# John von Neumann

Last updated

John von Neumann
John von Neumann in the 1940s
Member of the United States Atomic Energy Commission
In office
March 15, 1955 February 8, 1957

Von Neumann was the first to establish a rigorous mathematical framework for quantum mechanics, known as the Dirac–von Neumann axioms, in his widely influential 1932 work Mathematical Foundations of Quantum Mechanics . [225] After having completed the axiomatization of set theory, he began to confront the axiomatization of quantum mechanics. He realized in 1926 that a state of a quantum system could be represented by a point in a (complex) Hilbert space that, in general, could be infinite-dimensional even for a single particle. In this formalism of quantum mechanics, observable quantities such as position or momentum are represented as linear operators acting on the Hilbert space associated with the quantum system. [226]

The physics of quantum mechanics was thereby reduced to the mathematics of Hilbert spaces and linear operators acting on them. For example, the uncertainty principle, according to which the determination of the position of a particle prevents the determination of its momentum and vice versa, is translated into the non-commutativity of the two corresponding operators. This new mathematical formulation included as special cases the formulations of both Heisenberg and Schrödinger. [226] When Heisenberg was informed von Neumann had clarified the difference between an unbounded operator that was a self-adjoint operator and one that was merely symmetric, Heisenberg replied "Eh? What is the difference?" [227]

Von Neumann's abstract treatment permitted him also to confront the foundational issue of determinism versus non-determinism, and in the book he presented a proof that the statistical results of quantum mechanics could not possibly be averages of an underlying set of determined "hidden variables," as in classical statistical mechanics. In 1935, Grete Hermann published a paper arguing that the proof contained a conceptual error and was therefore invalid. [228] Hermann's work was largely ignored until after John S. Bell made essentially the same argument in 1966. [229] In 2010, Jeffrey Bub argued that Bell had misconstrued von Neumann's proof, and pointed out that the proof, though not valid for all hidden variable theories, does rule out a well-defined and important subset. Bub also suggests that von Neumann was aware of this limitation and did not claim that his proof completely ruled out hidden variable theories. [230] The validity of Bub's argument is, in turn, disputed. [231] In any case, Gleason's theorem of 1957 fills the gaps in von Neumann's approach.

Von Neumann's proof inaugurated a line of research that ultimately led, through Bell's theorem and the experiments of Alain Aspect in 1982, to the demonstration that quantum physics either requires a notion of reality substantially different from that of classical physics, or must include nonlocality in apparent violation of special relativity. [232]

In a chapter of The Mathematical Foundations of Quantum Mechanics, von Neumann deeply analyzed the so-called measurement problem. He concluded that the entire physical universe could be made subject to the universal wave function. Since something "outside the calculation" was needed to collapse the wave function, von Neumann concluded that the collapse was caused by the consciousness of the experimenter. He argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective consciousness" of the human observer. Although this view was accepted by Eugene Wigner, [233] the Von Neumann–Wigner interpretation never gained acceptance among the majority of physicists. [234] The Von Neumann–Wigner interpretation has been summarized as follows: [235]

The rules of quantum mechanics are correct but there is only one system which may be treated with quantum mechanics, namely the entire material world. There exist external observers which cannot be treated within quantum mechanics, namely human (and perhaps animal) minds, which perform measurements on the brain causing wave function collapse. [235]

Though theories of quantum mechanics continue to evolve, there is a basic framework for the mathematical formalism of problems in quantum mechanics underlying most approaches that can be traced back to the mathematical formalisms and techniques first used by von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations. [225]

Viewing von Neumann's work on quantum mechanics as a part of the fulfilment of Hilbert's sixth problem, noted mathematical physicist A. S. Wightman said in 1974 his axiomization of quantum theory was perhaps the most important axiomization of a physical theory to date. In the publication of his 1932 book, quantum mechanics became a mature theory in the sense it had a precise mathematical form, which allowed for clear answers to conceptual problems. [236] Nevertheless, von Neumann in his later years felt he had failed in this aspect of his scientific work as despite all the mathematics he developed (operator theory, von Neumann algebras, continuous geometries, etc.), he did not find a satisfactory mathematical framework for quantum theory as a whole (including quantum field theory). [237] [238]

#### Von Neumann entropy

Von Neumann entropy is extensively used in different forms (conditional entropy, relative entropy, etc.) in the framework of quantum information theory. [239] Entanglement measures are based upon some quantity directly related to the von Neumann entropy. Given a statistical ensemble of quantum mechanical systems with the density matrix ${\displaystyle \rho }$, it is given by ${\displaystyle S(\rho )=-\operatorname {Tr} (\rho \ln \rho ).\,}$ Many of the same entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and conditional quantum entropy.

#### Quantum mutual information

Quantum information theory is largely concerned with the interpretation and uses of von Neumann entropy. The von Neumann entropy is the cornerstone in the development of quantum information theory, while the Shannon entropy applies to classical information theory. This is considered a historical anomaly, as Shannon entropy might have been expected to be discovered before Von Neumann entropy, given the latter's more widespread application to quantum information theory. But Von Neumann discovered von Neumann entropy first, and applied it to questions of statistical physics. Decades later, Shannon developed an information-theoretic formula for use in classical information theory, and asked von Neumann what to call it. Von Neumann said to call it Shannon entropy, as it was a special case of von Neumann entropy. [240]

#### Density matrix

The formalism of density operators and matrices was introduced by von Neumann [241] in 1927 and independently, but less systematically by Lev Landau [242] and Felix Bloch [243] in 1927 and 1946 respectively. The density matrix is an alternative way to represent the state of a quantum system, which could otherwise be represented using the wavefunction. The density matrix allows the solution of certain time-dependent problems in quantum mechanics.

#### Von Neumann measurement scheme

The von Neumann measurement scheme, the ancestor of quantum decoherence theory, represents measurements projectively by taking into account the measuring apparatus which is also treated as a quantum object. The 'projective measurement' scheme introduced by von Neumann led to the development of quantum decoherence theories. [244] [245]

#### Quantum logic

Von Neumann first proposed a quantum logic in his 1932 treatise Mathematical Foundations of Quantum Mechanics , where he noted that projections on a Hilbert space can be viewed as propositions about physical observables. The field of quantum logic was subsequently inaugurated, in a famous paper of 1936 by von Neumann and Garrett Birkhoff, the first work ever to introduce quantum logics, [246] wherein von Neumann and Birkhoff first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work, but in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters that are polarized perpendicularly (e.g., horizontally and vertically), and therefore, a fortiori , it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession, but if the third filter is added between the other two, the photons will indeed pass through. This experimental fact is translatable into logic as the non-commutativity of conjunction ${\displaystyle (A\land B)\neq (B\land A)}$. It was also demonstrated that the laws of distribution of classical logic, ${\displaystyle P\lor (Q\land R)=(P\lor Q)\land (P\lor R)}$ and ${\displaystyle P\land (Q\lor R)=(P\land Q)\lor (P\land R)}$, are not valid for quantum theory. [247]

The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is in turn attributable to the fact that it is frequently the case in quantum mechanics that a pair of alternatives are semantically determinate, while each of its members is necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (spin angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which ${\displaystyle A\land (B\lor C)=A\land 1=A}$, while ${\displaystyle (A\land B)\lor (A\land C)=0\lor 0=0}$. [247] As Hilary Putnam writes, von Neumann replaced classical logic with a logic constructed in orthomodular lattices (isomorphic to the lattices of subspaces of the Hilbert space of a given physical system). [248]

Nevertheless, he was never satifisied with his work on quantum logic. He intended it to be a joint synthesis of formal logic and probability theory and when he attempted to write up a paper for the Henry Joseph Lecture he gave at the Washington Philosophical Society in 1945 he found that he could not, especially given that he was busy with war work at the time. He just could not make himself write something he did not fully understand to his satisfaction. During his address at the 1954 International Congress of Mathematicians he gave this issue as one of the unsolved problems that future mathematicians could work on. [249] [250]

### Fluid dynamics

Von Neumann made fundamental contributions in the field of fluid dynamics.

Von Neumann's contributions to fluid dynamics included his discovery of the classic flow solution to blast waves, [251] and the co-discovery (independently of Yakov Borisovich Zel'dovich and Werner Döring) of the ZND detonation model of explosives. [252] During the 1930s, von Neumann became an authority on the mathematics of shaped charges. [253]

Later with Robert D. Richtmyer, von Neumann developed an algorithm defining artificial viscosity that improved the understanding of shock waves. When computers solved hydrodynamic or aerodynamic problems, they tried to put too many computational grid points at regions of sharp discontinuity (shock waves). The mathematics of artificial viscosity smoothed the shock transition without sacrificing basic physics. [254]

Von Neumann soon applied computer modelling to the field, developing software for his ballistics research. During WW2, he arrived one day at the office of R.H. Kent, the Director of the US Army's Ballistic Research Laboratory, with a computer program he had created for calculating a one-dimensional model of 100 molecules to simulate a shock wave. Von Neumann then gave a seminar on his computer program to an audience which included his friend Theodore von Kármán. After von Neumann had finished, von Kármán said "Well, Johnny, that's very interesting. Of course you realize Lagrange also used digital models to simulate continuum mechanics." It was evident from von Neumann's face, that he had been unaware of Lagrange's Mécanique analytique. [255]

