations play a very 

great role in hydrodynamics.’’ 

Computers: Neumann-Architecture and Scientific 

Computing 

If the applied problems mentioned required a tremendous 

amount of computations, this appeared most spectacularly 

in Los Alamos, where for the work on the atomic bomb it 

was absolutely essential to replace experiments by mathematical 

modeling requiring a lot of computations. It is well 

known that in August 1944, on the railroad platform of 

Aberdeen, von Neumann by chance met Herman Goldstine, 

a main figure working in Philadelphia on the 

development of ENIAC. He immediately received a first 

lesson on the actual level of progress, soon got into the 

work, and presented groundbreaking new ideas (his first 

fundamental works were [vN45a] and [BGvN46]). So he not 

only became a devoted advocate of the project, but substantially 

participated in it in various ways; his two most 

important contributions were, first, the elaboration of the 

principles of the programmable computer, of the so-called 

von Neumann-computers, and second, the implementation 

of these principles in the construction of the computer IAS 

to be built at the Princeton Institute. This whole story is 

described in Aspray [A] and Legendi-Szentiványi [LSz], for 

instance. Let me just mention some key words about von 

Neumann’s main implementations: RAM, parallel computations, 

flow-diagrams, program libraries, subroutines. I 

would say that if problems of hydrodynamics and of the 

construction of computers were a necessity and were in the 

air, so to speak, the importance of the use of computers for 

scientific research was not equally obvious. Von Neumann 


On Reliable Computation 

Peter Gács 

Von Neumann’s best-known contribution to computer 

science is the conceptual design of general-purpose computers. 

Ideas of a stored-program computer had already 

been circulating at the time among engineers, but von 

Neumann’s familiarity with the concept of a universal 

Turing machine helped the clean formulations in John von 

Neumann’s ‘‘First draft of a report on the EDVAC’’ (Technical 

report, University of Pennsylvania Moore School of 

Engineering, Philadelphia, 1945). Control unit, arithmetic 

unit, and memory (which also holds the program) still form 

the major organizational division of most computers. The 

arithmetic and control units are modelled well as logic 

circuits with connections such as ‘‘and’’, ‘‘or’’, ‘‘not’’, and bit 

memory (von Neumann called these ‘‘artificial neurons’’ 

after McCulloch and Pitts). 

The report mentions the problem of physical errors, 

without proposing any solution. Von Neumann returned to 

the error correction problem in ‘‘Probabilistic logics and the 

synthesis of reliable organisms from unreliable components’’ 

(in C. Shannon and McCarthy (eds.), Automata 

Studies. Princeton University Press, Princeton, NJ, 1956). 

The most important theorem of this paper rich in ideas can 

be formulated as follows. Call a logic component ideal if it 

functions perfectly, and real if it malfunctions with probability, 

say, 10 -6 . Suppose that a given logic circuit C is built 

with N ideal components. Then one can build out of O(N 

log N) real components another logic circuit C 0 that computes, 

from every input, the same output as C, with 

probability C 0.99. 

In the solution, some parts (the so-called ‘‘restoring 

organs’’) of the circuit C 0 are built using a random permutation. 

This method was rather new at the time, though 

Erd}os’s random graphs and Shannon’s random codes 

existed already. The proof of the main theorem is somewhat 

sketchy: a rigorous proof appeared almost 20 years 

later (by Dobrushin and Ortyukov, using somewhat different 

constructions). 

There have been some more advances since (for example, 

making the circuits completely constructive with the 

help of ‘‘expanders’’), but the log N redundancy has not 

been improved in the general case. For a recent publication 

containing references see Andrei Romashchenko, ‘‘Reliable 

computation based on locally decodable codes’’ (in 

Proc. of STACS, LNCS 3884, (2006) pp. 537–548). 

Error-correcting computation, in the general setting 

introduced by von Neumann, has not seen significant 

industrial applications. Surprisingly, the decrease in the 

size of basic computing components (now transistors on a 

chip) has been accompanied with an increase in their 

reliability: the error rate now is one in maybe 10 20 executions. 

