Von Neumann's contributions and intellectual ability drew praise from colleagues in physics, mathematics, and beyond. His accolades include aMedal of Freedom and acrater on the Moon named in his honor.
Von Neumann was born inBudapest, Kingdom of Hungary (then part of Austria-Hungary),[13][14][15] on December 28, 1903, to a wealthy, non-observantJewish family. His birth name wasNeumann 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 adoctorate in law. He had moved to Budapest fromPécs at the end of the 1880s.[18] Miksa's father and grandfather were born in Ond (now part ofSzerencs),Zemplén County, northern Hungary. John's mother was Kann Margit (Margaret Kann);[19] her parents were Kann Jákab and Meisels Katalin of theMeisels 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 appellationMargittai, meaning "of Margitta" (todayMarghita, Romania). The family had no connection with the town; the appellation was chosen in reference to Margaret, as was their chosencoat of arms depicting threemarguerites. Neumann János became margittai Neumann János (John Neumann de Margitta), which he later changed to the German Johann von Neumann.[23]
Von Neumann was achild prodigy who at six years old could divide two eight-digit numbers in his head[24][25] and converse inAncient Greek.[26] He, his brothers and his cousins were instructed by governesses. Von Neumann's father believed that knowledge of languages other than their nativeHungarian was essential, so the children were tutored inEnglish,French,German andItalian.[27] By age eight, von Neumann was familiar withdifferential andintegral calculus, and by twelve he had readBorel'sLa Théorie des Fonctions.[28] He was also interested in history, readingWilhelm Oncken's 46-volume world history seriesAllgemeine Geschichte in Einzeldarstellungen (General History in Monographs).[29] One of the rooms in the apartment was converted into a library and reading room.[30]
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 analystGábor Szegő.[32] On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that, as recalled by his wife, he came back home with tears in his eyes.[33] By 19, von Neumann had published two major mathematical papers, the second of which gave the modern definition ofordinal numbers, which supersededGeorg Cantor's definition.[34] At the conclusion of his education at the gymnasium, he applied for and won the Eötvös Prize, a national award for mathematics.[35]
According to his friendTheodore 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.[36] Von Neumann and his father decided that the best career path waschemical 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 theUniversity of Berlin, after which he sat for the entrance exam toETH Zurich,[37] which he passed in September 1923.[38] Simultaneously von Neumann enteredPázmány Péter University, then known as the University of Budapest, as aPh.D. candidate inmathematics.[39] For his thesis, he produced anaxiomatization ofCantor's set theory.[40][41] In 1926, he graduated as achemical engineer from ETH Zurich and simultaneously passed his final examinationssumma cum laude for his Ph.D. in mathematics (with minors inexperimental physics and chemistry) at the University of Budapest.[42][43]
Excerpt from the university calendars for 1928 and 1928/29 of theFriedrich-Wilhelms-Universität Berlin announcing Neumann's lectures on the theory of functions II, axiomatic set theory and mathematical logic, the mathematical colloquium, review of recent work in quantum mechanics, special functions of mathematical physics and Hilbert's proof theory. He also lectured on the theory of relativity, set theory, integral equations and analysis of infinitely many variables.
Von Neumann'shabilitation was completed on December 13, 1927, and he began to give lectures as aPrivatdozent at the University of Berlin in 1928.[46] He was the youngest person electedPrivatdozent in the university's history.[47] He began writing nearly one major mathematics paper per month.[48] In 1929, he briefly became aPrivatdozent at theUniversity of Hamburg, where the prospects of becoming a tenured professor were better,[49] then in October of that year moved toPrinceton University as a visiting lecturer inmathematical physics.[50]
Von Neumann was baptized a Catholic in 1930.[51] Shortly afterward, he married Marietta Kövesi, who had studied economics at Budapest University.[50] Von Neumann and Marietta had a daughter,Marina, born in 1935; she would become a professor.[52] The couple divorced on November 2, 1937.[53] On November 17, 1938, von Neumann marriedKlára Dán.[54][55]
In 1933 Von Neumann accepted a tenured professorship at theInstitute for Advanced Study in New Jersey, when that institution's plan to appointHermann Weyl appeared to have failed.[56] His mother, brothers and in-laws followed von Neumann to the United States in 1939.[57] Von Neumannanglicized his name to John, keeping the German-aristocratic surnamevon Neumann.[23] Von Neumann became anaturalized U.S. citizen in 1937, and immediately tried to become alieutenant in the U.S. Army'sOfficers Reserve Corps. He passed the exams but was rejected because of his age.[58]
Klára and John von Neumann were socially active within the local academic community.[59] His whiteclapboard house on Westcott Road was one of Princeton's largest private residences.[60] He always wore formal suits.[61] He enjoyedYiddish and"off-color" humor.[28] In Princeton, he received complaints for playing extremely loud Germanmarch music;[62] Von Neumann did some of his best work in noisy, chaotic environments.[63] According toChurchill Eisenhart, von Neumann could attend parties until the early hours of the morning and then deliver a lecture at 8:30.[64]
He was known for always being happy to provide others of all ability levels with scientific and mathematical advice.[4][65][66] Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician.[67] His daughter wrote that he was very concerned with his legacy in two aspects: his life and the durability of his intellectual contributions to the world.[68]
Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing on technical ones.Herbert York described the many "Von Neumann Committees" that he participated in as "remarkable in style as well as output". The way the committees von Neumann chaired worked directly and intimately with the necessary military or corporate entities became a blueprint for allAir Force long-range missile programs.[69] Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general.[70]Stanisław Ulam suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others.[71]
He also maintained his knowledge of languages learnt in his youth. He knew Hungarian, French, German and English fluently, and maintained a conversational level of Italian, Yiddish, Latin and Ancient Greek. His Spanish was less perfect.[72] He had a passion for and encyclopedic knowledge of ancient history,[73][74] and he enjoyed readingAncient Greek historians in the original Greek. Ulam suspected they may have shaped his views on how future events could play out and how human nature and society worked in general.[75]
Von Neumann's closest friend in the United States was the mathematicianStanisław Ulam.[76] Von Neumann believed that much of his mathematical thought occurred intuitively; he would often go to sleep with a problem unsolved and know the answer upon waking up.[63] Ulam noted that von Neumann's way of thinking might not be visual, but more aural.[77] Ulam recalled, "Quite independently of his liking for abstract wit, he had a strong appreciation (one might say almost a hunger) for the more earthy type of comedy and humor".[78]
In 1955, a mass was found near von Neumann's collarbone, which turned out to be cancer originating in theskeleton,pancreas orprostate. (While there is general agreement that the tumor hadmetastasised, sources differ on the location of the primary cancer.)[79][80] The malignancy may have been caused byexposure toradiation atLos Alamos National Laboratory.[81] As death neared he asked for a priest, though the priest later recalled that von Neumann found little comfort in receiving thelast rites – he remained terrified of death and unable to accept it.[82][83][84][85] Of his religious views, von Neumann reportedly said, "So long as there is the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end," referring toPascal's wager. He confided to his mother, "There probably has to be a God. Many things are easier to explain if there is than if there isn't."[86][87]
At the beginning of the 20th century, efforts to base mathematics onnaive set theory suffered a setback due toRussell's paradox (on the set of all sets that do not belong to themselves).[90] The problem of an adequate axiomatization ofset theory was resolved implicitly about twenty years later byErnst Zermelo andAbraham Fraenkel.Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his 1925 doctoral thesis, von Neumann demonstrated two techniques to exclude such sets—theaxiom of foundation and the notion ofclass.[91]
The axiom of foundation proposed that every set can be constructed from the bottom up in an ordered succession of steps by way of the Zermelo–Fraenkel principles. If one set belongs to another, then the first must necessarily come before the second in the succession. This excludes the possibility of a set belonging to itself. To demonstrate that the addition of this new axiom to the others did not produce contradictions, von Neumann introduced themethod ofinner models, which became an essential demonstration instrument in set theory.[91]
The second approach to the problem of sets belonging to themselves took as its base the notion ofclass, and defines a set as a class that belongs to other classes, while aproper class is defined as a class that does not belong to other classes. On the Zermelo–Fraenkel approach, the axioms impede the construction of a set of all sets that do not belong to themselves. In contrast, on von Neumann's approach, the class of all sets that do not belong to themselves can be constructed, but it is aproper class, not a set.[91]