### Other work in physics

While not as prolific in physics as he was in mathematics, he nevertheless made several other notable contributions to it. His pioneering papers with Subrahmanyan Chandrasekhar on the statistics of a fluctuating gravitational field generated by randomly distributed stars were considered a tour de force. [256] In this paper they developed a theory of two-body relaxation [257] and used the Holtsmark distribution to model [258] the dynamics of stellar systems. [259] He wrote several other unpublished manuscripts on topics in stellar structure, some of which were included in Chandresekhar's other works. [260] [261] In some earlier work led by Oswald Veblen von Neumann helped develop basic ideas involving spinors that would lead to Roger Penrose's twistor theory. [262] [263] Much of this was done in seminars conducted at the IAS during the 1930s. [264] From this work he wrote a paper with A. H. Taub and Veblen extending the Dirac equation to projective relativity, maintaining invariance with regards to coordinate, spin, and gauge transformations, as a part of early research into potential theories of quantum gravity in the 1930s. [265] Additionally in the same time period he made several proposals to colleagues for dealing with the problems in the newly created quantum theory of fields and for quantizing spacetime, however both his colleagues and he himself did not consider the ideas fruitful and he did not work on them further. [266] [267] [268] Nevertheless, he maintained at least some interest in these ideas as he had as late as 1940 written a manuscript on the Dirac equation in De Sitter space. [269]

## Economics

### Game theory

Von Neumann founded the field of game theory as a mathematical discipline. [270] He proved his minimax theorem in 1928. It establishes that in zero-sum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss. [271]

Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). He improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944 Theory of Games and Economic Behavior , written with Oskar Morgenstern. Morgenstern wrote a paper on game theory and thought he would show it to von Neumann because of his interest in the subject. He read it and said to Morgenstern that he should put more in it. This was repeated a couple of times, and then von Neumann became a coauthor and the paper became 100 pages long. Then it became a book.[ citation needed ] The public interest in this work was such that The New York Times ran a front-page story. [272] In this book, von Neumann declared that economic theory needed to use functional analysis, especially convex sets and the topological fixed-point theorem, rather than the traditional differential calculus, because the maximum-operator did not preserve differentiable functions. [270]

Independently, Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and vector lattices. Von Neumann's functional-analytic techniques—the use of duality pairings of real vector spaces to represent prices and quantities, the use of supporting and separating hyperplanes and convex sets, and fixed-point theory—have been the primary tools of mathematical economics ever since. [273]

### Mathematical economics

Von Neumann raised the intellectual and mathematical level of economics in several influential publications. For his model of an expanding economy, he proved the existence and uniqueness of an equilibrium using his generalization of the Brouwer fixed-point theorem. [270] Von Neumann's model of an expanding economy considered the matrix pencil  A  λB with nonnegative matrices A and B; von Neumann sought probability vectors  p and q and a positive number λ that would solve the complementarity equation

${\displaystyle p^{T}(A-\lambda B)q=0}$

along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the growth factor which is 1 plus the rate of growth of the economy; the rate of growth equals the interest rate. [274] [275]

Von Neumann's results have been viewed as a special case of linear programming, where his model uses only nonnegative matrices. The study of his model of an expanding economy continues to interest mathematical economists with interests in computational economics. [276] [277] [278] This paper has been called the greatest paper in mathematical economics by several authors, who recognized its introduction of fixed-point theorems, linear inequalities, complementary slackness, and saddlepoint duality. In the proceedings of a conference on von Neumann's growth model, Paul Samuelson said that many mathematicians had developed methods useful to economists, but that von Neumann was unique in having made significant contributions to economic theory itself. [279]

Von Neumann's famous 9-page paper started life as a talk at Princeton and then became a paper in German that was eventually translated into English. His interest in economics that led to that paper began while he was lecturing at Berlin in 1928 and 1929. He spent his summers back home in Budapest, as did the economist Nicholas Kaldor, and they hit it off. Kaldor recommended that von Neumann read a book by the mathematical economist Léon Walras. Von Neumann found some faults in the book and corrected them–for example, replacing equations by inequalities. He noticed that Walras's General Equilibrium Theory and Walras's law, which led to systems of simultaneous linear equations, could produce the absurd result that profit could be maximized by producing and selling a negative quantity of a product. He replaced the equations by inequalities, introduced dynamic equilibria, among other things, and eventually produced the paper. [280]

### Linear programming

Building on his results on matrix games and on his model of an expanding economy, von Neumann invented the theory of duality in linear programming when George Dantzig described his work in a few minutes, and an impatient von Neumann asked him to get to the point. Dantzig then listened dumbfounded while von Neumann provided an hourlong lecture on convex sets, fixed-point theory, and duality, conjecturing the equivalence between matrix games and linear programming. [281]

Later, von Neumann suggested a new method of linear programming, using the homogeneous linear system of Paul Gordan (1873), which was later popularized by Karmarkar's algorithm. Von Neumann's method used a pivoting algorithm between simplices, with the pivoting decision determined by a nonnegative least squares subproblem with a convexity constraint (projecting the zero-vector onto the convex hull of the active simplex). Von Neumann's algorithm was the first interior point method of linear programming. [282]

## Computer science

Von Neumann was a founding figure in computing. [283] Von Neumann was the inventor, in 1945, of the merge sort algorithm, in which the first and second halves of an array are each sorted recursively and then merged. [284] [285] Von Neumann wrote the 23 pages long sorting program for the EDVAC in ink. On the first page, traces of the phrase "TOP SECRET", which was written in pencil and later erased, can still be seen. [285] He also worked on the philosophy of artificial intelligence with Alan Turing when the latter visited Princeton in the 1930s. [286]

Von Neumann's hydrogen bomb work was played out in the realm of computing, where he and Stanisław Ulam developed simulations on von Neumann's digital computers for the hydrodynamic computations. During this time he contributed to the development of the Monte Carlo method, which allowed solutions to complicated problems to be approximated using random numbers. [287]

Von Neumann's algorithm for simulating a fair coin with a biased coin is used in the "software whitening" stage of some hardware random number generators. [288] Because using lists of "truly" random numbers was extremely slow, von Neumann developed a form of making pseudorandom numbers, using the middle-square method. Though this method has been criticized as crude, von Neumann was aware of this: he justified it as being faster than any other method at his disposal, writing that "Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin." [289] Von Neumann also noted that when this method went awry it did so obviously, unlike other methods which could be subtly incorrect. [289]

While consulting for the Moore School of Electrical Engineering at the University of Pennsylvania on the EDVAC project, von Neumann wrote an incomplete First Draft of a Report on the EDVAC . The paper, whose premature distribution nullified the patent claims of EDVAC designers J. Presper Eckert and John Mauchly, described a computer architecture in which the data and the program are both stored in the computer's memory in the same address space. This architecture is the basis of most modern computer designs, unlike the earliest computers that were "programmed" using a separate memory device such as a paper tape or plugboard. Although the single-memory, stored program architecture is commonly called von Neumann architecture as a result of von Neumann's paper, the architecture was based on the work of Eckert and Mauchly, inventors of the ENIAC computer at the University of Pennsylvania. [290]

Von Neumann consulted for the Army's Ballistic Research Laboratory, most notably on the ENIAC project, [291] as a member of its Scientific Advisory Committee. [292] The electronics of the new ENIAC ran at one-sixth the speed, but this in no way degraded the ENIAC's performance, since it was still entirely I/O bound. Complicated programs could be developed and debugged in days rather than the weeks required for plugboarding the old ENIAC. Some of von Neumann's early computer programs have been preserved. [293]

The next computer that von Neumann designed was the IAS machine at the Institute for Advanced Study in Princeton, New Jersey. He arranged its financing, and the components were designed and built at the RCA Research Laboratory nearby. Von Neumann recommended that the IBM 701, nicknamed the defense computer, include a magnetic drum. It was a faster version of the IAS machine and formed the basis for the commercially successful IBM 704. [294] [295]

Stochastic computing was first introduced in a pioneering paper by von Neumann in 1953. [296] However, the theory could not be implemented until advances in computing of the 1960s. [297] [298] Around 1950 he was also among the first people to talk about the time complexity of computations, which eventually evolved into the field of computational complexity theory. [299]

Herman Goldstine once described how he felt that even in comparison to all his technical achievements in computer science, it was the fact that he was held in such high esteem, had such a reputation, that the digital computer was accepted so quickly and worked on by others. [300] As an example, he talked about Tom Watson, Jr.'s meetings with von Neumann at the Institute for Advanced Study, whom he had come to see after having heard of von Neumann's work and wanting to know what was happening for himself personally. IBM, which Watson Jr. later became CEO and president of, would play an enormous role in the forthcoming computer industry. The second example was that once von Neumann was elected Commissioner of the Atomic Energy Commission, he would exert great influence over the commission's laboratories to promote the use of computers and to spur competition between IBM and Sperry-Rand, which would result in the Stretch and LARC computers that lead to further developments in the field. Goldstine also notes how von Neumann's expository style when speaking about technical subjects, particularly to non-technical audiences, was very attractive. [301] This view was held not just by him but by many other mathematicians and scientists of the time too. [302]

### Cellular automata, DNA and the universal constructor

Von Neumann's rigorous mathematical analysis of the structure of self-replication (of the semiotic relationship between constructor, description and that which is constructed), preceded the discovery of the structure of DNA. [304]

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

The detailed proposal for a physical non-biological self-replicating system was first put forward in lectures von Neumann delivered in 1948 and 1949, when he first only proposed a kinematic self-reproducing automaton. [305] [306] While qualitatively sound, von Neumann was evidently dissatisfied with this model of a self-replicator due to the difficulty of analyzing it with mathematical rigor. He went on to instead develop a more abstract model self-replicator based on his original concept of cellular automata. [307]

Subsequently, the concept of the Von Neumann universal constructor based on the von Neumann cellular automaton was fleshed out in his posthumously published lectures Theory of Self Reproducing Automata. [308] Ulam and von Neumann created a method for calculating liquid motion in the 1950s. The driving concept of the method was to consider a liquid as a group of discrete units and calculate the motion of each based on its neighbors' behaviors. [309] Like Ulam's lattice network, von Neumann's cellular automata are two-dimensional, with his self-replicator implemented algorithmically. The result was a universal copier and constructor working within a cellular automaton with a small neighborhood (only those cells that touch are neighbors; for von Neumann's cellular automata, only orthogonal cells), and with 29 states per cell. Von Neumann gave an existence proof that a particular pattern would make infinite copies of itself within the given cellular universe by designing a 200,000 cell configuration that could do so.