This trend has physical limits, however. (Von 

Neumann himself contemplated these limits in a lecture in 

1949. He was not the first one, Szilárd probably preceded 

him, and the nature of these limits has been a subject of 


lively discussion among physicists ever since.) Restricting 

attention to logic circuits sidesteps the problem of errors 

in the memory and the issue of the bottleneck between 

memory and control unit. There are parallel computing 

models without division between memory and computation: 

the simplest of these, cellular automata, was first 

proposed (for other purposes) by von Neumann and also 

by Ulam. By now, universal reliable cellular automata 

have also been constructed. 

http://www.cs.bu.edu/*gacs 

saw this very quickly, his brain was completely prepared 

for it, and he devoted a substantial amount of energy to it. 

Reading his papers on this theme, one feels that he was 

not only aware of the great perspectives that computers had 

opened up, he was at the same time discovering and 

enjoying the new horizons with a lively, childish curiosity. A 

simple ‘‘game’’ was, for instance, to calculate the first 2000 

decimal digits of e and of p by the ENIAC in 1950. Aspray 

discusses von Neumann’s accomplishments in scientific 

computing in detail. For the record I add that about the 

principles of construction of computers he had reports in 

1945 (1), 1946 (1), 1947 (1), 1948 (1), 1951 (2), 1954 (2), and 

1963 (1). On scientific computing he had the first paper, 

joint with V. Bargmann and D. Montgomery [BMvN46], in 

1946 (1), on solving large systems of linear equations by 

computers. It was followed by works in 1947 (4), 1950 (3), 

1951 (2), 1953 (1), 1954 (1), and 1963 (2). He made computer 

experiments in number theory, ergodic theory, and 

stellar astronomy. He also had several suggestions for random-number 

generators (via the middle-square method or 

the logistic map), and, with Ulam, he was also working on 

inventing the Monte-Carlo method. (I emphasize that the 

publication counts are only for providing a feeling. The 

topic of a work is sometimes ambiguous; some manuscripts 

only appeared after his death; and some reports may have 

appeared as articles (I have not checked overlap).) Had he 

lived longer, he certainly would have greeted the amalgamation 

of computation with science and would have 

Commemoration of the creation of the EDVAC computer. (Credit: 

http://www.computer-stamps.com/country/22/Hungary/)

een one of the leaders. This is certainly true for computerassisted 

proofs. He would have been glad to see the birth of 

computational sciences: computer science, computational 

physics, chemistry, etc. He perhaps would not have been so 

glad to see that often computations replace creative and 

logical thinking, but this is only a guess. 

Last, but not least, I want to turn to a very intriguing 

interest of von Neumann. Having analyzed the functioning 

of computers, he formulated several related questions. He 

initiated research on cellular automata and probabilistic 

automata (i.e., automata with unreliable components). He 

raised the question of constructing self-correcting automata. 

Moreover, he envisioned the need to compare the 

functioning of computers with that of our brains. His articles 

on the subject appeared in 1958 in the collection 

[vN58], and the analysis of his thoughts on the topic would 

deserve a separate discussion. 

Epilogue 

All in all, fate, often combined with his own decisions, put 

before John von Neumann the most extraordinary and diverse 

scientific challenges. In all cases, he did his job in an ingenious 

and superb way. Factors such as his genes, family, education, 

the intellectual ferment that surrounded him in Budapest, and 

the intellectual level of his teachers and of Hungarian mathematics 

of the time, gave him an excellent launch. Besides his 

intelligence, knowledge, deductive power, and fantastic mental 

speed, he showed depth, perspective, and taste; his diverse 

scientific and human interests, complemented by his education 

in chemical engineering, contributed to his unrivalled courage, 

openness, and flexibility. By excelling in an unusually broad 

spectrum of mathematical activities, he became an outstanding 

representative of twentieth-century mathematics, whose influence 

was unbelievably wide within and outside mathematics. 