Overall, von Neumann's major achievement in set theory was an "axiomatization of set theory and (connected with that) elegant theory of theordinal andcardinal numbers as well as the first strict formulation of principles of definitions by thetransfinite induction".[92]
Building on theHausdorff paradox ofFelix Hausdorff (1914),Stefan Banach andAlfred Tarski in 1924 showed how to subdivide a three-dimensionalball intodisjoint sets, then translate and rotate these sets to form two identical copies of the same ball; this is theBanach–Tarski paradox. They also proved that a two-dimensional disk has no such paradoxical decomposition. But in 1929,[93] von Neumann subdivided the disk into finitely many pieces and rearranged them into two disks, using area-preservingaffine transformations instead of translations and rotations. The result depended on findingfree groups of affine transformations, an important technique extended later by von Neumann inhis work on measure theory.[94]
With the contributions of von Neumann to sets, the axiomatic system of the theory of sets avoided the contradictions of earlier systems and became usable as a foundation for mathematics, despite the lack of a proof of itsconsistency. The next question was whether it provided definitive answers to all mathematical questions that could be posed in it, or whether it might be improved by adding strongeraxioms that could be used to prove a broader class of theorems.[95]
By 1927, von Neumann was involving himself in discussions in Göttingen on whetherelementary arithmetic followed fromPeano axioms.[96] Building on the work ofAckermann, he began attempting to prove (using thefinistic methods ofHilbert's school) the consistency offirst-order arithmetic. He succeeded in proving the consistency of a fragment of arithmetic of natural numbers (through the use of restrictions oninduction).[97] He continued looking for a more general proof of the consistency of classical mathematics using methods fromproof theory.[98]
A strongly negative answer to whether it was definitive arrived in September 1930 at theSecond Conference on the Epistemology of the Exact Sciences, in whichKurt Gödel announced hisfirst theorem of incompleteness: the usual axiomatic systems are incomplete, in the sense that they cannot prove every truth expressible in their language. Moreover, every consistent extension of these systems necessarily remains incomplete.[99] At the conference, von Neumann suggested to Gödel that he should try to transform his results for undecidable propositions about integers.[100]
Less than a month later, von Neumann communicated to Gödel an interesting consequence of his theorem: the usual axiomatic systems are unable to demonstrate their own consistency.[99] Gödel replied that he had already discovered this consequence, now known as hissecond incompleteness theorem, and that he would send a preprint of his article containing both results, which never appeared.[101][102][103] Von Neumann acknowledged Gödel's priority in his next letter.[104] However, von Neumann's method of proof differed from Gödel's, and he was also of the opinion that the second incompleteness theorem had dealt a much stronger blow to Hilbert's program than Gödel thought it did.[105][106] With this discovery, which drastically changed his views on mathematical rigor, von Neumann ceased research in thefoundations of mathematics andmetamathematics and instead spent time on problems connected with applications.[107]
In a series of papers published in 1932, von Neumann made foundational contributions toergodic theory, a branch of mathematics that involves the states ofdynamical systems with aninvariant measure.[108] Of the 1932 papers on ergodic theory,Paul Halmos wrote that even "if von Neumann had never done anything else, they would have been sufficient to guarantee him mathematical immortality".[109] By then von Neumann had already written his articles onoperator theory, and the application of this work was instrumental in hismean ergodic theorem.[110]
The theorem is about arbitraryone-parameterunitary groups and states that for every vector in theHilbert space, exists in the sense of the metric defined by the Hilbert norm and is a vector which is such that for all. This was proven in the first paper. In the second paper, von Neumann argued that his results here were sufficient for physical applications relating toBoltzmann'sergodic hypothesis. He also pointed out thatergodicity had not yet been achieved and isolated this for future work.[111]
Later in the year he published another influential paper that began the systematic study of ergodicity. He gave and proved a decomposition theorem showing that the ergodicmeasure preserving actions of the real line are the fundamental building blocks from which all measure preserving actions can be built. Several other key theorems are given and proven. The results in this paper and another in conjunction withPaul Halmos have significant applications in other areas of mathematics.[111][112]
Inmeasure theory, the "problem of measure" for ann-dimensionalEuclidean spaceRn may be stated as: "does there exist a positive, normalized, invariant, and additive set function on the class of all subsets ofRn?"[113] The work ofFelix Hausdorff andStefan Banach had implied that the problem of measure has a positive solution ifn = 1 orn = 2 and a negative solution (because of theBanach–Tarski paradox) in all other cases. Von Neumann's work argued that the "problem is essentially group-theoretic in character": the existence of a measure could be determined by looking at the properties of thetransformation group of the given space. The positive solution for spaces of dimension at most two, and the negative solution for higher dimensions, comes from the fact that theEuclidean group is asolvable group for dimension at most two, and is not solvable for higher dimensions. "Thus, according to von Neumann, it is the change of group that makes a difference, not the change of space."[114] Around 1942 he toldDorothy Maharam how to prove that everycompleteσ-finitemeasure space has a multiplicative lifting; he did not publish this proof and she later came up with a new one.[115]
In a number of von Neumann's papers, the methods of argument he employed are considered even more significant than the results. In anticipation of his later study of dimension theory in algebras of operators, von Neumann used results on equivalence by finite decomposition, and reformulated the problem of measure in terms of functions.[116] A major contribution von Neumann made to measure theory was the result of a paper written to answer a question ofHaar regarding whether there existed analgebra of all bounded functions on the real number line such that they form "a complete system of representatives of the classes of almost everywhere-equal measurable bounded functions".[117] He proved this in the positive, and in later papers withStone discussed various generalizations and algebraic aspects of this problem.[118] He also proved by new methods the existence ofdisintegrations for various general types of measures. Von Neumann also gave a new proof on the uniqueness of Haar measures by using the mean values of functions, although this method only worked forcompact groups.[117] He had to create entirely new techniques to apply this tolocally compact groups.[119] He also gave a new, ingenious proof for theRadon–Nikodym theorem.[120] His lecture notes on measure theory at the Institute for Advanced Study were an important source for knowledge on the topic in America at the time, and were later published.[121][122][123]
Using his previous work on measure theory, von Neumann made several contributions to the theory oftopological groups, beginning with a paper on almost periodic functions on groups, where von Neumann extendedBohr's theory ofalmost periodic functions to arbitrarygroups.[124] He continued this work with another paper in conjunction withBochner that improved the theory of almostperiodicity to includefunctions that took on elements oflinear spaces as values rather than numbers.[125] In 1938, he was awarded theBôcher Memorial Prize for his work inanalysis in relation to these papers.[126][127]
Von Neumann was the first to axiomatically define an abstractHilbert space. He defined it as acomplex vector space with aHermitian scalar product, with the correspondingnorm being both separable and complete. In the same papers he also proved the general form of theCauchy–Schwarz inequality that had previously been known only in specific examples.[131] He continued with the development of thespectral theory of operators in Hilbert space in three seminal papers between 1929 and 1932.[132] This work cumulated in hisMathematical Foundations of Quantum Mechanics which alongside two other books byStone andBanach in the same year were the first monographs on Hilbert space theory.[133] Previous work by others showed that a theory ofweak topologies could not be obtained by usingsequences. Von Neumann was the first to outline a program of how to overcome the difficulties, which resulted in him defininglocally convex spaces andtopological vector spaces for the first time. In addition several other topological properties he defined at the time (he was among the first mathematicians to apply new topological ideas fromHausdorff from Euclidean to Hilbert spaces)[134] such asboundness andtotal boundness are still used today.[135] For twenty years von Neumann was considered the 'undisputed master' of this area.[117] These developments were primarily prompted by needs inquantum mechanics where von Neumann realized the need to extendthe spectral theory of Hermitian operators from the bounded to theunbounded case.[136] Other major achievements in these papers include a complete elucidation of spectral theory fornormal operators, the first abstract presentation of thetrace of apositive operator,[137][138] a generalisation ofRiesz's presentation ofHilbert's spectral theorems at the time, and the discovery ofHermitian operators in a Hilbert space, as distinct fromself-adjoint operators, which enabled him to give a description of all Hermitian operators which extend a given Hermitian operator. He wrote a paper detailing how the usage ofinfinite matrices, common at the time in spectral theory, was inadequate as a representation for Hermitian operators. His work on operator theory lead to his most profound invention in pure mathematics, the study of von Neumann algebras and in general ofoperator algebras.[139]
