John von Neumann

How to use this course

John von Neumann worked in more fields than almost any mathematician of the twentieth century. He began in the foundations of set theory, built the mathematical framework of quantum mechanics, created the theory of operator algebras, proved the first ergodic theorem, founded game theory, and wrote the design document from which most computers descend. In his last decade he turned to the logic of self-reproducing machines. He also spent much of his time as a consultant on weapons and as a government adviser.

This course follows him through each area in turn. The technical lessons are independent of each other, so you can read them in any order.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Readers who want only the life story can read Parts I and III and the opening paragraph of each lesson in Part II.

Timeline

Budapest, 1903–1921

Neumann János Lajos was born in Budapest on 28 December 1903, the eldest of three sons of Max Neumann, a lawyer turned banker, and Margit Kann. The family was Jewish, prosperous and not strictly observant. In 1913 the emperor Franz Joseph ennobled Max Neumann for services to the economy, and the family gained the predicate margittai. In German the son became Johann von Neumann, and in America, Johnny.

Budapest before the First World War was a rich, fast-growing city with an exceptional school system. The household employed governesses who taught the boys German and French, and stories of the young János dividing eight-digit numbers in his head were told for decades. Many of these stories cannot be checked. What is documented is the education he received.

The Lutheran gymnasium

He attended the Lutheran gymnasium in the Fasor district, one of the strongest schools in Central Europe. Eugene Wigner, a year ahead of him, became a lifelong friend. The mathematics teacher László Rátz saw his talent quickly and arranged university tutoring. Gábor Szegő taught him calculus as a teenager, and Michael Fekete became his regular tutor. Their joint paper on the zeros of certain minimal polynomials appeared in 1922, when von Neumann was eighteen. He won the Eötvös Prize, the national competition, as he finished school.

These were also the years of war, revolution and counter-revolution. In 1919 the short-lived Hungarian Soviet Republic frightened the family, who left Budapest for a time. The experience left him with a lasting hostility to communism that shaped his politics thirty years later.

Berlin, Zürich, Göttingen, 1921–1927

His father wanted a practical profession. The compromise was a double education. In 1921 von Neumann enrolled in mathematics at the University of Budapest, but he hardly attended; he sat the examinations. At the same time he studied chemistry in Berlin from 1921 to 1923, then passed the entrance examination of the Eidgenössische Technische Hochschule in Zürich in 1923. He graduated as a chemical engineer in 1926.

In Zürich he came to know Hermann Weyl and George Pólya. His real work was in set theory. In 1923 he published his definition of the ordinal numbers, and in 1925 an axiomatization of set theory that introduced classes alongside sets (Lesson 04). The Budapest doctorate, under Lipót Fejér, was awarded in 1926 for a thesis on the axiomatic construction of general set theory. The expanded German version appeared in Mathematische Zeitschrift in 1928.

Göttingen

A Rockefeller fellowship took him to Göttingen in 1926–27 to work with David Hilbert. Two of Hilbert’s projects absorbed him. The first was proof theory: Hilbert hoped to prove the consistency of arithmetic by finitary means, and von Neumann wrote a paper on the problem in 1927. The second was the new quantum mechanics. Werner Heisenberg’s matrix mechanics and Erwin Schrödinger’s wave mechanics had appeared within months of each other, and their equivalence was unclear. Hilbert lectured on the subject in 1926–27. With Hilbert and Lothar Nordheim, von Neumann wrote a first paper on the foundations, and then alone, in 1927, three papers that recast the theory in terms of Hilbert space (Lesson 05).

Privatdozent to Princeton, 1927–1933

Von Neumann completed his habilitation in Berlin on 13 December 1927 and began lecturing there in 1928, the youngest Privatdozent the university had appointed. In 1929 he moved to Hamburg. These were his most productive years in pure mathematics: the minimax theorem (1928), the definition of transfinite recursion (1928), the spectral theory of unbounded operators and the double commutant theorem (1929), and the notion of an amenable group (1929).

Germany had few chairs and many candidates. In 1929 Oswald Veblen invited him to Princeton, and he began lecturing there as a visiting lecturer in 1930, with Wigner on a similar appointment. Before leaving he married Mariette Kövesi in Budapest. He became a professor at Princeton University in 1931.

Königsberg, 1930

In September 1930, at a conference in Königsberg, Kurt Gödel announced his first incompleteness theorem. Von Neumann grasped it at once. Within weeks he wrote to Gödel that it implied a second result: a consistent system containing arithmetic cannot prove its own consistency. Gödel replied that he had already found this. Von Neumann acknowledged the priority and stopped working on the foundations of mathematics. Hilbert’s programme, in its original form, was finished.