Von Neumann’s achievements demonstrate convincingly the 

strength of the mathematical approach, ‘‘unreasonable effectiveness 

of mathematics,’’ as Wigner put it. It is not 

overstatement to compare the achievements of John von 

Neumann to those of Archimedes, Newton, Euler, or Gauss. 

Von Neumann was exceptionally widely known among 

mathematicians, and there are plenty of anecdotes related to 

him. I think that as a student, I heard from my professor A. 

Rényi the saying: ‘‘Other mathematicians prove what they can, 

Neumann what he wants.’’ (It fits, though I have mentioned 

that he was not happy that it was not himself who found 

Gödel’s incompleteness theorem or Haar’s construction of the 

invariant measure of a locally compact group or Birkhoff’s 

proof of the individual ergodic theorem.) 

Not long ago, Dan Stroock from MIT mentioned to me the 

following at least half-serious opinion: ‘‘A genius either does 

things better than other people, or does them completely 

differently (orthogonally) from other people.’’ I think it is 

true about von Neumann that he did most things better than 

other mathematicians would have done. An example of 

an ‘‘orthogonal’’ scholar could well be Einstein. Nevertheless, 

I think of times when von Neumann also did things 

orthogonally. 

Let me finish by citing a well-known anecdote. Question 

to Wigner: ‘‘How is it that Hungary produced so many 

geniuses in the early 20th century?’’ Wigner’s answer: ‘‘That 

many geniuses? I don’t understand the question. There was 

only one genius: John von Neumann.’’ 

Brief Biography 

1903, December 28: Born in Budapest 

1914-1921: Lutheran Gymnasium, Budapest (Teacher of 

maths: László Rátz. Tutored by Gábor Szego; afterward by 

M. Fekete and L. Fejér.) 

1921 Winter term - 1923 Summer term: Berlin University, 

student in chemistry, attends lectures in physics and 

mathematics (Professor of Mathematics: Erhard Schmidt, 

Professor of Chemistry: Nobel Laureate Fritz Haber). 

1923 Summer term - 1926 Summer term: ETH, Zürich, student 

in chemical engineering, attends lectures in physics 

and mathematics, too (Hermann Weyl, György Pólya); 

October 1926: Diploma in chemical engineering. 

1921 Winter term - 1925 Summer term: registered student of 

mathematics at Budapest University; March 1926: Ph.D. in 

mathematics (Supervisor: L. Fejér?). 

1926 Fall term - 1927 Summer term: visiting Göttingen 

University, grant from Rockefeller Foundation (D. Hilbert, 

L. Nordheim, R. Courant, K. Friedrichs, P. Jordan, future 

Nobel Laureate physicists Max Born, J. Franck, W. Pauli, 

W. Heisenberg). 

1927 Fall - 1929: Privatdozent at Berlin University. 

1929: Privatdozent at Hamburg University. 

1930 - 1938: Marriage with Marietta Kövesi. 

1930: Visiting Lecturer at Princeton University. 

1931 - 1933: Visiting Professor at Princeton University. 

1933 - 1957: Professor, Member of the Institute for 

Advanced Study, Princeton. 

1935: Daughter Marina von Neumann born. 

1938: Marriage with Klári Dán. 

1957, February 8: Dies in Walter Reed Hospital, Washington, 

buried in Princeton Cemetery 

A complete bibliography of John von Neumann can be 

found on the Internet: http://www.info.omikk.bme.hu/ 

tudomany/neumann/javnbibl.htm. Most papers, and manuscripts 

are in [T], and some in [BV]. 

ACKNOWLEDGMENTS 

I express my sincere gratitude to István Hargittai and Peter 

Lax for their most valuable remarks and to P. Gács and 

András Simonovits for their inserts. Special thanks are due 

to Réka Szász for polishing my English and to J. Fritz, I. 

Juhász, D. Petz, L. Rónyai, A. Simonovits, and A. Szenes for 

stimulating discussions. 

REFERENCES 

[A] W. Aspray, John von Neumann and the Origins of Modern 

Computing. MIT Press, Cambridge, MA, 1990. 