His later work on rings of operators lead to him revisiting his work on spectral theory and providing a new way of working through the geometric content by the use of direct integrals of Hilbert spaces.[136] Like in his work on measure theory he proved several theorems that he did not find time to publish. He toldNachman Aronszajn and K. T. Smith that in the early 1930s he proved the existence of proper invariant subspaces for completely continuous operators in a Hilbert space while working on theinvariant subspace problem.[140]
WithPascual Jordan he wrote a short paper giving the first derivation of a given norm from aninner product by means of theparallelogram identity.[143] Histrace inequality is a key result of matrix theory used in matrix approximation problems.[144] He also first presented the idea that the dual of a pre-norm is a norm in the first major paper discussing the theory of unitarily invariant norms and symmetric gauge functions (now known as symmetric absolute norms).[145][146][147] This paper leads naturally to the study of symmetricoperator ideals and is the beginning point for modern studies of symmetricoperator spaces.[148]
Von Neumann founded the study of rings of operators, through thevon Neumann algebras (originally called W*-algebras). While his original ideas forrings ofoperators existed already in 1930, he did not begin studying them in depth until he metF. J. Murray several years later.[157][158] A von Neumann algebra is a*-algebra of bounded operators on aHilbert space that is closed in theweak operator topology and contains theidentity operator.[159] Thevon Neumann bicommutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as being equal to thebicommutant.[160] After elucidating the study of thecommutative algebra case, von Neumann embarked in 1936, with the partial collaboration of Murray, on thenoncommutative case, the general study offactors classification of von Neumann algebras. The six major papers in which he developed that theory between 1936 and 1940 "rank among the masterpieces of analysis in the twentieth century";[161] they collect many foundational results and started several programs in operator algebra theory that mathematicians worked on for decades afterwards. An example is the classification offactors.[162] In addition in 1938 he proved that every von Neumann algebra on a separable Hilbert space is a direct integral of factors; he did not find time to publish this result until 1949.[163][164] Von Neumann algebras relate closely to a theory of noncommutative integration, something that von Neumann hinted to in his work but did not explicitly write out.[165][166] Another important result onpolar decomposition was published in 1932.[167]
Between 1935 and 1937, von Neumann worked onlattice theory, the theory ofpartially ordered sets in which every two elements have a greatest lower bound and a least upper bound. AsGarrett Birkhoff wrote, "John von Neumann's brilliant mind blazed over lattice theory like a meteor".[168] Von Neumann combined traditional projective geometry with modern algebra (linear algebra,ring theory, lattice theory). Many previously geometric results could then be interpreted in the case of generalmodules over rings. His work laid the foundations for some of the modern work in projective geometry.[169]
His biggest contribution was founding the field ofcontinuous geometry.[170] It followed his path-breaking work on rings of operators. In mathematics, continuous geometry is a substitute of complexprojective geometry, where instead of thedimension of asubspace being in a discrete set it can be an element of theunit interval. Earlier,Menger and Birkhoff had axiomatizedcomplex projective geometry in terms of the properties of itslattice of linear subspaces. Von Neumann, following his work on rings of operators, weakened thoseaxioms to describe a broader class of lattices, the continuous geometries.
While the dimensions of the subspaces of projective geometries are a discrete set (thenon-negative integers), the dimensions of the elements of a continuous geometry can range continuously across the unit interval. Von Neumann was motivated by his discovery ofvon Neumann algebras with a dimension function taking a continuous range of dimensions, and the first example of a continuous geometry other than projective space was theprojections of thehyperfinite type II factor.[171][172]
In more pure lattice theoretical work, he solved the difficult problem of characterizing the class of (continuous-dimensional projective geometry over an arbitrarydivision ring) in abstract language of lattice theory.[173] Von Neumann provided an abstract exploration of dimension in completedcomplementedmodular topological lattices (properties that arise in thelattices of subspaces ofinner product spaces):
Dimension is determined, up to a positive linear transformation, by the following two properties. It is conserved by perspective mappings ("perspectivities") and ordered by inclusion. The deepest part of the proof concerns the equivalence of perspectivity with "projectivity by decomposition"—of which a corollary is the transitivity of perspectivity.
For any integer every-dimensional abstract projective geometry isisomorphic to the subspace-lattice of an-dimensionalvector space over a (unique) corresponding division ring. This is known as theVeblen–Young theorem. Von Neumann extended this fundamental result in projective geometry to the continuous dimensional case.[174] Thiscoordinatization theorem stimulated considerable work in abstract projective geometry and lattice theory, much of which continued using von Neumann's techniques.[169][175] Birkhoff described this theorem as follows:
Any complemented modular latticeL having a "basis" ofn ≥ 4 pairwise perspective elements, is isomorphic with the latticeℛ(R) of all principalright-ideals of a suitableregular ringR. This conclusion is the culmination of 140 pages of brilliant and incisive algebra involving entirely novel axioms. Anyone wishing to get an unforgettable impression of the razor edge of von Neumann's mind, need merely try to pursue this chain of exact reasoning for himself—realizing that often five pages of it were written down before breakfast, seated at a living room writing-table in a bathrobe.[176]
This work required the creation ofregular rings.[177] A von Neumann regular ring is aring where for every, an element exists such that.[176] These rings came from and have connections to his work on von Neumann algebras, as well asAW*-algebras and various kinds ofC*-algebras.[178]
Many smaller technical results were proven during the creation and proof of the above theorems, particularly regardingdistributivity (such as infinite distributivity), von Neumann developing them as needed. He also developed a theory of valuations in lattices, and shared in developing the general theory ofmetric lattices.[179]
Birkhoff noted in his posthumous article on von Neumann that most of these results were developed in an intense two-year period of work, and that while his interests continued in lattice theory after 1937, they became peripheral and mainly occurred in letters to other mathematicians. A final contribution in 1940 was for a joint seminar he conducted with Birkhoff at the Institute for Advanced Study on the subject where he developed a theory of σ-complete lattice ordered rings. He never wrote up the work for publication.[180]
Von Neumann made fundamental contributions tomathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identicallynormally distributed variables.[181] This ratio was applied to the residuals from regression models and is commonly known as theDurbin–Watson statistic[182] for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first orderautoregression.[182]
Subsequently,Denis Sargan andAlok Bhargava extended the results for testing whether the errors on a regression model follow a Gaussianrandom walk (i.e., possess aunit root) against the alternative that they are a stationary first order autoregression.[183]
Von Neumann was the first to establish a rigorous mathematical framework forquantum mechanics, known as theDirac–von Neumann axioms, in his influential 1932 workMathematical Foundations of Quantum Mechanics.[214] After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented aslinear operators acting on the Hilbert space associated with the quantum system.[215]
Thephysics of quantum mechanics was thereby reduced to themathematics of Hilbert spaces and linear operators acting on them. For example, theuncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into thenon-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger.[215]
Von Neumann's abstract treatment permitted him to confront the foundational issue of determinism versus non-determinism, and in the book he presented aproof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables", as in classical statistical mechanics. In 1935,Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid.[216] Hermann's work was largely ignored until afterJohn S. Bell made essentially the same argument in 1966.[217] In 2010,Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for allhidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories.[218] The validity of Bub's argument is, in turn, disputed.Gleason's theorem of 1957 provided an argument against hidden variables along the lines of von Neumann's, but founded on assumptions seen as better motivated and more physically meaningful.[219][220]