The Institute for Advanced Study

In 1933 the new Institute for Advanced Study in Princeton appointed its first mathematics professors. Von Neumann, not yet thirty, joined Albert Einstein, Veblen, James Alexander and Weyl, and Marston Morse followed in 1935. He held the post for the rest of his life. In the same year the Nazi government began dismissing Jewish academics, and von Neumann resigned his German positions. He became a US citizen in 1937. His daughter Marina was born in 1935. The marriage ended in 1937, and in 1938 he married Klára Dán, also from Budapest.

Ordinals, axioms, paradoxes

For the general reader. Georg Cantor counted past infinity: after 0, 1, 2, … comes \(\omega\), then \(\omega+1\), and so on. But what is the number \(\omega\)? Von Neumann, aged nineteen, gave the answer now in every textbook: each ordinal number is simply the set of all smaller ordinals. He then wrote axioms that let set theory talk about collections too large to be sets without falling into paradox.

Ordinals (1923)

Set \(0=\varnothing\) and \(\alpha+1=\alpha\cup\{\alpha\}\). Then \(1=\{0\}\), \(2=\{0,1\}\), \(3=\{0,1,2\}\), and \(\omega=\{0,1,2,\dots\}\). A set \(T\) is transitive if \(x\in T\) implies \(x\subseteq T\).

Proof idea. Define \(f(w)=\{f(v):v<w\}\) by recursion along \(W\). Each \(f(w)\) is an ordinal, \(f\) preserves order, and the image \(f[W]\) is a transitive set of ordinals, hence an ordinal. Replacement guarantees that \(f[W]\) is a set. Von Neumann stressed that Ernst Zermelo’s 1908 axioms cannot prove this theorem; replacement, proposed by Abraham Fraenkel and Thoralf Skolem in 1922, is essential. In 1928 he also gave the first rigorous justification of definition by transfinite recursion.

Classes and limitation of size (1925)

Von Neumann’s 1925 system took functions as primitive, but in modern terms it distinguishes sets from classes. The universe \(V\) of all sets and the class of all ordinals are classes that are not members of anything. His key axiom was:

The axiom implies separation, replacement and a global well-ordering of \(V\). Paul Bernays and Gödel later reshaped the system into the class theory NBG, which Gödel used in 1940 for his consistency proof of the continuum hypothesis.

Foundation and the cumulative hierarchy

Von Neumann also proposed excluding sets such as \(x=\{x\}\) by an axiom of foundation: every non-empty set has an \(\in\)-minimal element. Define

The hierarchy is now called the von Neumann universe. The least \(\alpha\) with \(x\subseteq V_\alpha\) is the rank of \(x\).

Paradoxical decompositions (1929)

In 1924 Stefan Banach and Alfred Tarski showed that a solid ball in \(\mathbb R^3\) can be cut into finitely many pieces and reassembled by rotations and translations into two balls of the same size. No such paradox exists in the plane. Von Neumann found the reason in the group of motions, not in the space.

The rotation group \(\mathrm{SO}(3)\) contains a copy of \(F_2\), which drives the Banach–Tarski paradox. The isometry group of the plane is solvable, so the plane has no paradox under isometries. Von Neumann showed that a paradox in the plane reappears once area-preserving affine maps are allowed, because \(\mathrm{SL}(2,\mathbb R)\) also contains \(F_2\).

Hilbert space and quantum mechanics

For the general reader. In 1925 and 1926 physicists had two quantum theories that gave the same answers in different languages: Heisenberg’s infinite matrices and Schrödinger’s waves. Paul Dirac unified them with formal symbols such as delta functions that had no rigorous meaning. Von Neumann showed that both theories describe operators on a single infinite-dimensional space, Hilbert space, and gave quantum mechanics the mathematical form it still has.

The axioms

Von Neumann’s 1927 papers, and the 1932 book Mathematische Grundlagen der Quantenmechanik, set out the following framework. Let \(\mathcal H\) be a separable complex Hilbert space.