[T]here exists a critical size below which the process of synthesis is degenerative, but above which the phenomenon of synthesis, if properly arranged, can become explosive, in other words, where syntheses of automata can proceed in such a manner that each automaton will produce other automata which are more complex and of higher potentialities than itself.

—von Neumann, 1948 [308]

Von Neumann addressed the evolutionary growth of complexity amongst his self-replicating machines. [310] His "proof-of-principle" designs showed how it is logically possible, by using a general purpose programmable ("universal") constructor, to exhibit an indefinitely large class of self-replicators, spanning a wide range of complexity, interconnected by a network of potential mutational pathways, including pathways from the most simple to the most complex. This is an important result, as prior to that it might have been conjectured that there is a fundamental logical barrier to the existence of such pathways; in which case, biological organisms, which do support such pathways, could not be "machines", as conventionally understood. Von Neumann considers the potential for conflict between his self-reproducing machines, stating that "our models lead to such conflict situations", [311] indicating it as a field of further study. [308] :147

The cybernetics movement highlighted the question of what it takes for self-reproduction to occur autonomously, and in 1952, John von Neumann designed an elaborate 2D cellular automaton that would automatically make a copy of its initial configuration of cells. The von 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. Von Neumann proved that the most effective way of performing large-scale mining operations such as mining an entire moon or asteroid belt would be by using self-replicating spacecraft, taking advantage of their exponential growth. [312]

Von Neumann investigated the question of whether modelling evolution on a digital computer could solve the complexity problem in programming. [311]

Beginning in 1949, von Neumann's design for a self-reproducing computer program is considered the world's first computer virus, and he is considered to be the theoretical father of computer virology. [313]

### Scientific computing and numerical analysis

Considered to be possibly "the most influential researcher in scientific computing of all time", [314] von Neumann made several contributions to the field, both on the technical side and on the administrative side. He was one of the key developers of the stability analysis procedure that now bears his name, [315] a scheme used to ensure that when linear partial differential equation are solved numerically, the errors at each time step of the calculation do not build up. This scheme is still the mostly commonly used technique for stability analysis today. [316] His paper with Herman Goldstine in 1947 was the first to describe backward error analysis, although only implicitly. [317] He was also among the first researchers to write about the Jacobi method. [318] During his time at Los Alamos, he was the first to consider how to solve various problems of gas dynamics numerically, writing several classified reports on the topic. However, he was frustrated by the lack of progress with analytic methods towards solving these problems, many of which were nonlinear. As a result, he turned towards computational methods in order to break the deadlock. [319] While von Neumann only occasionally worked there as a consultant, under his influence Los Alamos became the undisputed leader in computational science during the 1950s and early 1960s. [320]

From his work at Los Alamos von Neumann realized that computation was not just a tool to brute force the solution to a problem numerically, but that computation could also provide insight for solving problems analytically too, [321] through heuristic hints, and that there was an enormous variety of scientific and engineering problems towards which computers would be useful, most significant of which were nonlinear problems. [322] 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, which would defeat the current stalemate there was when trying to solve them by classical analysis methods. [323] Titled "High-speed Computing Devices and Mathematical Analysis", he also described how wind tunnels, which at the time were being constructed at heavy cost, were actually analog computers, and how digital computers, which he was developing, would replace them and dawn a new era of fluid dynamics. He was given a very warm reception, with Garrett Birkhoff describing it as "an unforgettable sales pitch". Instead of publishing this talk in the proceedings of the congress, he expanded on it with Goldstine into the manuscript "On the Principles of Large Scale Computing Machines", which he would present to the US Navy and other audiences in the hopes of drumming up their support for scientific computing using digital computers. In his papers, many in conjunction with others, he developed the concepts of inverting matrices, random matrices and automated relaxation methods for solving elliptic boundary value problems. [324]

### Weather systems and global warming

As part of his research into possible applications of computers, von Neumann became interested in weather prediction, noting the similarities between the problems in the field and previous problems he had worked on during the Manhattan Project, both of which involved nonlinear fluid dynamics. [325] In 1946 von Neumann founded the "Meteorological Project" at the Institute for Advanced Study, securing funding for his project from the Weather Bureau along with the US Air Force and US Navy weather services. [326] With Carl-Gustaf Rossby, considered the leading theoretical meteorologist at the time, he gathered a twenty strong group of metereologists who began to work on various problems in the field. However, as other postwar work took up considerable portions of his time he was not able to devote enough of it to proper leadership of the project and little was done during this time period. However this changed when a young Jule Gregory Charney took up co-leadership of the project from Rossby. [327] By 1950 von Neumann and Charney wrote the world's first climate modelling software, and used it to perform the world's first numerical weather forecasts on the ENIAC computer that von Neumann had arranged to be used; [326] von Neumann and his team published the results as Numerical Integration of the Barotropic Vorticity Equation. [328] Together they played a leading role in efforts to integrate sea-air exchanges of energy and moisture into the study of climate. [329] Though primitive, news of the ENIAC forecasts quickly spread around the world and a number of parallel projects in other locations were initiated. [330] In 1955 von Neumann, Charney and their collaborators convinced their funders to open up the Joint Numerical Weather Prediction Unit (JNWPU) in Suitland, Maryland which began routine real-time weather forecasting. [331] 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." [332] Positive results of Norman 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 in order to ensure continued support from the Weather Bureau and the military, leading to the creation of the General Circulation Research Section (now known as the Geophysical Fluid Dynamics Laboratory) next to the JNWPU in Suitland, Maryland. [333] He continued work both on technical issues of modelling and in ensuring continuing funding for these projects, which, like many others, were enormously helped by von Neumann's unwavering support to legitimize them. [334]

His research into weather systems and meteorological prediction led him to propose manipulating the environment by spreading colorants on the polar ice caps to enhance absorption of solar radiation (by reducing the albedo), [335] [336] thereby inducing global warming. [335] [336] Von Neumann proposed a theory of global warming as a result of the activity of humans, noting that the Earth was only 6 °F (3.3 °C) colder during the last glacial period, he wrote in 1955: "Carbon dioxide released into the atmosphere by industry's burning of coal 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." [337] [338] However, von Neumann urged a degree of caution in any program of intentional human weather manufacturing: "What could be done, of course, is no index to what should 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 one could carry out the necessary analyses needed to predict the results, intervene on any desired scale, and ultimately achieve rather fantastic results." [338] He also warned that weather and climate control could have military uses, telling Congress in 1956 that they could pose an even bigger risk than ICBMs. Although he died the next year, this continuous advocacy ensured that during the Cold War there would be continued interest and funding for research. [339]

"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."

—von Neumann, 1955 [338]

### Technological singularity hypothesis

The first use of the concept of a singularity in the technological context is attributed to von Neumann, [340] 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." [341] This concept was fleshed out later in the book Future Shock by Alvin Toffler.

## Defense work

### Manhattan Project

Beginning in the late 1930s, von Neumann developed an expertise in explosions—phenomena that are difficult to model mathematically. During this period, von Neumann was the leading authority of the mathematics of shaped charges. This led him to a large number of military consultancies, primarily for the Navy, which in turn led to his involvement in the Manhattan Project. The involvement included frequent trips by train to the project's secret research facilities at the Los Alamos Laboratory in a remote part of New Mexico. [53]

Von Neumann made his principal contribution to the atomic bomb in the concept and design of the explosive lenses that were needed to compress the plutonium core of the Fat Man weapon that was later dropped on Nagasaki. 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". [342]

When it turned out that there would not be enough uranium-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 the plutonium-239 that was available from the Hanford Site. [343] He established the design of the explosive lenses required, but there remained concerns about "edge effects" and imperfections in the explosives. [344] His calculations showed that implosion would work if it did not depart by more than 5% from spherical symmetry. [345] After a series of failed attempts with models, this was achieved by George Kistiakowsky, and the construction of the Trinity bomb was completed in July 1945. [346]

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. [347] [348]

Von Neumann, four other scientists, and various military personnel were included in the target selection committee that was responsible for choosing the Japanese cities of Hiroshima and Nagasaki as the first 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 and thus maximum effect. The cultural capital Kyoto, which had been spared the bombing inflicted upon militarily significant cities, was von Neumann's first choice, [349] a selection seconded by Manhattan Project leader General Leslie Groves. However, this target was dismissed by Secretary of War Henry L. Stimson. [350]

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-named Trinity. The event was conducted as a test of the implosion method device, at the bombing range near Alamogordo Army Airfield, 35 miles (56 km) southeast of Socorro, New Mexico. Based on his observation alone, von Neumann estimated the test had resulted in a blast equivalent to 5 kilotons of TNT (21  TJ ) but Enrico 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. [351] It was in von Neumann's 1944 papers that the expression "kilotons" appeared for the first time. [352] After the war, Robert Oppenheimer remarked that the physicists involved in the Manhattan project had "known sin". Von Neumann's response was that "sometimes someone confesses a sin in order to take credit for it." [353]