Von Neumann's proof inaugurated a line of research that ultimately led, throughBell's theorem and theexperiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires anotion of reality substantially different from that of classical physics, or must includenonlocality in apparent violation of special relativity.[221]
In a chapter ofThe Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-calledmeasurement problem. He concluded that the entire physical universe could be made subject to the universalwave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. In other words, while the line between observer and observed could be drawn in different places, the theory only makes sense if an observer exists somewhere.[222] Although the idea ofconsciousness causing collapse was accepted by Eugene Wigner,[223] this interpretation never gained acceptance among the majority of physicists.[224]
Though theories of quantum mechanics continue to evolve, a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches can be traced back to the mathematical formalisms and techniques first used by von Neumann. Discussions aboutinterpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.[214]
Viewing von Neumann's work on quantum mechanics as a part of the fulfilment ofHilbert's sixth problem, mathematical physicistArthur Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. With his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems.[225] Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed, he did not find a satisfactory mathematical framework for quantum theory as a whole.[226][227]
Von Neumann entropy is extensively used in different forms (conditional entropy,relative entropy, etc.) in the framework ofquantum information theory.[228] Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given astatistical ensemble of quantum mechanical systems with thedensity matrix, it is given by Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy[229] andconditional quantum entropy. Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy, a cornerstone in the former's development; theShannon entropy applies to classical information theory.[230]
Thevon Neumann measurement scheme, the ancestor of quantumdecoherence theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories.[235][236]
Von Neumann first proposed aquantum logic in his 1932 treatiseMathematical Foundations of Quantum Mechanics, where he noted that projections on aHilbert space can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated in a 1936 paper by von Neumann and Garrett Birkhoff, the first to introduce quantum logics,[237] wherein von Neumann and Birkhoff first proved that quantum mechanics requires apropositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., horizontally and vertically), and therefore,a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is addedbetween the other two, the photons will indeed pass through. This experimental fact is translatable into logic as thenon-commutativity of conjunction. It was also demonstrated that the laws of distribution of classical logic, and, are not valid for quantum theory.[238]
The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is in turn attributable to the fact that it is frequently the case in quantum mechanics that a pair of alternatives are semantically determinate, while each of its members is necessarily indeterminate. Consequently, thedistributive law of classical logic must be replaced with a weaker condition.[238] Instead of a distributive lattice, propositions about a quantum system form anorthomodular lattice isomorphic to the lattice of subspaces of the Hilbert space associated with that system.[239]
Nevertheless, he was never satisfied with his work on quantum logic. He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at theWashington Philosophical Society in 1945 he found that he could not, especially given that he was busy with war work at the time. During his address at the 1954International Congress of Mathematicians he gave this issue as one of the unsolved problems that future mathematicians could work on.[240][241]
Later withRobert D. Richtmyer, von Neumann developed an algorithm definingartificialviscosity that improved the understanding ofshock waves. When computers solved hydrodynamic or aerodynamic problems, they put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics.[245]
Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During World War II, he approached R. H. Kent, the director of the US Army'sBallistic Research Laboratory, with a computer program for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann gave a seminar on his program to an audience which included his friendTheodore von Kármán. After von Neumann had finished, von Kármán said "Of course you realizeLagrange also used digital models to simulatecontinuum mechanics." Von Neumann had been unaware of Lagrange'sMécanique analytique.[246]
Von Neumann's memorial plaque on the wall of his birthplace in Budapest, 5th district Báthory u. 26.
While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions. His pioneering papers withSubrahmanyan Chandrasekhar on the statistics of a fluctuatinggravitational field generated byrandomly distributedstars were considered atour de force.[247] In this paper they developed a theory of two-body relaxation[248] and used theHoltsmark distribution to model[249] thedynamics of stellar systems.[250] He wrote several other unpublished manuscripts on topics instellar structure, some of which were included in Chandrasekhar's other works.[251][252] In earlier work led byOswald Veblen von Neumann helped develop basic ideas involvingspinors that would lead toRoger Penrose'stwistor theory.[253][254] Much of this was done in seminars conducted at theIAS during the 1930s.[255] From this work he wrote a paper withA. H. Taub and Veblen extending theDirac equation toprojective relativity, with a key focus on maintaininginvariance with regards to coordinate,spin, andgauge transformations, as a part of early research into potential theories ofquantum gravity in the 1930s.[256] In the same time period he made several proposals to colleagues for dealing with the problems in the newly createdquantum field theory and forquantizing spacetime; however, both his colleagues and he did not consider the ideas fruitful and did not pursue them.[257][258][259] Nevertheless, he maintained at least some interest, in 1940 writing a manuscript on the Dirac equation inde Sitter space.[260]
Von Neumann founded the field ofgame theory as a mathematical discipline.[261] He proved hisminimax theorem in 1928. It establishes that inzero-sum games withperfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair ofstrategies for both players that allows each to minimize their maximum losses.[262] Such strategies are calledoptimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended theminimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944Theory of Games and Economic Behavior, written withOskar Morgenstern. The public interest in this work was such thatThe New York Times ran a front-page story.[263] In this book, von Neumann declared that economic theory needed to usefunctional analysis, especiallyconvex sets and thetopologicalfixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions.[261]
Von Neumann's functional-analytic techniques—the use ofduality pairings of realvector spaces to represent prices and quantities, the use ofsupporting andseparating hyperplanes and convex sets, and fixed-point theory—have been primary tools of mathematical economics ever since.[264]
Von Neumann raised themathematical 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 theBrouwer fixed-point theorem.[261] Von Neumann's model of an expanding economy considered thematrix pencilA − λB with nonnegative matrices A andB; von Neumann soughtprobabilityvectorsp and q and a positive number λ that would solve thecomplementarity equation along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vectorp represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solutionλ represents the growth factor which is 1 plus therate of growth of the economy; the rate of growth equals theinterest rate.[265][266]
Von Neumann's results have been viewed as a special case oflinear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists.[267][268] 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, andsaddlepoint duality.[269] 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.[270] The lasting importance of the work on general equilibria and the methodology of fixed point theorems is underscored by the awarding ofNobel prizes in 1972 toKenneth Arrow, in 1983 toGérard Debreu, and in 1994 toJohn Nash who used fixed point theorems to establish equilibria fornon-cooperative games and forbargaining problems in his Ph.D. thesis. Arrow and Debreu also used linear programming, as did Nobel laureatesTjalling Koopmans,Leonid Kantorovich,Wassily Leontief,Paul Samuelson,Robert Dorfman,Robert Solow, andLeonid Hurwicz.[271]
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 economistNicholas Kaldor; Kaldor recommended that von Neumann read a book by the mathematical economistLéon Walras. Von Neumann noticed that Walras'sGeneral Equilibrium Theory andWalras'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.[272]
Building on his results on matrix games and on his model of an expanding economy, von Neumann invented thetheory of duality in linear programming whenGeorge 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.[273]
Later, von Neumann suggested a new method oflinear programming, using the homogeneous linear system ofPaul Gordan (1873), which was later popularized byKarmarkar's algorithm. Von Neumann's method used a pivoting algorithm betweensimplices, with the pivoting decision determined by a nonnegativeleast squares subproblem with a convexity constraint (projecting the zero-vector onto theconvex hull of the active simplex). Von Neumann's algorithm was the firstinterior point method of linear programming.[274]
TheAVIDAC computer was partially based on the architecture of theIAS machine developed by von Neumann.