1. A pure state is a unit vector \(\psi\in\mathcal H\), up to a phase. A general state is a density operator: a positive trace-class operator \(\rho\) with \(\operatorname{Tr}\rho=1\).
2. An observable is a self-adjoint operator \(A\), possibly unbounded, with spectral measure \(E_A\).
3. The probability that a measurement of \(A\) in state \(\rho\) gives a value in a Borel set \(\Delta\) is \(\operatorname{Tr}(\rho\,E_A(\Delta))\); the expectation is \(\langle A\rangle=\operatorname{Tr}(\rho A)\).
4. Between measurements, \(\rho(t)=e^{-iHt/\hbar}\rho\,e^{iHt/\hbar}\), where \(H\) is the Hamiltonian.

Heisenberg’s matrices are operators written in the basis \(\ell^2(\mathbb N)\); Schrödinger’s waves are vectors in \(L^2(\mathbb R^3)\). The Riesz–Fischer theorem says these two spaces are isomorphic, which explains the equivalence of the two theories.

The spectral theorem

The position and momentum operators are unbounded and defined only on dense subspaces. Von Neumann distinguished symmetric operators, with \(\langle A\phi,\psi\rangle=\langle\phi,A\psi\rangle\) on the domain, from self-adjoint ones, with \(A=A^*\) including equality of domains. Only the latter have a spectral decomposition.

Uniqueness of the commutation relations

Set \(\hbar=1\). Heisenberg’s relation \([X,P]=i\) cannot hold for bounded operators, so von Neumann worked with the unitary groups \(U(a)=e^{iaX}\) and \(V(b)=e^{ibP}\), which satisfy Weyl’s form of the relation:

Every quantum system with finitely many degrees of freedom therefore has essentially one representation of position and momentum. The theorem fails for infinitely many degrees of freedom, which is one source of the difficulties of quantum field theory.

Entropy, measurement, hidden variables

Von Neumann defined the entropy of a state as \(S(\rho)=-k\operatorname{Tr}(\rho\ln\rho)\). He separated two kinds of change: continuous unitary evolution, and the discontinuous change caused by measurement, which takes \(\rho\) to \(\sum_i P_i\rho P_i\) for the eigenprojections \(P_i\) of the measured observable. He showed that the boundary between the measured system and the observing apparatus can be moved along the chain of instruments without changing predictions. Both ideas frame the measurement problem to this day.

The book also contained a proof that no hidden-variable theory could reproduce quantum mechanics. Its key assumption was that expectation values add for all observables, including those that do not commute. Grete Hermann criticized this assumption in 1935, and John Bell made the same point in 1966. The proof excludes a narrower class of theories than was long believed. In 1936, with Garrett Birkhoff, von Neumann proposed that the closed subspaces of \(\mathcal H\) form a non-distributive “logic” of quantum propositions.

Rings of operators

For the general reader. The observables of a quantum system form an algebra: you can add and multiply them. Von Neumann asked which algebras of operators can occur and found a striking answer with Francis Murray. Some algebras measure “dimension” in whole numbers, as ordinary spaces do. Others measure it on a continuous scale from 0 to 1, as if a space could be two-thirds dimensional. A third kind cannot measure size at all.

The double commutant theorem (1929)

For a set \(S\subseteq B(\mathcal H)\), the commutant is \(S'=\{T\in B(\mathcal H):TX=XT\ \forall X\in S\}\). It is always closed in the weak operator topology.

Proof sketch. Let \(x\in M''\) and \(\xi\in\mathcal H\). The closed subspace \(K=\overline{M\xi}\) contains \(\xi\) (because \(1\in M\)) and is invariant under \(M\) and under \(M^*=M\), so its projection \(p\) lies in \(M'\). Then \(x\) commutes with \(p\), so \(x\xi=xp\xi=px\xi\in K\): some \(m\in M\) has \(m\xi\) close to \(x\xi\). For finitely many vectors \(\xi_1,\dots,\xi_n\), apply the same argument to \(\xi_1\oplus\cdots\oplus\xi_n\) in \(\mathcal H^n\), where \(M\) acts diagonally and \(x\oplus\cdots\oplus x\) lies in its double commutant. This gives strong density.

A weakly closed unital \(*\)-algebra is now called a von Neumann algebra. Von Neumann proved in 1938 (published 1949) that every von Neumann algebra on a separable space is a direct integral of factors, algebras with centre \(M\cap M'=\mathbb C1\). Factors are the building blocks.

Comparing projections

In a von Neumann algebra \(M\), projections \(e,f\) are equivalent, \(e\sim f\), if there is a partial isometry \(u\in M\) with \(u^*u=e\) and \(uu^*=f\). A projection is finite if it is not equivalent to a proper subprojection of itself. In \(B(\mathcal H)\), \(e\sim f\) exactly when the ranges have the same dimension, so dimension is recovered from the algebra alone.