Von Neumann continued unperturbed in his work and became, along with Edward Teller, one of those who sustained the hydrogen bomb project. He collaborated with Klaus Fuchs on further development of the bomb, and in 1946 the two filed a secret patent on "Improvement in Methods and Means for Utilizing Nuclear Energy", which outlined a scheme for using a fission bomb to compress fusion fuel to initiate nuclear fusion. [354] The Fuchs–von Neumann patent used radiation implosion, but not in the same way as is used in what became the final hydrogen bomb design, the Teller–Ulam design. Their work was, however, incorporated into the "George" shot of Operation Greenhouse, which was instructive in testing out concepts that went into the final design. [355] The Fuchs–von Neumann work was passed on to the Soviet Union by Fuchs as part of his nuclear espionage, but it was not used in the Soviets' own, independent development of the Teller–Ulam design. The historian Jeremy 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." [355]

For his wartime services, von Neumann was awarded the Navy Distinguished Civilian Service Award in July 1946, and the Medal for Merit in October 1946. [356]

### Post war

In 1950, von Neumann became a consultant to the Weapons Systems Evaluation Group (WSEG), [357] whose function was to advise the Joint Chiefs of Staff and the United States Secretary of Defense on the development and use of new technologies. [358] He also became an adviser to the Armed Forces Special Weapons Project (AFSWP), which was responsible for the military aspects on nuclear weapons. Over the following two years, he became a consultant to the Central Intelligence Agency (CIA), a member of the influential General Advisory Committee of the Atomic Energy Commission, a consultant to the newly established Lawrence Livermore National Laboratory, and a member of the Scientific Advisory Group of the United States Air Force [357] among a host of other agencies. Beside the Coast Guard, there was not a single US military or intelligence organization which von Neumann did not advise. [359] During this time he became the "superstar" defense scientist at the Pentagon. His authority was considered infalliable at the highest levels including the secretary of defense and Joint Chiefs of Staff. [21] This applied not just to US government agencies but to private companies too, he was hired as a consultant to the RAND Corporation with the equivalent salary for an average full time analyst, yet his job was only to write down his thoughts each morning while shaving. [360]

During several meetings of the advisory board of the US Air Force von Neumann and Edward Teller predicted that by 1960 the US would be able to build a hydrogen bomb, one not only powerful but light enough too to fit on top of a rocket. In 1953 Bernard Schriever, who was present at the meeting with Teller and von Neumann, paid a personal visit to von Neumann at Princeton in order to confirm this possibility. [361] Schriever would then enlist Trevor Gardner, who in turn would also personally visit von Neumann several weeks later in order to fully understand the future possibilities before beginning his campaign for such a weapon in Washington. [362] 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 potential Soviet advancements in both these areas and in strategic defenses against American bombers into reports prepared for the Department of Defense (DoD) in order to argue for the creation of ICBMs. [363] Gardner on several occasions would bring von Neumann to the Pentagon in order to discuss with various senior officials his reports. [364] Several design decisions in these reports such as inertial guidance mechanisms would form the basis for all ICBMs thereafter. [365] By 1954 von Neumann was also regularly testifying to various Congressional military subcommittees to ensure continued support for the ICBM program, which would later expand to include senior officials from all over the US government including those from the State Department and National Security Council (NSC). [366]

However, this was not enough. In order to have the ICBM program run at full throttle they needed direct action by the President. [367] On July 28, 1955, Schriever, Gardner, and von Neumann had managed to arrange a direct meeting with President Eisenhower at the White House in order to relay their concerns. While the other two would focus on the introduction and conclusion, von Neumann would present the technical meat of the argument. White House staff had told them all three presentations could take up a maximum of half an hour and could only include "straightforward and factual" information, with no attempts to "sell" to the President their specific needs. Dillon Anderson, who was head of the NSC staff, was skeptical of the wide-ranging solutions that the trio posed as they could downgrade attention given to other defense projects. General Tommy Power, who was there with them that day, did not think there was enough time to get a subject of such importance across given the restrictions however the three thought they could compress their arguments enough to do so. At 10:00 AM their meeting was set to begin. They were to address not only President Eisenhower, but a whole host of the top civilian and military leaders of the country including Vice President Richard Nixon, Admiral Arthur Radford, chairman of the Joint Chiefs of Staff, the secretaries of State, Defense and Treasury, and the head of the CIA among others. The program officially belonged to Tommy Power as Commander in Chief of the Strategic Air Command yet he was considered a lesser figure.

Gardner began by describing the strategic consequences of ICBMs and briefly what the other two presenters would say. Von Neumann then began his speech, with no notes as he often did, speaking as the nation's preeminent scientist in matters of nuclear weaponry. He discussed technical matters, from the base nuclear engineering to the intricacies of missile targeting. Within these discussions, he once again mixed warnings that there were no known defenses against such weapons, and the fifteen minutes of warning that would be provided with the available radar system technology was all so little. One of the participants at the meeting, Vince Ford, was keeping track of the faces of all those listening to try to see if anyone was confused or lost. He saw no one, and thought that von Neumann had "knocked the ball out of the park." Now it was General Schriever's turn to speak. However, it was already 11:05 AM, and the meeting was supposed to finish five minutes before. Von Neumann had spoken for much longer than was originally planned however there was no restlessness or desire from anyone to leave; everyone was paying close attention to the speakers. Schriever spoke on how to realize the technology physically, in terms of manpower and what organizations were working on it, and the strategic plans for how to complete the project in the fastest way if it would be approved. He smartly attributed all the work being done to the recommendations of the earlier Teapot Committee that von Neumann chaired, and hence capitalized on the credibility of all the distinguished scientists that served on it too. The early restriction by Anderson also no longer mattered as much, as his proposed solutions were no longer his own but the solutions proposed in the final report of the Teapot Committee.

### Atomic Energy Commission

In 1955, von Neumann became a commissioner of the Atomic Energy Commission (AEC). He accepted this position and used it to further the production of compact hydrogen bombs suitable for intercontinental ballistic missile (ICBM) delivery. He involved himself in correcting the severe shortage of tritium and lithium 6 needed for these compact weapons, and he argued against settling for the intermediate-range missiles that the Army wanted. He was adamant that H-bombs delivered into the heart of enemy territory by an ICBM would be the most effective weapon possible, and that the relative inaccuracy of the missile wouldn't 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. [372] [373] Despite his disagreement with Oppenheimer over the need for a crash program to develop the hydrogen bomb, he testified on the latter's behalf at the 1954 Oppenheimer security hearing, at which he asserted that Oppenheimer was loyal, and praised him for his helpfulness once the program went ahead. [374]

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 in time. The SM-65 Atlas passed its first fully functional test in 1959, two years after his death. [375] The more advanced Titan rockets were deployed in 1962. Both had been proposed in the ICBM committees von Neumann chaired. [371] 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. [375] [371]

### Mutual assured destruction

Von Neumann is credited with developing the equilibrium strategy of mutual assured destruction (MAD). He also "moved heaven and earth" to bring MAD about. His goal was to quickly develop ICBMs and the compact hydrogen bombs that they could deliver to the USSR, and he knew the Soviets were doing similar work because the CIA interviewed German rocket scientists who were allowed to return to Germany, and von Neumann had planted a dozen technical people in the CIA. The Soviets considered that bombers would soon be vulnerable, and they shared von Neumann's view that an H-bomb in an ICBM was the ne plus ultra of weapons; they believed that whoever had superiority in these weapons would take over the world, without necessarily using them. [376] He was afraid of a "missile gap" and took several more steps to achieve his goal of keeping up with the Soviets:

• He modified the ENIAC by making it programmable and then wrote programs for it to do the H-bomb calculations (which further further the feasibility of the Teller-Ulam design).
• Under the aegis of the AEC he promoted the development of a compact H-bomb which could fit in an ICBM.
• He personally interceded to speed up the production of lithium-6 and tritium needed for the compact bombs.
• He caused several separate missile projects to be started, because he felt that competition combined with collaboration got the best results. [377]

Von Neumann's assessment that the Soviets had a lead in missile technology, considered pessimistic at the time, was soon proven correct in the Sputnik crisis. [378]

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 from Nazism, Fascism and Soviet Communism. [76] During a Senate committee hearing he described his political ideology as "violently anti-communist, and much more militaristic than the norm". He was quoted in 1950 remarking, "If you say why not bomb [the Soviets] tomorrow, I say, why not today? If you say today at five o'clock, I say why not one o'clock?" [379] [380]

On February 15, 1956, von Neumann was presented with the Medal of Freedom by President Dwight D. Eisenhower. His citation read:

Dr. von Neumann, in a series of scientific study projects of major national significance, has materially increased the scientific progress of this country in the armaments field. Through his work on various highly classified missions performed outside the continental limits of the United States in conjunction with critically important international programs, Dr. von Neumann has resolved some of the most difficult technical problems of national defense. [381]

Even when dying of cancer, von Neumann continued his work while he still could. Lewis Strauss, who at the time was chairman of the AEC and a close friend, described some of his final memories of von Neumann in his memoir.