Von Neumann consulted for the Army'sBallistic Research Laboratory, most notably on theENIAC project,[276] as a member of its Scientific Advisory Committee.[277] Although the single-memory, stored-program architecture is commonly calledvon Neumann architecture, the architecture was based on the work ofJ. Presper Eckert andJohn Mauchly, inventors of ENIAC and its successor,EDVAC.While consulting for the EDVAC project at theUniversity of Pennsylvania, von Neumann wrote an incompleteFirst Draft of a Report on the EDVAC. The paper, whose premature distribution nullified the patent claims of Eckert and Mauchly, described a computer that stored both its data and its program in the same address space, unlike the earliest computers which stored their programs separately onpaper tape orplugboards. This architecture became the basis of most modern computer designs.[278]
Next, von Neumann designed theIAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at theRCA Research Laboratory nearby. Von Neumann recommended that theIBM 701, nicknamedthe defense computer, include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successfulIBM 704.[279][280]
Flow chart from von Neumann's "Planning and coding of problems for an electronic computing instrument", published in 1947
Von Neumann was the inventor, in 1945, of themerge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged.[281][282]
As part of von Neumann's hydrogen bomb work, he and Stanisław Ulam developed simulations for hydrodynamic computations. He also contributed to the development of theMonte Carlo method, which usedrandom numbers to approximate the solutions to complicated problems.[283]
Von Neumann's algorithm for simulating afair coin with a biased coin is used in the "software whitening" stage of somehardware random number generators.[284] Because obtaining "truly" random numbers was impractical, von Neumann developed a form ofpseudorandomness, using themiddle-square method. He justified this crude method as faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin."[284] He also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect.[284]
The first implementation of von Neumann's self-reproducing universal constructor.[289] Three generations of machine are shown: the second has nearly finished constructing the third. The lines running to the right are the tapes of genetic instructions, which are copied along with the body of the machines.A simple configuration in von Neumann's cellular automaton. A binary signal is passed repeatedly around the blue wire loop, using excited and quiescentordinary transmission states. A confluent cell duplicates the signal onto a length of red wire consisting ofspecial transmission states. The signal passes down this wire and constructs a new cell at the end. This particular signal (1011) codes for an east-directed special transmission state, thus extending the red wire by one cell each time. During construction, the new cell passes through several sensitised states, directed by the binary sequence.
Von Neumann's mathematical analysis of the structure ofself-replication preceded the discovery of the structure of DNA.[290] Ulam and von Neumann are also generally credited with creating the field ofcellular automata, beginning in the 1940s, as a simplified mathematical model of biological systems.[291]
In lectures in 1948 and 1949, von Neumann proposed akinematic self-reproducing automaton.[292][293] By 1952, he was treating the problem more abstractly. He designed an elaborate 2Dcellular automaton that would automatically make a copy of its initial configuration of cells.[294] TheVon Neumann universal constructor based on thevon Neumann cellular automaton was fleshed out in his posthumousTheory of Self Reproducing Automata.[295]Thevon Neumann neighborhood, in which each cell in a two-dimensional grid has the four orthogonally adjacent grid cells as neighbors, continues to be used for other cellular automata.[296]
From this work von Neumann realized that computation was not just a tool tobrute force the solution to a problem numerically, but could also provide insight for solving problems analytically,[304] and that there was an enormous variety of scientific and engineering problems towards which computers would be useful, most significant of which werenonlinear problems.[305] In June 1945 at the First Canadian Mathematical Congress he gave his first talk on general ideas of how to solve problems, particularly of fluid dynamics numerically.[246] He also described howwind tunnels were actuallyanalog computers, and how digital computers would replace them and bring a new era of fluid dynamics.Garrett Birkhoff described it as "an unforgettable sales pitch". He expanded this talk with Goldstine into the manuscript "On the Principles of Large Scale Computing Machines" and used it to promote the support of scientific computing. His papers also developed the concepts ofinverting matrices,random matrices and automatedrelaxation methods for solvingelliptic boundary value problems.[306]
As part of his research into possible applications of computers, von Neumann became interested in weather prediction, noting similarities between the problems in the field and those he had worked on during the Manhattan Project.[307] In 1946 von Neumann founded the "Meteorological Project" at the Institute for Advanced Study, securing funding for his project fromthe Weather Bureau, theUS Air Force and US Navy weather services.[308] WithCarl-Gustaf Rossby, considered the leading theoretical meteorologist at the time, he gathered a group of twenty meteorologists to work on various problems in the field. However, given his other postwar work he was not able to devote enough time to proper leadership of the project and little was accomplished.
This changed when a youngJule Gregory Charney took up co-leadership of the project from Rossby.[309] By 1950 von Neumann and Charney wrote the world's first climate modelling software, and used it to perform the world's first numericalweather forecasts on the ENIAC computer that von Neumann had arranged to be used;[308] von Neumann and his team published the results asNumerical Integration of the Barotropic Vorticity Equation.[310] Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate.[311] Though primitive, news of the ENIAC forecasts quickly spread around the world and a number of parallel projects in other locations were initiated.[312]
In 1955 von Neumann, Charney and their collaborators convinced their funders to open the Joint Numerical Weather Prediction Unit (JNWPU) inSuitland, Maryland, which began routine real-time weather forecasting.[313] Next up, von Neumann proposed a research program for climate modeling:
The approach is to first try short-range forecasts, then long-range forecasts of those properties of the circulation that can perpetuate themselves over arbitrarily long periods of time, and only finally to attempt forecast for medium-long time periods which are too long to treat by simple hydrodynamic theory and too short to treat by the general principle of equilibrium theory.[314]
Positive results ofNorman A. Phillips in 1955 prompted immediate reaction and von Neumann organized a conference at Princeton on "Application of Numerical Integration Techniques to the Problem of the General Circulation". Once again he strategically organized the program as a predictive one to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now theGeophysical Fluid Dynamics Laboratory) next to the JNWPU.[315] He continued work both on technical issues of modelling and in ensuring continuing funding for these projects.[316]During the late 19th century,Svante Arrhenius suggested that human activity could causeglobal warming by addingcarbon dioxide to the atmosphere.[317] In 1955, von Neumann observed that this may already have begun: "Carbon dioxide released into the atmosphere by industry's burning ofcoal and oil – more than half of it during the last generation – may have changed the atmosphere's composition sufficiently to account for a general warming of the world by about one degree Fahrenheit."[318][319] His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on thepolar ice caps to enhance absorption of solar radiation (by reducing thealbedo).[320][321][320][321] However, he urged caution in any program of atmosphere modification:
Whatcould be done, of course, is no index to whatshould be done... In fact, to evaluate the ultimate consequences of either a general cooling or a general heating would be a complex matter. Changes would affect the level of the seas, and hence the habitability of the continental coastal shelves; the evaporation of the seas, and hence general precipitation and glaciation levels; and so on... But there is little doubt that onecould carry out the necessary analyses needed to predict the results, intervene on any desired scale, and ultimately achieve rather fantastic results.[319]
He also warned that weather and climate control could have military uses, tellingCongress in 1956 that they could pose an even bigger risk thanICBMs.[322]
"The technology that is now developing and that will dominate the next decades is in conflict with traditional, and, in the main, momentarily still valid, geographical and political units and concepts. This is a maturing crisis of technology... The most hopeful answer is that the human species has been subjected to similar tests before and it seems to have a congenital ability to come through, after varying amounts of trouble."
The first use of the concept of asingularity in the technological context is attributed to von Neumann,[323] who according to Ulam discussed the "ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue."[324] This concept was later fleshed out in the 1970 bookFuture Shock byAlvin Toffler.
Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, he was the leading authority of the mathematics ofshaped charges, leading him to a large number of military consultancies and consequently his involvement in theManhattan Project. The involvement included frequent trips to the project's secret research facilities at theLos Alamos Laboratory in New Mexico.[39]
Von Neumann made his principal contribution to theatomic bomb in the concept and design of theexplosive lenses that were needed to compress theplutonium core of theFat Man weapon that was later dropped onNagasaki.[325] While von Neumann did not originate the "implosion" concept, he was one of its most persistent proponents, encouraging its continued development against the instincts of many of his colleagues, who felt such a design to be unworkable. He also eventually came up with the idea of using more powerful shaped charges and less fissionable material to greatly increase the speed of "assembly".[326]
When it turned out that there would not be enoughuranium-235 to make more than one bomb, the implosive lens project was greatly expanded and von Neumann's idea was implemented. Implosion was the only method that could be used with theplutonium-239 that was available from theHanford Site.[327] He established the design of theexplosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives.[328] His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry.[329] After a series of failed attempts with models, this was achieved byGeorge Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945.[330]
In a visit to Los Alamos in September 1944, von Neumann showed that the pressure increase from explosion shock wave reflection from solid objects was greater than previously believed if the angle of incidence of the shock wave was between 90° and some limiting angle. As a result, it was determined that the effectiveness of an atomic bomb would be enhanced with detonation some kilometers above the target, rather than at ground level.[331][332]
Implosion mechanism
Von Neumann was included in the target selection committee that was responsible for choosing the Japanese cities ofHiroshima and Nagasaki as thefirst targets of the atomic bomb. Von Neumann oversaw computations related to the expected size of the bomb blasts, estimated death tolls, and the distance above the ground at which the bombs should be detonated for optimum shock wave propagation. The cultural capitalKyoto was von Neumann's first choice,[333] a selection seconded by Manhattan Project leader GeneralLeslie Groves. However, this target was dismissed bySecretary of WarHenry L. Stimson.[334]
On July 16, 1945, von Neumann and numerous other Manhattan Project personnel were eyewitnesses to the first test of an atomic bomb detonation, which was code-namedTrinity. The event was conducted as a test of the implosion method device, at theAlamogordo Bombing Range in New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5kilotons of TNT (21 TJ) butEnrico Fermi produced a more accurate estimate of 10 kilotons by dropping scraps of torn-up paper as the shock wave passed his location and watching how far they scattered. The actual power of the explosion had been between 20 and 22 kilotons.[335] It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time.[336]
Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained thehydrogen bomb project. He collaborated withKlaus Fuchs on further development of the bomb, and in 1946 the two filed a secret patent outlining a scheme for using a fission bomb to compress fusion fuel to initiatenuclear fusion.[337] The Fuchs–von Neumann patent usedradiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, theTeller–Ulam design. Their work was, however, incorporated into the "George" shot ofOperation Greenhouse, which was instructive in testing out concepts that went into the final design.[338] The Fuchs–von Neumann work was passed on to the Soviet Union by Fuchs as part of hisnuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historianJeremy Bernstein has pointed out that ironically, "John von Neumann and Klaus Fuchs, produced a brilliant invention in 1946 that could have changed the whole course of the development of the hydrogen bomb, but was not fully understood until after the bomb had been successfully made."[338]
During several meetings of the advisory board of the US Air Force, von Neumann andEdward Teller predicted that by 1960 the US would be able to build a hydrogen bomb light enough to fit on top of a rocket. In 1953Bernard Schriever, who was present at the meeting, paid a personal visit to von Neumann at Princeton to confirm this possibility.[344] Schriever enlistedTrevor Gardner, who in turn visited von Neumann several weeks later to fully understand the future possibilities before beginning his campaign for such a weapon in Washington.[345] Now either chairing or serving on several boards dealing with strategic missiles and nuclear weaponry, von Neumann was able to inject several crucial arguments regarding potentialSoviet advancements in both these areas and in strategic defenses against American bombers into government reports to argue for the creation ofICBMs.[346] Gardner on several occasions brought von Neumann to meetings with the US Department of Defense to discuss with various senior officials his reports.[347] Several design decisions in these reports such as inertial guidance mechanisms would form the basis for all ICBMs thereafter.[348] By 1954, von Neumann was also regularly testifying to variousCongressional military subcommittees to ensure continued support for the ICBM program.[349]
However, this was not enough. To have the ICBM program run at full throttle they needed direct action by the President of the United States.[350] They convincedPresident Eisenhower in a direct meeting in July 1955, which resulted in a presidential directive on September 13, 1955. It stated that "there would be the gravest repercussions on the national security and on the cohesion of the free world" if the Soviet Union developed the ICBM before the US and therefore designated the ICBM project "a research and development program of the highest priority above all others." The Secretary of Defense was ordered to commence the project with "maximum urgency".[351] Evidence would later show that the Soviets indeed were already testing their ownintermediate-range ballistic missiles at the time.[352] Von Neumann would continue to meet the President, including at his home inGettysburg, Pennsylvania, and other high-level government officials as a key advisor on ICBMs until his death.[353]
In 1955, von Neumann became a commissioner of theAtomic Energy Commission (AEC), which at the time was the highest official position available to scientists in the government.[354] (While his appointment formally required that he sever all his other consulting contracts,[355] an exemption was made for von Neumann to continue working with several critical military committees after theAir Force and several keysenators raised concerns.[353]) He used this position to further the production of compact hydrogen bombs suitable forintercontinental ballistic missile (ICBM) delivery. He involved himself in correcting the severe shortage oftritium andlithium 6 needed for these weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered deep into enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile would not be a problem with an H-bomb. He said the Russians would probably be building a similar weapon system, which turned out to be the case.[356][357] WhileLewis Strauss was away in the second half of 1955 von Neumann took over as acting chairman of the commission.[358]
In his final years before his death from cancer, von Neumann headed the United States government's top-secret ICBM committee, which would sometimes meet in his home. Its purpose was to decide on the feasibility of building an ICBM large enough to carry a thermonuclear weapon. Von Neumann had long argued that while the technical obstacles were sizable, they could be overcome. TheSM-65 Atlas passed its first fully functional test in 1959, two years after his death.[359] The more advancedTitan rockets were deployed in 1962. Both had been proposed in the ICBM committees von Neumann chaired.[353] The feasibility of the ICBMs owed as much to improved, smaller warheads that did not have guidance or heat resistance issues as it did to developments in rocketry, and his understanding of the former made his advice invaluable.[359][353]
Von Neumann entered government service primarily because he felt that, if freedom and civilization were to survive, it would have to be because the United States would triumph over totalitarianism fromNazism,Fascism andSoviet Communism.[61] During aSenate committee hearing he described his political ideology as "violentlyanti-communist, and much more militaristic than the norm".[360][361]
Herman Goldstine commented on von Neumann's ability to intuit hidden errors and remember old material perfectly.[362][363] When he had difficulties he would not labor on them; instead, he would go home and sleep on it and come back later with a solution.[364] 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.[365] At times he could be ignorant of the standard mathematical literature, finding it easier to rederive basic information he needed rather than chase references.[366]
AfterWorld War II began, he became extremely busy with both academic and military commitments. His habit of not writing up talks or publishing results worsened.[367] 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".[368]
The mathematicianJean 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".[161] 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 inlinear algebra andgeneral topology. While von Neumann had an encyclopedic background, his range in pure mathematics was not as wide asPoincaré,Hilbert or evenWeyl: von Neumann never did significant work innumber theory,algebraic topology,algebraic geometry ordifferential geometry. However, in applied mathematics his work equalled that ofGauss,Cauchy orPoincaré.[117]
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."[369] 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 atorus.[370] 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 toHermann Weyl, who he thought was deeper and broader.[364]
In his biography of von Neumann,Salomon Bochner wrote that much of von Neumann's works in pure mathematics involved finite and infinite dimensionalvector 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 astopology,differential geometry andharmonic integrals,algebraic geometry and other such fields. Von Neumann rarely worked in these fields and, as Bochner saw it, had little affinity for them.[130]
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.[371] 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.[372]