Type I factors are the algebras \(B(\mathcal K)\). Type II₁ factors carry a faithful normal trace \(\tau\) with \(\tau(1)=1\), and \(D(e)=\tau(e)\) gives projections of every real dimension in \([0,1]\). Von Neumann showed that the lattice of projections of a II₁ factor is a continuous geometry, a projective geometry with a continuous dimension, and developed that theory in 1936–37.

Factors from groups

Let \(G\) be a countable group, and let \(\lambda(g)\delta_h=\delta_{gh}\) on \(\ell^2(G)\). The group von Neumann algebra \(L(G)=\lambda(G)''\) has the trace \(\tau(x)=\langle x\delta_e,\delta_e\rangle\). If every non-trivial conjugacy class of \(G\) is infinite (an ICC group), \(L(G)\) is a II₁ factor. In 1943 Murray and von Neumann showed that all approximately finite-dimensional II₁ factors are isomorphic, and that \(L(F_2)\) is not one of them. Whether \(L(F_2)\cong L(F_3)\) is still unknown.

Type III factors remained mysterious until the Tomita–Takesaki theory around 1970 and Alain Connes’s classification of the amenable factors in the 1970s. Vaughan Jones’s index theory for subfactors (1983) led to a new knot polynomial. Israel Gelfand and Mark Naimark’s C*-algebras (1943) grew from the same source.

The mean ergodic theorem

For the general reader. Statistical mechanics assumes that the long-run time average of a quantity, followed along one trajectory, equals its average over all possible states. Ludwig Boltzmann had made this “ergodic hypothesis” plausible but never proved anything like it. In 1931 von Neumann proved the first precise theorem of this kind, using the Hilbert-space methods he had built for quantum mechanics.

Koopman’s operator

Let \((X,\mu)\) be a probability space and \(T:X\to X\) measure-preserving: \(\mu(T^{-1}A)=\mu(A)\). In 1931 Bernard Koopman observed that \(Uf=f\circ T\) is an isometry of \(L^2(X,\mu)\), unitary if \(T\) is invertible. Von Neumann saw that the time averages of \(f\) are averages of powers of \(U\).

Proof. Write \(A_N=\frac1N\sum_{n<N}U^n\), so \(\|A_N\|\le1\). If \(Ux=x\), then \(A_Nx=x\). If \(x=y-Uy\), the sum telescopes: \(A_Nx=\frac1N(y-U^Ny)\), of norm at most \(2\|y\|/N\to0\). For unitary \(U\), \(Ux=x\) iff \(U^*x=x\), so the orthogonal complement of the range of \(I-U\) is \(\ker(I-U^*)=\ker(I-U)\). Thus \(\mathcal H=\ker(I-U)\oplus\overline{\operatorname{ran}(I-U)}\). Since the \(A_N\) are uniformly bounded, convergence to 0 on \(\operatorname{ran}(I-U)\) extends to its closure. (The argument in this form is due to Frigyes Riesz; von Neumann’s original used the spectral theorem.)

For a measure-preserving \(T\), the system is ergodic when the only invariant functions are constants. Then \(Pf=\int f\,d\mu\), and the time averages \(\frac1N\sum f(T^nx)\) converge in \(L^2\) to the space average.

Birkhoff and priority

Von Neumann found his theorem first and communicated it to George David Birkhoff. Birkhoff then proved the stronger pointwise version, and his note appeared in the Proceedings of the National Academy of Sciences in December 1931, before von Neumann’s paper of 1932. In 1932 Birkhoff and Koopman published a note in the same journal setting out the order of discovery.

Von Neumann continued with Koopman (1932) and with his assistant Paul Halmos. The Halmos–von Neumann theorem (1942) says that an ergodic transformation with pure point spectrum is isomorphic to a rotation of a compact abelian group, and is determined up to isomorphism by its eigenvalues. Spectral invariants became the first tool for classifying dynamical systems.

Compact groups and Hilbert’s fifth problem

Invariant averaging also lies behind another 1933 result. Using the invariant measure that Alfréd Haar had just constructed, von Neumann proved that every compact, locally Euclidean group is a Lie group, settling Hilbert’s fifth problem for compact groups. The full problem was solved in 1952 by Andrew Gleason, Deane Montgomery and Leo Zippin. His 1934 memoir on almost periodic functions on groups won the Bôcher Prize.