Until the last, he continued to be a member of the Commission and chairman of an important advisory committee to the Defense Department. On one dramatic occasion, I was present at a meeting at Walter Reed Hospital, where, gathered around Johnny's bedside were the Secretary of Defense and his deputies, the Secretaries of the three Armed Services, and all the military Chiefs of Staff. The central figure was a young man who but a few years before had come to the United States as an immigrant. [382]

### Consultancies

A list of consultancies given by various sources is as follows: [383] [384] [385] [20] [386] [387] [388]

While his appointment as full Atomic Energy Commissioner in late 1954 formally required he sever all his other consulting contracts, [389] an exemption was made for von Neumann to continue working with several critical military committees after the Air Force and several key senators raised concerns. [371]

## Personality

Gian-Carlo Rota wrote in his famously controversial book, Indiscrete Thoughts, that von Neumann was a lonely man who had trouble relating to others except on a strictly formal level. [390] Françoise Ulam described how she never saw von Neumann in anything but a formal suit and tie. [391] His daughter wrote in her memoirs that she believed her father was motivated by two key convictions, one, that every person had the responsibility to make full use of their intellectual capacity, and two, that there is a critical importance of an environment of political freedom in order to pursue the first conviction. She added that he "enjoyed the good life, liked to live well, and counted a number of celebrities among his friends and colleagues". He was also very concerned with his legacy, in two aspects, the first being the durability of his intellectual contributions to the world, [392] and secondly the life of his daughter. His brother, Nicholas noted that John tended to take a statistical view of the world, and that characterized many of his views. [393] His encyclopedic knowledge of history did not help him in this point of view, nor did his work in game theory. He often liked to discuss the future in world events and politics and compare them with events in the past, predicting in 1936 that war would break out in Europe and that the French army was weak and would not matter in any conflict. [394] On the other hand, Stan Ulam described his warmth this way, "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". He delighted in gossip and dirty jokes. Conversations with friends on scientific topics could go on for hours without respite, never being a shortage of things to discuss, even when leaving von Neumann's specialty in mathematics. [395] He would mix in casual jokes, anecdotes and observations of people into his conversations, which allowed him to release any tension or wariness if there were disagreements, especially on questions of politics. [396] Von Neumann was not a quiet person either; he enjoyed going to and hosting parties several times a week, [397] Churchill Eisenhart recalls in an interview that von Neumann could attend parties until the early hours of the morning, then the next day right at 8:30 could be there on time and deliver clear, lucid lectures. Graduate students would try to copy von Neumann in his ways; however, they did not have any success. [398]

He was also known for always being happy to provide others with scientific and mathematical advice, [5] even when the recipient did not later credit him, which he did on many occasions with mathematicians and scientists of all ability levels. [399] [400] Wigner wrote that he perhaps supervised more work (in a casual sense) than any other modern mathematician. Collected works of colleagues at Princeton are full of references to hints or results from casual conversations with him. [401] [402] However, he did not particularly like it when he felt others were challenging him and his brilliance, being a very competitive person. [403] [404] A story went at the Aberdeen Proving Ground how a young scientist had pre-prepared a complicated expression with solutions for several cases. When von Neumann came to visit, he asked him to evaluate them, and for each case would give his already calculated answer just before Johnny did. By the time they came to the third case it was too much for Johnny and he was upset until the joker confessed. [405] Nevertheless, he would put in an effort to appear modest and did not like boasting or appearing in a self-effacing manner. [406] Towards the end of his life on one occasion his wife Klari chided him for his great self-confidence and pride in his intellectual achievements. He replied only to say that on the contrary he was full of admiration for the great wonders of nature compared to which all we do is puny and insignificant. [407]

In addition to his speed in mathematics, he was also a quick speaker, with Banesh Hoffmann noting that it made it very difficult to take notes, even in shorthand. [408] Many considered him an excellent chairman of committees, deferring rather easily on personal or organizational matters but pressing strongly 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 all Air Force long-range missile programs. [409] He also maintained his knowledge of languages he learnt in his youth, becoming somewhat of a linguist. He knew Hungarian, French, German and English fluently, and maintained at least a conversational level of Italian, Yiddish, Ancient Latin and Greek. His Spanish was less perfect, but once on a trip to Mexico he tried to create his own "neo-Castilian" mix of English and Spanish. [410]

Even from a young age he was somewhat emotionally distant, and some women felt that he was lacking curiosity in subjective and personal feelings. Despite this the person he was confided to most was his mother. [411] Ulam felt he did not devote enough time to ordinary family affairs and that in some conversations with him Johnny was shy about such topics. The fact he was constantly working on all kinds of intellectual, academic and advisery affairs probably meant he could not be a very attentive husband. This may show in the fact his personal life was not so smooth compared to his working one. Friendship wise he felt most at ease with those of similar background, third or fourth generation wealthy Jews like himself, and was quite conscious of his position in society. [412] As a child he was poor in athletics and thus did not make friends this way (but he did join in on class pranks). [413]

In general he did not disagree with people, if someone was inclined to think or do things in a certain way he would not try to contradict or dissuade them. His manner was just to go along, even when asked for advice. Ulam said he had an innocent little trick that he used where he would suggest to someone that something he [von Neumann] wanted done had in fact originated from that person in order to get them to do it. Nevertheless, he held firm on scientific matters he believed in. [414] [8]

Many people who had known von Neumann were puzzled by his relationship to the military and to power structures in general. [415] He seemed to admire generals and admirals and more generally those who wielded power in society. Ulam suspected that he had a hidden admiration for people or organizations that could influence the thoughts and decision making of others. During committee meetings he was not a particularly strong debater and as a whole preferred to avoid controversy and yield to those more forceful in their approaches. [416] When hospitalized at the end of his life Ulam told him on one occasion he was on the same floor as president Dwight Eisenhower after the president suffered a heart attack, and von Neumann was greatly amused by this. [417]

As a whole he was overwhelmingly, universally, curious. Compared to other mathematicians or scientists of the time he had a broader view of the world and more 'common sense' outside of academics. Mathematics and the sciences, history, literature, and politics were all major interests of his. In particular his knowledge of ancient history was encyclopedic and at the level of a professional historian. [418] One of the many things he enjoyed reading was the precise and wonderful way Greek historians such as Thucydides and Herodotus wrote, which he could of course read in the original language. Ulam suspected these may have shaped his views on how future events could play out and how human nature and society worked in general. [419]

### Mathematical style

Rota, in describing von Neumann's relationship with his friend Stanislaw Ulam, wrote that von Neumann had "deep-seated and recurring self-doubts". [390] As an example on one occasion he said in the future he would be forgotten while Gödel would be remembered with Pythagoras. [420] Ulam suggests that some of his self-doubts with regard for his own creativity may have come from the fact he had not himself discovered several important ideas that others had even though he was more than capable of doing so, giving the incompleteness theorems and Birkhoff's pointwise ergodic theorem as examples. Johnny had a virtuosity in following complicated reasoning and had supreme insights, yet he perhaps felt he did have the gift for seemingly irrational proofs and theorems or intuitive insights that came from nowhere. 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 Johnny was not convinced of the importance of his work, and only when finding some ingenious technical trick or new approach that he took some pleasure from his work that satiated his concerns. [421] However, according to Rota, von Neumann still had an "incomparably stronger technique" compared to his friend, despite describing Ulam as the more creative mathematician. [390] Ulam, in his obituary of von Neumann, described how he was adept in dimensional estimates and did algebraical or numerical computations in his head without the need for pencil and paper, often impressing physicists who needed the help of physical utensils. His impression of the way von Neumann thought was that he did not visualise things physically, instead he thought abstractly, treated properties of objects as some logical consequence of an underlying fundamental physical assumption. [422] Albert Tucker described von Neumann's overall interest in things as problem oriented, not even that, but as he "would deal with the point that came up as a thing by itself." [423]

Herman Goldstine compared his lectures to being on glass, smooth and lucid. You would sit down and listen to them and not even feel the need to write down notes because everything was so clear and obvious, however once one would come home and try understand the subject, you would suddenly realize it was not so easy. By comparison, Goldstine thought his scientific articles were written in a much harsher manner, and with much less insight. [399] Another person who attended his lectures, Albert Tucker, described his lecturing as "terribly quick" and said that people often had to ask von Neumann questions in order to slow him down so they could think through the ideas he was going through, even if his presentation was clear they would still be thinking of the previous idea when von Neumann moved on to the next one. Von Neumann knew about this and was grateful for the assistance of his audience in telling him when he was going too quickly. [423] Halmos described his lectures as "dazzling", with his speech clear, rapid, precise and all encompassing. He would cover all approaches to the subject he was speaking on and relate them to each other. Like Goldstine, he also described how everything seemed "so easy and natural" in lectures and a puzzled feeling once one tried to think over it at home. [424]

His work habits were rather methodical, after waking up and having breakfast at the Nassau Club, he would visit the Institute for Advanced Study and begin work for the day. He would continue working for the entirety of the day, including after going home at five. Even if he was entertaining guests or hosting a party he could still spend some time in his work room working away, still following the conversation in the other room where guests were. Although he went to bed at a reasonable time he would awaken late in the night, two or three in the morning by which time his brain had thought through problems he had in the previous day and begin working again and writing things down. He placed great importance on writing down ideas he had in detail, [425] [426] and if he had a new one he would sometimes drop what he was doing to write them down. [427]

Von Neumann was asked to write an essay for the layman describing what mathematics is. He explained that mathematics straddles the world between the empirical and logical, arguing that geometry was originally empirical, but Euclid constructed a logical, deductive theory. However, he argued that there is always the danger of straying too far from the real world and becoming irrelevant sophistry. [431] [432] [433]