Ulam remarked that most mathematicians could master one technique that they then used repeatedly, whereas von Neumann had mastered three:
A facility with the symbolic manipulation of linear operators;
An intuitive feeling for the logical structure of any new mathematical theory;
An intuitive feeling for the combinatorial superstructure of new theories.[373]
Although he was commonly described as an analyst, he once classified himself an algebraist,[374] and his style often displayed a mix of algebraic technique and set-theoretical intuition.[375] He loved obsessive detail and had no issues with excess repetition or overly explicit notation. An example of this was a paper of his on rings of operators, where he extended the normal functional notation, to. However, this process ended up being repeated several times, where the final result were equations such as. The 1936 paper became known to students as "von Neumann's onion"[376] because the equations "needed to be peeled before they could be digested". Overall, although his writings were clear and powerful, they were not clean or elegant.[377] Although powerful technically, his primary concern was more with the clear and viable formation of fundamental issues and questions of science rather than just the solution of mathematical puzzles.[376]
According to Ulam, von Neumann surprised physicists by doing dimensional estimates and algebraic computations in his head with fluency Ulam likened toblindfold chess. His impression was that von Neumann analyzed physical situations by abstract logical deduction rather than concrete visualization.[378]
Goldstine compared his lectures to being on glass, smooth and lucid. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight.[65]Halmos described his lectures as "dazzling", with his speech clear, rapid, precise and all encompassing. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures but puzzling on later reflection.[366] He was a quick speaker:Banesh Hoffmann found it very difficult to take notes, even inshorthand,[379] andAlbert Tucker said that people often had to ask von Neumann questions to slow him down so they could think through the ideas he was presenting. Von Neumann knew about this and was grateful for his audience telling him when he was going too quickly.[380] Although he did spend time preparing for lectures, he rarely used notes, instead jotting down points of what he would discuss and for how long.[366]
One of his remarkable abilities was his power of absolute recall. As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover, he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me howA Tale of Two Cities started. Whereupon, without any pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.[381]
Von Neumann was reportedly able to memorize the pages of telephone directories. He entertained friends by asking them to randomly call out page numbers; he then recited the names, addresses and numbers therein.[29][382]Stanisław Ulam believed that von Neumann's memory was auditory rather than visual.[383]
Von Neumann's mathematical fluency, calculation speed, and general problem-solving ability were widely noted by his peers.Paul Halmos called his speed "awe-inspiring."[384]Lothar Wolfgang Nordheim described him as the "fastest mind I ever met".[385]Enrico Fermi told physicistHerbert L. Anderson: "You know, Herb, Johnny can do calculations in his head ten times as fast as I can! And I can do them ten times as fast as you can, Herb, so you can see how impressive Johnny is!"[386]Edward Teller admitted that he "never could keep up with him",[387] andIsrael Halperin described trying to keep up as like riding a "tricycle chasing a racing car."[388]
He had an unusual ability to solve novel problems quickly.George Pólya, whose lectures atETH Zürich von Neumann attended as a student, said, "Johnny was the only student I was ever afraid of. If in the course of a lecture I stated an unsolved problem, the chances were he'd come to me at the end of the lecture with the complete solution scribbled on a slip of paper."[389] WhenGeorge Dantzig brought von Neumann an unsolved problem in linear programming "as I would to an ordinary mortal", on which there had been no published literature, he was astonished when von Neumann said "Oh, that!", before offhandedly giving a lecture of over an hour, explaining how to solve the problem using the hitherto unconceivedtheory of duality.[390]
A story about von Neumann's encounter with the famousfly puzzle[391] has enteredmathematical folklore. In this puzzle, two bicycles begin 20 miles apart, and each travels toward the other at 10 miles per hour until they collide; meanwhile, a fly travels continuously back and forth between the bicycles at 15 miles per hour until it is squashed in the collision. The questioner asks how far the fly traveled in total; the "trick" for a quick answer is to realize that the fly's individual transits do not matter, only that it has been traveling at 15 miles per hour for one hour. AsEugene Wigner tells it,[392]Max Born posed the riddle to von Neumann. The other scientists to whom he had posed it had laboriously computed the distance, so when von Neumann was immediately ready with the correct answer of 15 miles, Born observed that he must have guessed the trick. "What trick?" von Neumann replied. "All I did was sum thegeometric series."[393]
Rota wrote that von Neumann had "deep-seated and recurring self-doubts".[394]John L. Kelley reminisced in 1989 that "Johnny von Neumann has said that he will be forgotten whileKurt Gödel is remembered withPythagoras, but the rest of us viewed Johnny with awe."[395] Ulam suggests that some of his self-doubts with regard for his own creativity may have come from the fact he had not discovered several important ideas that others had, even though he was more than capable of doing so, giving theincompleteness theorems andBirkhoff'spointwise ergodic theorem as examples. Von Neumann had a virtuosity in following complicated reasoning and had supreme insights, yet he perhaps felt he did not have the gift for seemingly irrational proofs and theorems or intuitive insights. Ulam describes how during one of his stays at Princeton while von Neumann was working on rings of operators, continuous geometries and quantum logic he felt that von Neumann was not convinced of the importance of his work, and only when finding some ingenious technical trick or new approach did he take some pleasure in it.[396] However, according to Rota, von Neumann still had an "incomparably stronger technique" compared to his friend, despite describing Ulam as the more creative mathematician.[394]
Nobel LaureateHans 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."[397]Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics".[367]Eugene Wigner said, "He understood mathematical problems not only in their initial aspect, but in their full complexity."[398]Claude Shannon called him "the smartest person I've ever met", a common opinion.[399]Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. And he was a genius, in the sense that a genius is a man who hastwo great ideas".[400] In 2006, Tom Siegfried wrote that "If any one person in the previous century personified the wordpolymath, it was von Neumann" and that "His contributions to physics, mathematics, computer science, and economics rank him as one of the all-time intellectual giants of each field."[401]
Wigner noted the extraordinary mind that von Neumann had, and described von Neumann as having a mind faster than anyone he knew, stating that:[398]
I have known a great many intelligent people in my life. I knewMax Planck,Max von Laue, andWerner Heisenberg.Paul Dirac was my brother-in-law;Leo Szilard and Edward Teller have been among my closest friends; andAlbert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me.
"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," wroteMiklós Rédei.[402] Lax stated that von Neumann would have won aNobel 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."[403]Gian-Carlo Rota wrote that von Neumann "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."[404] He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century.[405] As a result of his wide reaching influence and contributions to many fields, he is widely considered apolymath.[406][407][408]
NeurophysiologistLeon 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."[409] 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. Innuclear missilery he was considered "the clearly dominant advisory figure" according toHerbert York.[410] EconomistNicholas Kaldor said he was "unquestionably the nearest thing to a genius I have ever encountered."[269] Likewise,Paul Samuelson wrote, "We economists are grateful for von Neumann's genius. It is not for us to calculate whether he was aGauss, or aPoincaré, or aHilbert. He was the incomparable Johnny von Neumann. He darted briefly into our domain and it has never been the same since."[411]
Von Neumann's first published paper wasOn the position of zeroes of certain minimum polynomials, co-authored withMichael Fekete and published when von Neumann was 18. At 19, his solo paperOn the introduction of transfinite numbers was published.[424] He expanded his second solo paper,An axiomatization of set theory, to create his PhD thesis.[425] His first book,Mathematical Foundations of Quantum Mechanics, was published in 1932.[426] Following this, von Neumann switched from publishing in German to publishing in English, and his publications became more selective and expanded beyond pure mathematics. His 1942Theory of Detonation Waves contributed to military research,[427] his work on computing began with the unpublished 1946On the principles of large scale computing machines, and his publications on weather prediction began with the 1950Numerical integration of the barotropic vorticity equation.[428] Alongside his later papers were informal essays targeted at colleagues and the general public, such as his 1947The Mathematician,[429] described as a "farewell to pure mathematics", and his 1955Can we survive technology?, which considered a bleak future including nuclear warfare and deliberate climate change.[430] His complete works have been compiled into a six-volume set.[424]
^While Israel Halperin's thesis advisor is often listed asSalomon Bochner, this may be because "Professors at the university direct doctoral theses but those at the Institute do not. Unaware of this, in 1934 I asked von Neumann if he would direct my doctoral thesis. He replied Yes."[4]
^Weyl, Hermann (2012). Pesic, Peter (ed.).Levels of Infinity: Selected Writings on Mathematics and Philosophy (1 ed.). Dover Publications. p. 55.ISBN978-0-486-48903-2.
^Rota, Gian-Carlo (1989). "The Lost Cafe". In Cooper, Necia Grant; Eckhardt, Roger; Shera, Nancy (eds.).From Cardinals To Chaos: Reflections On The Life And Legacy Of Stanisław Ulam. Cambridge University Press. pp. 23–32.ISBN978-0-521-36734-9.OCLC18290810.