Games, utility, growth

For the general reader. In a game of matching pennies, any fixed strategy can be exploited. Von Neumann proved in 1928 that if each player may randomize, every two-player game with opposed interests has a well-defined value and optimal strategies. With the economist Oskar Morgenstern he built on this a general theory of strategic behaviour, which became a foundation of modern economics.

The minimax theorem (1928)

A two-person zero-sum game is an \(m\times n\) real matrix \(A\). Player 1 picks a row, player 2 a column, and player 2 pays \(a_{ij}\) to player 1. Mixed strategies are probability vectors \(x\in\Delta_m\), \(y\in\Delta_n\), and the expected payoff is \(x^{\mathsf T}Ay\).

The inequality \(\le\) holds for any function: \(\min_y x^{\mathsf T}Ay\le x^{\mathsf T}Ay_0\le\max_x x^{\mathsf T}Ay_0\) for all \(x,y_0\). The 1928 proof of equality was topological. The 1944 book used convexity, as follows.

Proof. The set \(C=\{Ay+z:\ y\in\Delta_n,\ z\in\mathbb R^m_{\ge0}\}\) is closed and convex. Case (i) says \(0\in C\). If \(0\notin C\), a separating hyperplane gives \(x\neq0\) with \(x\cdot c>0\) for all \(c\in C\). Letting one coordinate of \(z\) grow shows \(x\ge0\); rescale so \(x\in\Delta_m\). Taking \(z=0\), \(y=e_j\) gives \((x^{\mathsf T}A)_j>0\) for every \(j\).

Proof of the theorem. Apply the lemma to \(A-tJ\), where \(J\) is the all-ones matrix and \(t\in\mathbb R\). In case (i), \(\max_x x^{\mathsf T}Ay\le t\), so the upper value is at most \(t\). In case (ii), \(\min_y x^{\mathsf T}Ay>t\), so the lower value exceeds \(t\). Hence no \(t\) satisfies lower \(\le t<\) upper, and the two values are equal.

The theorem is equivalent to linear programming duality. When George Dantzig described his simplex method to von Neumann in 1947, von Neumann recognized the connection at once.

The 1944 book and expected utility

Theory of Games and Economic Behavior (Princeton, 1944) developed zero-sum games, games with many players, coalitions and a solution concept now called a stable set. The second edition (1947) added an appendix proving that preferences over lotteries satisfying completeness, transitivity, continuity and independence can be represented by the expected value of a utility function \(u\), unique up to \(u\mapsto au+b\) with \(a>0\). This is the von Neumann–Morgenstern utility theorem. John Nash’s equilibrium (1950) extended the minimax idea to non-zero-sum games.

The expanding economy (1937)

In a 1937 paper, presented in Karl Menger’s Vienna colloquium, von Neumann modelled an economy with \(m\) processes and \(n\) goods. Process \(i\) run at unit intensity consumes \(a_{ij}\) and produces \(b_{ij}\) of good \(j\). He sought intensities \(x\ge0\), prices \(p\ge0\), a growth factor \(\alpha\) and an interest factor \(\beta\) with

Under his assumption that every entry of \(A+B\) is positive, he proved that a solution exists with \(\alpha=\beta\), and that this common value is unique: the maximal balanced growth rate equals the minimal interest rate. The proof used a generalization of Brouwer’s fixed-point theorem, which Shizuo Kakutani recast in 1941 as the Kakutani fixed-point theorem.

Numerical computation and the stored program

For the general reader. Wartime weapons work needed solutions of equations that no one could solve by hand. Von Neumann became the leading advocate of electronic computers for science. He wrote the report that defined how a computer should be organized, built one at Princeton, and invented methods to use it: random sampling, stability tests for numerical schemes and an efficient sorting algorithm.

The EDVAC report (1945)

In the summer of 1944 Herman Goldstine told von Neumann about the ENIAC, being built at the Moore School in Philadelphia by J. Presper Eckert and John Mauchly. The ENIAC was programmed by rewiring. Its designers already planned a successor, the EDVAC, that would hold its program in memory. Von Neumann joined their discussions, and in spring 1945 wrote the First Draft of a Report on the EDVAC, dated 30 June 1945. Goldstine circulated it widely.

The 101-page draft described the logical organization of a machine with a central arithmetic part, a central control, a memory, and input and output, built from idealized switching elements modelled on the McCulloch–Pitts neurons. Instructions and numbers share one binary memory. Because instructions are numbers, a program can compute and modify addresses, which makes loops and indexed arrays possible. This organization is now called the von Neumann architecture. The report named only von Neumann as author, and Eckert and Mauchly resented the loss of credit. Because the report was a public disclosure, it also made the EDVAC ideas unpatentable.