Although he was commonly described as an analyst, he once classified himself an algebraist, [434] and his style often displayed a mix of algebraic technique and set-theoretical intuition. [435] 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, ${\displaystyle \phi (x)}$ to ${\displaystyle \phi ((x))}$. However, this process ended up being repeated several times, where the final result were equations such as ${\displaystyle (\psi ((((a)))))^{2}=\phi ((((a))))}$. The 1936 paper became known to students as "von Neumann's onion" [436] 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. Von Neumann always saw the bigger picture and the trees never concealed the forest for him. [10] Although powerful technically his primary concern seemed to be more with the clear and viable formation of fundamental issues and questions of science rather than just the solution of mathematical puzzles. [437]

At times he could be ignorant of the standard mathematical literature, it would at times be easier to rederive basic information he needed rather than chase references. He did not 'write down' to a specific audience, but rather he wrote it exactly as he saw it. Although he did spend time preparing for lectures, often it was just before he was to present them, and he rarely used notes, instead jotting down points of what he would discuss and how long he would spend on it. [424]

After World War II began, he increasingly became extremely busy with a multitude of both academic and military commitments. He already had a bad habit of not writing up talks or publishing results he found, [438] which only worsened. Another factor was that he did not find it easy to discuss a topic formally in writing to others unless it was already mature in his mind. If it was, he could talk freely and without hesitation, but if it was not, he would, in his own words, "develop the worst traits of pedantism and inefficiency". [439]

## Recognition

### Cognitive abilities

Nobel Laureate Hans Bethe said "I have sometimes wondered whether a brain like von Neumann's does not indicate a species superior to that of man", [41] and later Bethe wrote that "[von Neumann's] brain indicated a new species, an evolution beyond man". [440] Paul Halmos states that "von Neumann's speed was awe-inspiring." [405] Israel Halperin said: "Keeping up with him was ... impossible. The feeling was you were on a tricycle chasing a racing car." [441] Edward Teller admitted that he "never could keep up with him". [442] Teller also said "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." [443] Peter Lax wrote "Von Neumann was addicted to thinking, and in particular to thinking about mathematics". [438] Claude Shannon called him "the smartest person I’ve ever met", a common opinion. [444]

When George 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 unconceived theory of duality. [445]

Lothar Wolfgang Nordheim described von Neumann as the "fastest mind I ever met", [446] and Jacob Bronowski wrote "He was the cleverest man I ever knew, without exception. He was a genius." [447] George Pólya, whose lectures at ETH 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." [448] Enrico Fermi told physicist Herbert 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!" [449]

Eugene Wigner described him in this way: "I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Werner Heisenberg. Paul Dirac was my brother-in-Law; Leo Szilard and Edward Teller have been among my closest friends; and Albert 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. You saw immediately the quickness and power of von Neumann's mind. He understood mathematical problems not only in their initial aspect, but in their full complexity. Swiftly, effortlessly, he delved deeply into the details of the most complex scientific problem. He retained it all. His mind seemed a perfect instrument, with gears machined to mesh accurately to one thousandth of an inch." [450]

Halmos recounts a story told by Nicholas Metropolis, concerning the speed of von Neumann's calculations, when somebody asked von Neumann to solve the famous fly puzzle: [451]

Two bicyclists start 20 miles apart and head toward each other, each going at a steady rate of 10 mph. At the same time a fly that travels at a steady 15 mph starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover? The slow way to find the answer is to calculate what distance the fly covers on the first, southbound, leg of the trip, then on the second, northbound, leg, then on the third, etc., etc., and, finally, to sum the infinite series so obtained.

The quick way is to observe that the bicycles meet exactly one hour after their start, so that the fly had just an hour for his travels; the answer must therefore be 15 miles.

When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick?" asked von Neumann, "All I did was sum the geometric series." [452]

Wigner told a similar story, only with a swallow instead of a fly, and says it was Max Born who posed the question to von Neumann in the 1920s. [453]

Similarly, when the first computers he was helping develop were completed, simple tests like "what is the lowest power of 2 that has the number 7 in the fourth position from the end?" were conducted to ensure their accuracy. For modern computers this would take only a fraction of a second but for the first computers Johnny would race against them in calculation, and win. [405]

Accolades and anecdotes were not limited to those from the physical or mathematical sciences either, neurophysiologist Leon Harmon, described him in a similar manner, "Von Neumann was a true genius, the only one I've ever known. I've met Einstein and Oppenheimer and Teller and—who's the mad genius from MIT? I don't mean McCulloch, but a mathematician. Anyway, a whole bunch of those other guys. Von Neumann was the only genius I ever met. The others were supersmart .... And great prima donnas. But von Neumann's mind was all-encompassing. He could solve problems in any domain. ... And his mind was always working, always restless." [454] US President Dwight D. Eisenhower considered him "the outstanding mathematician of the time". [455] 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, industrialists and scientists that no other scientist could match. In nuclear missilery he was considered "the clearly dominant advisory figure" according to Herbert York whose opinions "everyone took very seriously". [456]

Even for writer Arthur Koestler, who was not an academic, von Neumann was "one of the few people for whom Koestler entertained not only respect but reverence, and he shared Koestler's Central European addiction to abstruse philosophical discussions, political debate, and dirty jokes. The two of them derived considerable pleasure from discussing the state of American civilization (was it in crisis or simply at the stage of adolescence?), the likely future of Europe (would there be war?), free will versus determinism, and the definition of pregnancy (“the uterus taking seriously what was pointed at it in fun”)." [457]

He is often given as an example that mathematicians could do great work in the physical sciences too, however R. D. Richtmyer describes how during von Neumann's time at Los Alamos he functioned not as a mathematician applying his art to physics problems, but rather entirely as a physicist in the mind and thought (except faster). He describes him as a first-rate physicist who knew quantum mechanics, atomic, molecular, and nuclear physics, particle physics, astrophysics, relativity, and physical and organic chemistry. As such any mathematician who does not possess the same talent as von Neumann should not be fooled into thinking physics is easy just because they study mathematics. [458]

### Eidetic memory

Von Neumann was also noted for his eidetic memory, particularly of the symbolic kind. Herman Goldstine writes:

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 how A 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. [459]

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. [41] [460] In his autobiography Stanislaw Ulam writes that Johnny's memory was auditory rather than visual. He did not have to any extent an intuitive 'common sense' for guessing what may happen in a given physical situation. [461]

## Legacy

"It seems fair to say that if the influence of a scientist is interpreted broadly enough to include impact on fields beyond science proper, then John von Neumann was probably the most influential mathematician who ever lived," wrote Miklós Rédei in John von Neumann: Selected Letters. [462] James Glimm wrote: "he is regarded as one of the giants of modern mathematics". [463] The mathematician Jean Dieudonné said that von Neumann "may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions", [14] while Peter Lax described him as possessing the "most scintillating intellect of this century". [464] In the foreword of Miklós Rédei's Selected Letters, Peter Lax wrote, "To gain a measure of von Neumann's achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honored by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a 3+12-fold winner, for his work in physics, in particular, quantum mechanics". [465] Rota writes that "he was the first to have a vision of the boundless possibilities of computing, and he had the resolve to gather the considerable intellectual and engineering resources that led to the construction of the first large computer" and consequently that "No other mathematician in this century has had as deep and lasting an influence on the course of civilization." [466] He believed in the power of mathematical reasoning to influence modern civilization, an idea which expressed itself through his life work. He is widely regarded as one of the greatest and most influential mathematicians and scientists of the 20th century. [463] [467] [468] [302] [469] [470] [471]

### Mastery of mathematics

Stan Ulam, who knew von Neumann well, described his mastery of mathematics this way: "Most mathematicians know one method. For example, Norbert Wiener had mastered Fourier transforms. Some mathematicians have mastered two methods and might really impress someone who knows only one of them. John von Neumann had mastered three methods." He went on to explain that the three methods were:

1. A facility with the symbolic manipulation of linear operators;
2. An intuitive feeling for the logical structure of any new mathematical theory;
3. An intuitive feeling for the combinatorial superstructure of new theories. [472]

As an example of the last point Eugene Wigner described how once he did not understand a mathematical theorem and asked von Neumann for help. Von Neumann would ask Wigner whether he knew several other different but related theorems and then he would then explain the problematic theorem based off what Wigner already knew. Using such circular paths he could make even the most difficult concepts easy. [473] On another occasion he wrote, "Nobody knows all science, not even von Neumann did. But as for mathematics, he advanced every part of it except number theory and topology. That is, I think, something unique." [474] Likewise Halmos noted that while von Neumann knew lots of mathematics, the most notable gaps were in algebraic topology and number theory, describing a story of how von Neumann once was walking by and saw something on the blackboard he didn't understand. Upon asking Halmos told him it was just the usual identification for a torus. While elementary even for modern graduate students this kind of work never crossed his path and thus he did not know it. [475]

One time he admitted to Herman Goldstine that he had no facility at all in topology and he was never comfortable with it, with Goldstine later bringing this up when comparing him to Hermann Weyl, whom he thought was deeper and broader than von Neumann. [430] Similarly Albert Tucker said he never saw von Neumann work on anything he would call "topological" and described how once von Neumann was giving a proof of a topological theorem, which he thought, while ingenious, was the kind of proof an analyst would give rather than someone who worked on combinatorial topology. [423]

Towards the end of his life he deplored to Ulam the fact that it no longer felt possible for anyone to have more than passing knowledge of one-third of the field of pure mathematics. [476] In fact in the early 1940s Ulam himself concocted for him at his suggestion a doctoral style examination in various fields in order to find weaknesses in his knowledge. He did find them, with von Neumann being unable to answer satisfactorily a question each in differential geometry, number theory, and algebra. "This may also tend to show that doctoral exams have little permanent meaning" was their conclusion. However while 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 Johnny could have aspired to do so. [477]