^Read, Colin (2012).The Portfolio Theorists: von Neumann, Savage, Arrow and Markowitz. Great Minds in Finance. Palgrave Macmillan. p. 65.ISBN978-0-230-27414-3. Retrieved2017-09-29.When von Neumann realised he was incurably ill his logic forced him to realise that he would cease to exist... [a] fate which appeared to him unavoidable but unacceptable.
^Ayoub, Raymond George (2004).Musings Of The Masters: An Anthology Of Mathematical Reflections. Washington, D.C.: MAA. p. 170.ISBN978-0-88385-549-2.OCLC56537093.
^Von Plato, Jan (2018)."The Development of Proof Theory". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Winter 2018 ed.). Stanford University. Retrieved2023-09-25.
^von Neumann, John (1934). "Almost Periodic Functions in a Group. I.".Transactions of the American Mathematical Society.36 (3):445–492.doi:10.2307/1989792.JSTOR1989792.
^von Neumann, John; Bochner, Salomon (1935). "Almost Periodic Functions in Groups, II".Transactions of the American Mathematical Society.37 (1):21–50.doi:10.2307/1989694.JSTOR1989694.
^v. Neumann, J. (1929). "Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen".Mathematische Zeitschrift (in German).30 (1):3–42.doi:10.1007/BF01187749.S2CID122565679.
^Kar, Purushottam; Karnick, Harish (2013). "On Translation Invariant Kernels and Screw Functions". p. 2.arXiv:1302.4343 [math.FA].
^Alpay, Daniel; Levanony, David (2008). "On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions".Potential Analysis.28 (2):163–184.arXiv:0705.2863.doi:10.1007/s11118-007-9070-4.S2CID15895847.
^Sargan, J.D.; Bhargava, Alok (1983). "Testing residuals from least squares regression for being generated by the Gaussian random walk".Econometrica.51 (1):153–174.doi:10.2307/1912252.JSTOR1912252.
^Narkiewicz, Wladyslaw (2004).Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics (3rd ed.). Springer. p. 120.doi:10.1007/978-3-662-07001-7.ISBN978-3-662-07001-7.Narkiewicz, Władysław (2018).The Story of Algebraic Numbers in the First Half of the 20th Century: From Hilbert to Tate. Springer Monographs in Mathematics. Springer. p. 144.doi:10.1007/978-3-030-03754-3.ISBN978-3-030-03754-3.
^Harzheim, Egbert (2008). "A Construction of Subsets of the Reals which have a Similarity Decomposition".Order.25 (2):79–83.doi:10.1007/s11083-008-9079-3.S2CID45005704.
^Kronz, Fred; Lupher, Tracy (2021)."Quantum Theory and Mathematical Rigor". In Zalta, Edward N. (ed.).Stanford Encyclopedia of Philosophy (Winter 2021 ed.). Stanford University. Retrieved2022-12-21.
^Hall, Brian C. (2013). "Systems and Subsystems, Multiple Particles".Quantum Theory for Mathematicians. Graduate Texts in Mathematics. Vol. 267. pp. 419–440.doi:10.1007/978-1-4614-7116-5_19.ISBN978-1-4614-7115-8.
^Giulini, Domenico; Joos, Erich; Kiefer, Claus; Kupsch, Joachim; Stamatescu, Ion-Olimpiu;Zeh, H. Dieter (1996).Decoherence and the Appearance of a Classical World in Quantum Theory. Berlin, Heidelberg: Springer Berlin Heidelberg.ISBN978-3-662-03263-3.OCLC851393174.
^Bacciagaluppi, Guido (2020)."The Role of Decoherence in Quantum Mechanics". In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy (Fall 2020 ed.). Stanford University. Retrieved2023-09-25.
^Uchaikin, Vladimir V.; Zolotarev, Vladimir M. (1999).Chance and Stability: Stable Distributions and their Applications. De Gruyter. pp. xviii, 281, 424.doi:10.1515/9783110935974.ISBN978-3-11-063115-9.
where the nonnegative matrixA must be square and where thediagonal matrixIis theidentity matrix. Von Neumann's irreducibility condition was called the "whales andwranglers" hypothesis byD. G. Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given byDavid Gale and byJohn Kemeny, Morgenstern, andGerald L. Thompson in the 1950s and then by Stephen M. Robinson in the 1970s.
^Rockafellar, R. T. (1970).Convex analysis. Princeton University Press. pp. i, 74.ISBN978-0-691-08069-7.OCLC64619.Rockafellar, R. T. (1974). "Convex Algebra and Duality in Dynamic Models of production". In Loz, Josef; Loz, Maria (eds.).Mathematical Models in Economics. Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972. Amsterdam: Elsevier North-Holland Publishing and Polish Academy of Sciences. pp. 351–378.OCLC839117596.
^Bruckmann, Gerhart; Weber, Wilhelm, eds. (September 21, 1971).Contributions to von Neumann's Growth Model. Proceedings of a Conference Organized by the Institute for Advanced Studies Vienna, Austria, July 6 and 7, 1970. Springer–Verlag.doi:10.1007/978-3-662-24667-2.ISBN978-3-662-22738-1.
^Dantzig, G. B. (1983). "Reminiscences about the origins of linear programming.". In Bachem, A.; Grötschel, M.; Korte, B. (eds.).Mathematical Programming The State of the Art: Bonn 1982. Berlin, New York: Springer-Verlag. pp. 78–86.ISBN0-387-12082-3.OCLC9556834.
^Damerow, Julia, ed. (June 14, 2010)."John von Neumann's Cellular Automata".Embryo Project Encyclopedia. Arizona State University. School of Life Sciences. Center for Biology and Society. Retrieved2024-01-14.
^Birkhoff, Garrett (1990). "Fluid dynamics, reactor computations, and surface representation". In Nash, Stephen G. (ed.).A history of scientific computing. Association for Computing Machinery. pp. 64–69.doi:10.1145/87252.88072.ISBN978-0-201-50814-7.
^Herken, Gregg (2002).Brotherhood of the Bomb: The Tangled Lives and Loyalties of Robert Oppenheimer, Ernest Lawrence, and Edward Teller. New York: Holt. pp. 171, 374.ISBN978-0-8050-6589-3.OCLC48941348.
^Adami, Christoph (2024).The Evolution of Biological Information: How Evolution Creates Complexity, from Viruses to Brains. Princeton: Princeton University Press. pp. 189–190.ISBN978-0-691-24114-2.
^McCorduck, Pamela (2004).Machines Who Think: A Personal Inquiry into the History and Prospects of Artificial Intelligence (2nd ed.). Routledge. p. 81.ISBN978-1-56881-205-2.
Pais, Abraham (2000).The Genius of Science: A Portrait Gallery: A Portrait Gallery of Twentieth-Century Physicists. Oxford University Press.ISBN978-0-19-850614-0.
Taub, A. H., ed. (1976) [1963].John von Neumann Collected Works Volume VI: Theory of Games, Astrophysics, Hydrodynamics and Meteorology. New York: Pergamon Press.ISBN978-0-08-009566-0.OCLC493423386.
Hargittai, Balazs; Hargittai, Istvan (2015).Wisdom of the Martians of Science: In Their Own Words with Commentaries. World Scientific.doi:10.1142/9809.ISBN978-981-4723-81-7.
Leonard, Robert (2010).Von Neumann, Morgenstern, and the Creation of Game Theory: From Chess to Social Science, 1900–1960. Cambridge University Press.doi:10.1017/CBO9780511778278.ISBN978-1-107-60926-6.
Grafton, Samuel (September 1956). "Married to a Man Who Believes the Mind Can Move the World".Good Housekeeping Magazine (Interview with Klari von Neumann). pp. 80–81,282–292.
Oral History Project - The Princeton Mathematics Community in the 1930s, contains many interviews that describe contact and anecdotes of von Neumann and others at the Princeton University and Institute for Advanced Study community.
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