At the Institute for Advanced Study he then led the construction of his own machine, described in the 1946 Preliminary Discussion with Arthur Burks and Goldstine. The chief engineer was Julian Bigelow. The IAS machine used 40-bit words and 1,024 words of Williams-tube memory, and was fully operational on 10 June 1952. Its design was copied widely, in machines such as JOHNNIAC, MANIAC I and ILLIAC I.

Stability of difference schemes

Von Neumann developed at Los Alamos a test for whether rounding errors grow in a numerical scheme. It was published with Jule Charney and Ragnar Fjørtoft in 1950. The idea: errors obey the same linear difference equation as the solution, so decompose them into Fourier modes \(e^{ij\theta}\) and track each mode’s growth.

For the heat equation \(u_t=\kappa u_{xx}\), the explicit scheme with \(r=\kappa\,\Delta t/\Delta x^2\) is

Substituting \(u_j^n=g^n e^{ij\theta}\) gives the amplification factor

The worst mode is \(\theta=\pi\), with \(g=1-4r\); the condition \(1-4r\ge-1\) gives \(r\le\tfrac12\). Halving \(\Delta x\) forces \(\Delta t\) down by a factor of four. With Robert Richtmyer, von Neumann also introduced artificial viscosity (1950), a way to compute shock waves without tracking them explicitly.

Monte Carlo and random numbers

In 1946 Stanislaw Ulam, playing solitaire while recovering from an illness, proposed estimating probabilities by simulating many games. Von Neumann turned the idea into a method for neutron transport and, with Nicholas Metropolis, Klára von Neumann and others, ran the first fully automated Monte Carlo calculations on the ENIAC in spring 1948. To generate random numbers he proposed the middle-square method: square a \(2k\)-digit number and keep the middle \(2k\) digits. He knew its faults and argued that fast, reproducible arithmetic sequences were still preferable to physical randomness for computing. In the same 1949 lecture he described the rejection method for sampling from a distribution, and a trick for extracting fair bits from a biased coin.

Matrices, sorting and weather

With Goldstine, von Neumann published in 1947 an error analysis of inverting large matrices by elimination, often called the first paper of modern numerical analysis. In 1945 he wrote a merge sort program for the EDVAC: split the list, sort each half, merge. In 1950 the ENIAC produced the first numerical weather forecasts, integrating the barotropic vorticity equation, from the Institute’s Meteorology Project led by Charney.

Cellular automata and self-reproduction

For the general reader. Can a machine build a copy of itself? It seems that the machine would need a description of itself, which would need a description of the description, and so on forever. Von Neumann found the way out in 1948, five years before the structure of DNA was published, and then built a mathematical universe, a grid of cells with simple rules, in which a self-copying machine provably exists.

The logical argument (1948)

In his lecture at the Hixon Symposium in Pasadena in September 1948, published in 1951 as The General and Logical Theory of Automata, von Neumann proposed the following design. Let \(\phi(X)\) denote a written description of a machine \(X\).

• \(A\), a universal constructor: given \(\phi(X)\), it builds \(X\).
• \(B\), a copier: given any description, it makes a copy of it.
• \(C\), a controller: given \(\phi(X)\), it runs \(A\) to build \(X\), runs \(B\) to copy \(\phi(X)\), and attaches the copy to \(X\).

Let \(D=A+B+C\). Fed its own description, \(D+\phi(D)\) produces \(D+\phi(D)\). The regress is avoided because the description is used twice: once interpreted, as instructions, and once copied without interpretation, as data. Adding an arbitrary machine \(F\) to \(D\), and its description to the tape, gives a reproducing machine that also does something else, and mutations on the tape give heritable variation.

The 29-state automaton

Following a suggestion of Ulam, von Neumann replaced physical parts by a two-dimensional grid of cells, each in one of finitely many states, updated in discrete time steps by the same rule. Each cell sees itself and its four orthogonal neighbours, the von Neumann neighbourhood. His automaton has 29 states: one unexcitable ground state, 8 sensitized states used during construction, 4 confluent states, and 16 transmission states (ordinary and special, in four directions, quiescent or excited). Ordinary transmission cells carry signals like wires; special transmission cells carry signals that build or destroy cells.