In his biography of von Neumann, Salomon Bochner describes how much of von Neumann's works in pure mathematics involved finite and infinite dimensional vector spaces in one way or another, which at the time, covered much of the total area of mathematics. However he pointed out this still did not cover an important part of the mathematical landscape, in particular, anything that involved geometry "in the global sense", topics such as topology, differential geometry and harmonic integrals, algebraic geometry and other such fields. In these fields he said von Neumann worked on rarely, and had very little affinity for it in his thinking. [139]

Likewise Jean Dieudonné noted in his biographical article that while he had an encyclopedic background, his range in pure mathematics was not as wide as Poincaré, Hilbert or even Weyl. 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 a priori that previously seemed to have little connection with mathematics. His style in analysis was not of the traditional English or French schools but rather of the German one, where analysis is based extensively on foundations in linear algebra and general topology. As with Bochner, he noted von Neumann never did significant work in number theory, algebraic topology, algebraic geometry or differential geometry. However, for his limits in pure mathematics he made up for in applied mathematics, where his work certainly equalled that of legendary mathematicians such as Gauss, Cauchy or Poincaré. Dieudonné notes that during the 1930s when von Neumann's work in pure mathematics was at its peak, there was hardly an important area he didn't have at least passing acquaintance with. [126]

### Honors and awards

A list of the following awards and honors was drawn from various biographic statements given by von Neumann. [499] [500] [385]

Awards:

Co-Editorship:

Honorary societies:

Honorary doctorates:

Honorary positions:

Society memberships:

## Selected works

Collections of von Neumann's published works can be found on zbMATH and Google Scholar. A list of his known works as of 1995 can be found in The Neumann Compendium.

### Scholarly articles

• 1947. The Mathematician, The Works of the Mind. ed. by R. B. Heywood, University of Chicago Press, 180–196.
• 1951. The Future of High-Speed Computing, Digest of an address at the IBM Seminar on Scientific Computation, November 1949, Proc. Comp. Sem., IBM, 13.
• 1954. The Role of Mathematics in the Sciences and in Society. Address at 4th Conference of Association of Princeton Graduate Alumni, June, 16–29.
• 1954. The NORC and Problems in High Speed Computing, Address on the occasion of the first public showing of the IBM Naval Ordnance Research Calculator, December 2.
• 1955. Method in the Physical Sciences, The Unity of Knowledge, ed. by L. Leary, Doubleday, 157–164.
• 1955. Can We Survive Technology? Fortune, June.
• 1955. Impact of Atomic Energy on the Physical and Chemical Sciences, Speech at M.I.T. Alumni Day Symposium, June 13, Summary, Tech. Rev. 15–17.
• 1955. Defense in Atomic War, Paper delivered at a symposium in honor of Dr. R. H. Kent, December 7, 1955, The Scientific Bases of Weapons, Journ. Am. Ordnance Assoc., 21–23.
• 1956. The Impact of Recent Developments in Science on the Economy and on Economics, Partial text of a talk at the National Planning Assoc., Washington, D.C., December 12, 1955, Looking Ahead, 4:11.

### Collected works

• 1963. John von Neumann Collected Works (6 Volume Set), Taub, A. H., editor, Pergamon Press Ltd. ISBN   9780080095660
• 1961. Volume I: Logic, Theory of Sets and Quantum Mechanics
• 1961. Volume II: Operators, Ergodic Theory and Almost Periodic Functions in a Group
• 1961. Volume III: Rings of Operators
• 1962. Volume IV: Continuous Geometry and other topics
• 1963. Volume V: Design of Computers, Theory of Automata and Numerical Analysis
• 1963. Volume VI: Theory of Games, Astrophysics, Hydrodynamics and Meteorology