[AI] Sir M. Atiyah, D. Iagolnitzer (eds.), Fields Medallists’ Lectures. 

World Scientific, Singapore, 1997. 

[BV] F. Bródy, T. Vámos (eds.), The Neumann Compendium. World 

Scientific, Singapore, 1995. 


[Lax] P. Lax, Remembering John von Neumann. Proc. of Symposia 

in Pure Mathematics, 50:5-7, 1990. 

[LSz] T. Legendi, T. Szentiványi, Leben und Werk von John von 

Neumann. Mannheim, 1983. 

[Lor] E. N. Lorentz, Deterministic Nonperiodic Flow. J. Atmos. Sci., 

20:130-141, 1963. 

[Luk] J. Lukacs, Budapest 1900: A Historical Portrait of a City & Its 

Culture. Grove Press, 1988. 

[M] N. Macrae, John von Neumann. The Scientific Genius. Pantheon 

Books, New York, 1992. 

[S] Ya. G. Sinai, Dynamical Systems with Elastic Reflections. 

Russian Math. Surveys, 25:139-189, 1970. 

[T] A. H. Taub, John von Neumann: Collected Works I-VI. 

Pergamon Press, Oxford, 1961-1963. 

WORKS BY VON NEUMANN: 

[vN23] John von Neumann, Zur Einführung der transfiniten Zahlen. 

Acta Sci. Math. Szeged, 1:199-208, 1923. 

[HNvN27] D. Hilbert, L. Nordheim, J. von Neumann, Über die Grundlagen 

der Quantenmechanik. Math. Ann. 98:1-30, 1927. 

[vN28] John von Neumann. Zur Theorie der Gesellschaftsspiele. 

Math. Ann. 100:295-320, 1928. 

[vN29] John von Neumann, Zur Algebra der Funktionaloperatoren 

und Theorie der normalen Operatoren. Math. Ann. 102: 

370-427, 1929. 

[vN32a] John von Neumann, Proof of the Quasi-Ergodic Hypothesis. 

Proc. Nat. Acad. Sci., 18:70-82, 1932. 

[vN32b] John von Neumann, Mathematische Grundlagen der 

Quantenmechanik. Springer, Berlin, 1932. 


[vN34] John von Neumann, Almost Periodic Functions on a 

Group. Trans. Amer. Math. Soc. 36:445-492, 1934. 

[BvN35] S. Bochner, John von Neumann, Almost Periodic Functions 

on Groups, II. Trans. Amer. Math. Soc. 37:21-50, 

1935. 

[vN37] John von Neumann, Über ein ökonomisches Gleichungssystem 

und eine Verallgemeinerung des Browerschen 

Fixpunktsatzes. Erg. eines Math. Coll, Vienna, ed. by K. 

Menger 8:73-83, 1937. 

[vN42] John von Neumann, Theory of Detonation Waves. Progress 

Report, 1942. 

[vN43] John von Neumann, Theory of Shock Waves, Progress 

Report, 1943. 

[vNM44] John von Neumann, O. Morgenstern, Theory of Games and 

Economic Behaviour. Princeton University Press, 1944. 

[vN45a] John von Neumann, First Draft of a Report on EDVAC, pp. 

48, 1945. 

[vN45b] John von Neumann, Use of Variational Methods in Hydrodynamics. 

Memorandum to O. Veblen, March 26, 1945. 

[BGvN46] A. W. Burks, H. H. Goldstine, John von Neumann, 

Preliminary Discussion of the Logical Design of an Electronic 

Computing Instrument, Part I, 1946. 

[BMvN46] V. Bargman, D. Montgomery, John von Neumann, Solution 

of Linear Systems of High Order. Report, 1946. 

[vN47] John von Neumann, The Mathematician, in The Works of the 

Mind, R. B. Heywood (ed.) University of Chicago Press, 180- 

196, 1947. 

[vN58] John von Neumann, The Computer and the Brain (Silliman 

Lectures). Yale University Press, pp. 88, 1958.

John von Neumann, the Mathematician

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