Von Neumann worked on the construction from 1948 and lectured on it at Illinois in December 1949, but left it unfinished. Burks edited and completed the manuscript as Theory of Self-Reproducing Automata (1966). Later work simplified it: Edgar Codd found an 8-state version (1968), and Renato Nobili and Umberto Pesavento ran a complete self-reproducing configuration on a computer in 1995. John Conway’s Game of Life (1970) descends from this line.

Reliability from unreliable parts

In lectures given in January 1952, published in 1956, von Neumann asked how a brain or computer can work reliably when its components fail. His answer was redundancy with majority voting, organized so that error probabilities stay bounded at every stage of a long computation. The question, and the threshold phenomena he found, reappear in the theory of fault-tolerant quantum computation.

His last work, the Silliman lectures prepared for Yale in 1956 but never delivered, compared computers with the nervous system. It was published as The Computer and the Brain in 1958.

Los Alamos and Washington, 1937–1955

From the late 1930s von Neumann’s work moved steadily toward applications. He became an expert on shock waves and shaped charges, joined the scientific advisory committee of the Army’s Ballistic Research Laboratory at Aberdeen in 1940, and from 1941 consulted for the Navy’s Bureau of Ordnance. His wartime contributions to fluid dynamics include the blast-wave solution now named after Geoffrey Taylor, von Neumann and Leonid Sedov, and the theory of detonation known as the ZND model.

The atomic bomb

In September 1943 he began consulting at Los Alamos. Plutonium could not be assembled fast enough in a gun-type bomb, and Seth Neddermeyer had proposed instead to crush a subcritical sphere inward with explosives. Von Neumann showed how a symmetric implosion could work and contributed to the design of the explosive lenses that shaped the converging shock wave. This design was tested at Trinity on 16 July 1945 and used in the bomb dropped on Nagasaki. In spring 1945 he served on the Target Committee that recommended Japanese cities for attack, and he calculated the optimal height of burst.

After the war

The computers of Lesson 09 were built partly for weapons work. He supported Edward Teller’s push for the hydrogen bomb and worked on its calculations. He chaired the Air Force committee on strategic missiles (1953–54), whose report accelerated the American intercontinental ballistic missile programme. His politics were firmly anti-communist, and he favoured a strong nuclear posture. In the 1954 security hearing of J. Robert Oppenheimer he testified in Oppenheimer’s favour, even though they disagreed on the hydrogen bomb.

President Dwight Eisenhower appointed him to the Atomic Energy Commission, and he was sworn in on 15 March 1955. He moved to Washington and largely gave up research. He served as president of the American Mathematical Society from 1951 to 1953.

Death and legacy, 1955–1957

In 1955 doctors found a tumour near his collarbone; it was a metastatic cancer whose origin is disputed. He continued to work for the Atomic Energy Commission from a wheelchair and prepared the Silliman lectures. In 1956 he received the Medal of Freedom from President Eisenhower and the Enrico Fermi Award. He was admitted to Walter Reed Army Medical Center, where he received a Catholic priest; he had been baptized a Catholic in 1930. He died there on 8 February 1957, aged 53, and is buried in Princeton Cemetery.

Students and circle

He supervised few doctoral students. They include Béla Lengyel (Budapest, 1935), Israel Halperin (1936), Maurice Pryce (1937), Friedrich Mautner (1948), Donald Gillies (1953) and John Mayberry (1956) at Princeton. His influence passed more through assistants and collaborators: Paul Halmos, Francis Murray, Garrett Birkhoff, Oskar Morgenstern, Goldstine, Burks, Ulam and Charney. Peter Lax and many others counted him as a mentor.

Legacy

• Foundations: von Neumann ordinals, the cumulative hierarchy and class theory are standard set theory.
• Quantum theory: Hilbert space, self-adjoint operators, density matrices and von Neumann entropy are the language of quantum mechanics and quantum information.
• Operator algebras: von Neumann algebras underlie noncommutative geometry, subfactor theory and algebraic quantum field theory.
• Dynamics and analysis: the mean ergodic theorem began abstract ergodic theory; amenability became a central idea in group theory.
• Economics: game theory, expected utility and the growth model shaped mathematical economics.
• Computing: the stored-program architecture, Monte Carlo methods, stability analysis, numerical weather prediction and cellular automata.
• Honours: Bôcher Prize (1938), Medal for Merit (1947), Medal of Freedom (1956), Enrico Fermi Award (1956). The IEEE John von Neumann Medal, SIAM’s John von Neumann Prize lecture and the John von Neumann Theory Prize in operations research bear his name, as does a lunar crater.