## Notes

1. "Neumann de Margitta Miksa a Magyar Jelzálog-Hitelbank igazgatója n:Kann Margit gy:János-Lajos, Mihály-József, Miklós-Ágost | Libri Regii | Hungaricana". archives.hungaricana.hu (in Hungarian). Retrieved August 8, 2022.
2. Dyson 2012, p. 48.
3. Israel & Gasca 2009, p. 14.
4. Goldstine 1980, p. 169.
5. Halperin 1990, p. 16.
6. While Israel Halperin's thesis advisor is often listed as Salomon 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." [5]
7. John von Neumann at the Mathematics Genealogy Project. Retrieved March 17, 2015.
8. Szanton 1992, p. 130.
9. Dempster, M. A. H. (February 2011). "Benoit B. Mandelbrot (1924–2010): a father of Quantitative Finance" (PDF). Quantitative Finance. 11 (2): 155–156. doi:10.1080/14697688.2011.552332. S2CID   154802171.
10. Halmos 1973, p. 388.
11. Albers & Alexanderson 2008, p. 169.
12. Byrne 2010, pp. 56, 73.
13. Rédei 1999, p. 7.
14. Dieudonné 2008, p. 90.
15. Burke 2020, pp. 157.
16. Doran & Kadison 2004, p. 8.
17. Zund, Joseph D. (2002). "George David Birkhoff and John von Neumann: a question of priority and the ergodic theorems, 1931–1932". Historia Mathematica. 29 (2): 138–156. doi:10.1006/hmat.2001.2338. MR   1896971. See page 151.
18. Aspray 1990, p. 246.
19. Jacobsen 2015, p. 40.
20. Andrew Hill (December 14, 2015). "Person of the Year: Past winners". Financial Times . Archived from the original on September 18, 2019. Retrieved April 23, 2022.
21. Peter Martin (December 24, 1999). "VON NEUMANN: Architect of the computer age". Financial Times . Archived from the original on September 19, 2015. Retrieved April 23, 2022.
22. "Every FT Person of the Year since 1970". Financial Times . December 12, 2017. Archived from the original on December 12, 2021. Retrieved April 23, 2022.
23. Bhattacharya 2022, p. 4.
24. Doran & Kadison 2004, p. 1.
25. Myhrvold, Nathan (March 21, 1999). "John von Neumann". Time . Archived from the original on February 11, 2001.
26. Blair 1957, p. 104.
27. Dyson 1998, p. xxi.
28. Macrae 1992, pp. 38–42.
29. Macrae 1992, pp. 37–38.
30. Macrae 1992, p. 39.
31. Macrae 1992, pp. 44–45.
32. Macrae 1992, pp. 57–58.
33. Henderson 2007, p. 30.
34. Mitchell 2009, p. 124.
35. Macrae 1992, pp. 46–47.
36. Halmos 1973, p. 383.
37. Blair 1957, p. 90.
38. Macrae 1992, p. 52.
39. Macrae 1992, pp. 70–71.
40. Doran & Kadison 2004, p. 3.
41. Macrae 1992, pp. 32–33.
42. Lax 1990, p. 5.
43. Nasar 2001, p. 81.
44. Macrae 1992, p. 84.
45. von Kármán, T., & Edson, L. (1967). The wind and beyond. Little, Brown & Company.
46. Macrae 1992, pp. 85–87.
47. Macrae 1992, p. 97.
48. Regis, Ed (November 8, 1992). "Johnny Jiggles the Planet". The New York Times . Retrieved February 4, 2008.
49. von Neumann, J. (1928). "Die Axiomatisierung der Mengenlehre". Mathematische Zeitschrift (in German). 27 (1): 669–752. doi:10.1007/BF01171122. ISSN   0025-5874. S2CID   123492324.
50. Macrae 1992, pp. 86–87.
51. The Collected Works of Eugene Paul Wigner: Historical, Philosophical, and Socio-Political Papers. Historical and Biographical Reflections and Syntheses, By Eugene Paul Wigner, (Springer 2013), page 128
52. Macrae 1992, pp. 98–99.
53. The History Of Game Theory, Volume 1: From the Beginnings to 1945, By Mary-Ann Dimand, Robert W Dimand, (Routledge, 2002), page 129
54. Macrae 1992, p. 145.
55. Macrae 1992, pp. 143–144.
56. Macrae 1992, pp. 155–157.
57. "Marina Whitman". The Gerald R. Ford School of Public Policy at the University of Michigan. July 18, 2014. Retrieved January 5, 2015.
58. "Princeton Professor Divorced by Wife Here". Nevada State Journal. November 3, 1937.
59. Heims 1980, p. 178.
60. Macrae 1992, pp. 170–174.
61. Bochner 1958, p. 446.
62. Macrae 1992, pp. 43, 157.
63. Macrae 1992, pp. 167–168.
64. Macrae 1992, p. 371.
65. Macrae 1992, pp. 195–196.
66. Macrae 1992, pp. 190–195.
67. Ulam 1976, p. 70.
68. Macrae 1992, pp. 170–171.
69. Regis 1987, p. 103.
70. "Conversation with Marina Whitman". Gray Watson (256.com). Archived from the original on April 28, 2011. Retrieved January 30, 2011.
71. Poundstone, William (May 4, 2012). "Unleashing the Power". The New York Times .
72. Blair 1957, p. 93.
73. Ulam 1958, p. 6.
74. Ulam 1976, pp. 100–101.
75. Macrae 1992, p. 150.
76. Macrae 1992, p. 48.
77. Blair 1957, p. 94.
78. Stern, Nancy (January 20, 1981). "An Interview with Cuthbert C. Hurd" (PDF). Charles Babbage Institute, University of Minnesota. Retrieved June 3, 2010.
79. Rota 1989, pp. 26–27.
80. Macrae 1992, p. 75.
81. Pais, Abraham (1994). Einstein Lived Here. Oxford University Press. ISBN   0-19-853994-0.
82. While there is a general agreement that the initially discovered bone tumour was a secondary growth, sources differ as to the location of the primary cancer. While Macrae gives it as pancreatic, the Life magazine article says it was prostate. Sheehan's book gives it as testicular.
83. Veisdal, Jørgen (November 11, 2019). "The Unparalleled Genius of John von Neumann". Medium. Retrieved November 19, 2019.
84. Jacobsen 2015, p. 62.
85. Read, Colin (2012). The Portfolio Theorists: von Neumann, Savage, Arrow and Markowitz. Great Minds in Finance. Palgrave Macmillan. p. 65. ISBN   978-0230274143 . Retrieved September 29, 2017. 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.
86. Halmos 1973, p. 394.
87. Jacobsen 2015, p. 63.
88. Macrae 1992 , p. 379"
89. Dransfield & Dransfield 2003 , p. 124 "He was brought up in a Hungary in which anti-Semitism was commonplace, but the family were not overly religious, and for most of his adult years von Neumann held agnostic beliefs."
90. Ayoub 2004 , p. 170 "On the other hand, von Neumann, giving in to Pascal's wager on his death bed, received extreme unction."
91. Pais 2006 , p. 109 "He had been completely agnostic for as long as I had known him. As far as I could see this act did not agree with the attitudes and thoughts he had harbored for nearly all his life."
92. Poundstone 1993, p. 194.
93. Macrae 1992, p. 380.
94. Ulam 1976, pp. 242–243.
95. Van Heijenoort 1967, pp. 393–394.
96. Macrae 1992, pp. 104–105.
97. Murawski, Roman (2010). "John Von Neumann And Hilbert's School". Essays In The Philosophy And History Of Logic And Mathematics. Poznań Studies in the Philosophy of the Sciences and the Humanities. p. 196. ISBN   978-90-420-3091-6.
98. von Neumann, J. (1929), "Zur allgemeinen Theorie des Masses" (PDF), Fundamenta Mathematicae , 13: 73–116, doi:
99. van der Waerden 1975, p. 34.
100. Neumann, J. v. (1927). "Zur Hilbertschen Beweistheorie". Mathematische Zeitschrift (in German). 24: 1–46. doi:10.1007/BF01475439. S2CID   122617390.
101. Murawski, Roman (2010). "John Von Neumann And Hilbert's School". Essays In The Philosophy And History Of Logic And Mathematics. Poznań Studies in the Philosophy of the Sciences and the Humanities. pp. 204–206. ISBN   978-90-420-3091-6.
102. von Neumann 2005, p. 123.
103. Dawson 1997, p. 70.
104. von Neumann 2005, p. 124.
105. Macrae 1992, p. 182.
106. von Plato, Jan (2018). "In search of the sources of incompleteness" (PDF). Proceedings of the International Congress of Mathematicians 2018. 3: 4080. doi:10.1142/9789813272880_0212. ISBN   978-981-327-287-3. S2CID   203463751.
107. von Plato, Jan (2020). Can Mathematics Be Proved Consistent?. Springer International Publishing. p. 18. ISBN   978-3-030-50876-0.
108. Murawski, Roman (2010). "John Von Neumann And Hilbert's School". Essays In The Philosophy And History Of Logic And Mathematics. Poznań Studies in the Philosophy of the Sciences and the Humanities. p. 209. ISBN   978-90-420-3091-6.
109. Two of the papers are:
Hopf, Eberhard (1939). "Statistik der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung". Leipzig Ber. Verhandl. Sächs. Akad. Wiss. (in German). 91: 261–304.
110. Halmos 1958, p. 93.
111. Halmos 1958, p. 91.
112. Mackey 1990, p. 27.
113. Mackey 1990, pp. 28–30.
114. Ornstein 1990, p. 39.
115. Halmos 1958, p. 86.
116. Halmos 1958, p. 87.
117. Pietsch 2007, p. 168.
118. Halmos 1958, p. 88.
119. Dieudonné, Jean. "Von Neumann, Johann (or John)". Encyclopedia.com. Complete Dictionary of Scientific Biography. Retrieved August 7, 2021.
120. Ionescu-Tulcea, Alexandra; Ionescu-Tulcea, Cassius (1969). Topics in the Theory of Lifting. Springer-Verlag Berlin Heidelberg. p. V. ISBN   978-3-642-88509-9.
121. Halmos 1958, p. 89.
122. Neumann, J. v. (1940). "On Rings of Operators. III". Annals of Mathematics. 41 (1): 94–161. doi:10.2307/1968823. JSTOR   1968823.
123. Halmos 1958, p. 90.
124. Neumann, John von (1950). Functional Operators, Volume 1: Measures and Integrals. Princeton University Press. ISBN   9780691079660.
125. von Neumann, John (1999). Invariant Measures. American Mathematical Society. ISBN   978-0-8218-0912-9.
126. 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. JSTOR   1989792.
127. 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. JSTOR   1989694.
128. "AMS Bôcher Prize". AMS. January 5, 2016. Retrieved January 12, 2018.
129. Bochner 1958, p. 440.
130. von Neumann, J. (1933). "Die Einfuhrung Analytischer Parameter in Topologischen Gruppen". Annals of Mathematics . 2 (in German). 34 (1): 170–190. doi:10.2307/1968347. JSTOR   1968347.
131. 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. S2CID   122565679.
132. Bochner 1958, p. 441.
133. Pietsch 2007, p. 11.
134. Dieudonné 1981, p. 172.
135. Pietsch 2007, p. 14.
136. Dieudonné 1981, pp. 211, 218.
137. Pietsch 2007, pp. 58, 65–66.
138. Steen 1973, p. 370.
139. Pietsch 2014, p. 54.
140. Dieudonné 1981, pp. 175–176, 178–179, 181, 183.
141. Steen 1973, p. 373.
142. Pietsch 2007, p. 202.
143. Kar, Purushottam; Karnick, Harish (2013). "On Translation Invariant Kernels and Screw Functions". p. 2. arXiv: [math.FA].
144. 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:. doi:10.1007/s11118-007-9070-4. S2CID   15895847.
145. Horn & Johnson 2013, p. 320.
146. Horn & Johnson 2013, p. 458.
147. Horn & Johnson 2013, p. 335.
148. Horn & Johnson 1991, p. 139.
149. Bhatia, Rajendra (1997). Matrix Analysis. Graduate Texts in Mathematics. Vol. 169. New York: Springer. p. 109. doi:10.1007/978-1-4612-0653-8. ISBN   978-1-4612-0653-8.
150. Lord, Sukochev & Zanin 2021, p. 73.
151. Prochnoa, Joscha; Strzelecki, Michał (2022). "Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings". Journal of Approximation Theory. 277: 105736. arXiv:. doi:10.1016/j.jat.2022.105736. S2CID   232335769.
152. "Nuclear operator". Encyclopedia of Mathematics. Archived from the original on June 23, 2021. Retrieved August 7, 2022.
153. Pietsch 2007, p. 372.
154. Lord, Sukochev & Zanin 2012, p. 73.
155. Lord, Sukochev & Zanin 2021, p. 26.
156. Pietsch 2007, p. 272.
157. Pietsch 2007, pp. 272, 338.
158. Pietsch 2007, p. 140.
159. Murray 1990, pp. 57–59.
160. Petz & Rédei 1995, pp. 163–181.
161. "Von Neumann Algebras" (PDF). Princeton University. Retrieved January 6, 2016.
162. Pietsch 2007, pp. 151.
163. Kadison 1990, pp. 70, 86.
164. Pietsch 2007, p. 146.
165. "Direct Integrals of Hilbert Spaces and von Neumann Algebras" (PDF). University of California at Los Angeles. Archived from the original (PDF) on July 2, 2015. Retrieved January 6, 2016.
166. Kadison 1990, pp. 65, 71, 74.
167. Pietsch 2007, p. 148.
168. Birkhoff 1958, p. 50.
169. Lashkhi 1995, p. 1044.
170. Macrae 1992, p. 140.
171. von Neumann, John (1930). "Zur Algebra der Funktionaloperationen und Theorie der normalen Operatoren". Mathematische Annalen (in German). 102 (1): 370–427. Bibcode:1930MatAn.102..685E. doi:10.1007/BF01782352. S2CID   121141866.. The original paper on von Neumann algebras.
172. Birkhoff 1958, pp. 50–51.
173. Birkhoff 1958, p. 51.
174. Lashkhi 1995, pp. 1045–1046.
175. Wehrung 2006, p. 1.
176. Birkhoff 1958, p. 52.
177. Goodearl 1979, p. ix.
178. Birkhoff 1958, pp. 52–53.
179. Birkhoff 1958, pp. 55–56.
180. von Neumann, John (1941). "Distribution of the ratio of the mean square successive difference to the variance". Annals of Mathematical Statistics . 12 (4): 367–395. doi:. JSTOR   2235951.
181. Durbin, J.; Watson, G. S. (1950). "Testing for Serial Correlation in Least Squares Regression, I". Biometrika . 37 (3–4): 409–428. doi:10.2307/2332391. JSTOR   2332391. PMID   14801065.
182. 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. JSTOR   1912252.
183. Rédei 1959, p. 226.
184. von Neumann, J. (1925). "Egyenletesen sürü szämsorozatok (Gleichmässig dichte Zahlenfolgen)". Mat. Fiz. Lapok. 32: 32–40.