Connections

People linked to John von Neumann across this collection. A name without a link has no page here yet. See John von Neumann in the network of all 337 lives →

Teachers and advisors

• Lipót Fejér 1926

Students

• Donald Gillies 1953
• Friedrich Mautner 1948
• Israel Halperin 1936
• Maurice Pryce 1937
• Paul Halmos 1939-1942

Influenced by

• Alfred Tarski
• David Hilbert 1926-1927
• Frigyes Riesz
• Georg Cantor
• Stefan Banach
• Gábor Szegő
• László Rátz
• Michael Fekete 1920-1922

Influenced

• Stephen Cole Kleene
• Alain Connes
• John Nash
• Peter Lax

Colleagues and collaborators

• Alan Turing 1936-1938
• Albert Einstein 1933-1955
• Claude Shannon
• David Hilbert 1926-1929
• G. I. Taylor
• Hermann Weyl 1933-1955
• J. Robert Oppenheimer
• Kurt Gödel
• Leonid Kantorovich
• Norbert Wiener
• Olga Taussky-Todd 1947
• Solomon Lefschetz
• Stanislaw Ulam 1935-1957
• Arthur Burks 1946-1966
• Bernard Koopman 1932
• Edward Teller
• Francis Murray 1936-1943
• Garrett Birkhoff 1936
• Herman Goldstine 1944-1957
• Jule Charney 1948-1956
• Oskar Morgenstern 1940-1947

Rivals

• Émile Borel
• George David Birkhoff 1931-1932

Friends

• Eugene Wigner 1915-1957

Also mentioned in this life

• Ludwig Boltzmann
• L. E. J. Brouwer
• Richard Courant
• Paul Dirac
• Enrico Fermi
• Joseph Fourier
• Gottlob Frege
• Israel Gelfand
• Werner Heisenberg
• Henri Lebesgue
• Sophus Lie
• Erwin Schrödinger

Fields

• set theory 12 lives
• mathematical physics 45 lives
• functional analysis 11 lives
• ergodic theory 7 lives
• numerical analysis 6 lives

On CoursesHub

• Von Neumann & the Digital Age · History of Mathematics
• Hilbert Spaces & Quantum Mechanics · History of Mathematics & Physics
• Functional Analysis · Mathematics for Physics

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, John von Neumann, MacTutor History of Mathematics.
2. John von Neumann, Wikipedia (with extensive references).
3. John von Neumann, Mathematics Genealogy Project.
4. Atomic Heritage Foundation, John von Neumann.
5. S. Ulam, John von Neumann 1903–1957, Bull. Amer. Math. Soc. 64 (1958), no. 3, part 2.
6. N. Macrae, John von Neumann, Pantheon, 1992.
7. G. Dyson, Turing’s Cathedral, Pantheon, 2012 (on the IAS computer).

Mathematics

1. J. von Neumann, Zur Einführung der transfiniten Zahlen, Acta Sci. Math. (Szeged) 1 (1923), 199–208.
2. J. von Neumann, Eine Axiomatisierung der Mengenlehre, J. reine angew. Math. 154 (1925), 219–240. English translation in J. van Heijenoort, From Frege to Gödel, Harvard, 1967.
3. J. von Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), 295–320.
4. J. von Neumann, Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102 (1929), 49–131.
5. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, 1932; English translation, Princeton, 1955.
6. J. von Neumann, Proof of the quasi-ergodic hypothesis, Proc. Natl. Acad. Sci. 18 (1932), 70–82.
7. F. J. Murray and J. von Neumann, On rings of operators, Ann. of Math. 37 (1936), 116–229; IV, 44 (1943), 716–808.
8. G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. of Math. 37 (1936), 823–843.
9. J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, Princeton, 1944.
10. J. von Neumann, First Draft of a Report on the EDVAC, 1945; edited by M. D. Godfrey, IEEE Ann. Hist. Comput. 15 (1993), 27–75.
11. J. von Neumann and H. H. Goldstine, Numerical inverting of matrices of high order, Bull. Amer. Math. Soc. 53 (1947), 1021–1099.
12. J. G. Charney, R. Fjørtoft and J. von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2 (1950), 237–254.
13. J. von Neumann, The general and logical theory of automata, in Cerebral Mechanisms in Behavior: The Hixon Symposium, Wiley, 1951.
14. J. von Neumann, Theory of Self-Reproducing Automata, ed. A. W. Burks, University of Illinois Press, 1966.
15. J. von Neumann, The Computer and the Brain, Yale University Press, 1958.
16. List of scientific publications by John von Neumann, Wikipedia.