Notes

INTRODUCTION: WHO WAS JOHN VON NEUMANN?

1. Albert Einstein, 1922, Sidelights on Relativity, E. P. Dutton and Company, New York.

2. Freeman Dyson, 2018, personal communication.

CHAPTER 1: MADE IN BUDAPEST

1. Renamed Bajcsy-Zsilinsky Street in 1945 after a resistance hero.

2. John Lukacs, 1998, Budapest 1900: A Historical Portrait of a City and Its Culture, Grove Press, New York.

3. Robert Musil, 1931–3, Der Mann ohne Eigenschaften, Rowohlt Verlag, Berlin. English edition: 1997, The Man without Qualities, trans. Sophie Wilkins, Picador, London.

4. Nicholas A. Vonneuman, 1987, John von Neumann as Seen by His Brother, P.O. Box 3097, Meadowbrook, Pa.

5. Ibid.

6. Formidable but not infallible. His school friend Eugene Wigner recalled that when Johnny had once tried to multiply two five-digit numbers in his head, he got the answer wrong. ‘Then why on earth are you congratulating me?’ a bewildered Jancsi had asked him. Because, Wigner said, Johnny had been very close.

7. See, for example, Harry Henderson, 2007, Mathematics: Powerful Patterns into Nature and Society, Chelsea House, New York, p. 30.

8. Despite decades of research, whether chess ability is correlated with general intelligence or mathematical ability is still a fiercely contested question. A recent meta-analysis of past studies (Alexander P. Burgoyne et al., ‘The Relationship between Cognitive Ability and Chess Skill: A Comprehensive Meta-analysis’, Intelligence, 59 (2016), pp. 72–83) suggests some correlation exists and is strongest for numerical ability, and among younger players rather than highly skilled older masters of the game. One of the earliest studies in the area found no differences in intelligence between eight grandmasters and non-chess players (I. N. Djakow, N. W. Petrowski and P. A. Rudik, 1927, Psychologie des Schachspiels, deGruyter, Berlin).

9. Klára von Neumann, Johnny, quoted in George Dyson, 2012, Turing’s Cathedral, Pantheon Books, New York.

10. Vonneuman, John von Neumann as Seen by His Brother.

11. Ibid.

12. Ibid. Von Neumann was among several early computer pioneers to be inspired by this ingenious mechanism, which allowed fabrics with complex patterns to be mass-produced for the first time. Like the loom, the Analytical Engine, a mechanical computer designed in the 1830s by the English polymath Charles Babbage, was to be programmed with strings of punched cards. ‘We may say most aptly that the Analytical Engine weaves algebraic patterns just as the Jacquard loom weaves flowers and leaves,’ wrote Babbage’s collaborator, the mathematician Countess Ada Lovelace. Sadly, Babbage never managed to raise the sizeable sum he needed to build the machine. Jacquard’s place in the history of computing was assured by his loom’s influence on Herman Hollerith, an enterprising civil servant who worked for the United States Census Bureau. Less than twenty years after Babbage’s death in 1871, Hollerith was granted a patent for an electromechanical device ‘to simplify and thereby facilitate the compilation of statistics’ by recording information using holes in a roll of paper tape – though he quickly switched to more robust punched cards. Hollerith’s machines were used for the 1890 American census, and the improvements in efficiency they brought about led dozens of other countries to lease his equipment. In 1911, his company and three others were amalgamated to form what became IBM, a firm that von Neumann would influence when he was older, encouraging them to switch from mechanical computers to electronic ones.

13. Of Johnny’s brothers, Nicholas merged the ‘von’ with ‘Neumann’ after he arrived in the United States, becoming Vonneuman. Michael simply used Neumann.

14. Theodore von Kármán with Lee Edson, 1967, The Wind and Beyond: Theodore von Kármán, Pioneer in Aviation and Pathfinder in Space, Little, Brown and Co., Boston.

15. Tibor Frank, 2007, The Social Construction of Hungarian Genius (1867–1930), Eötvös Loránd University, Budapest.

16. George Klein, 1992, The Atheist and the Holy City: Encounters and Reflections, MIT Press, Cambridge, Mass.

17. Edward Teller (with Judith Shoolery), 2001, Memoirs: A Twentieth-Century Journey in Science and Politics, Perseus, Cambridge, Mass.

18. Stanisław M. Ulam, 1991, Adventures of a Mathematician, University of California Press, Berkeley.

19. Science rarely lives up to this ideal in the real world.

CHAPTER 2: TO INFINITY AND BEYOND

1. After being forced out of his job at the University of Königsberg by the Nazis in 1933.

2. George Pólya Papers, quoted in Frank, The Social Construction of Hungarian Genius.

3. Gabor Szego˝ Papers, quoted in ibid.

4. M. Fekete and J. L. von Neumann, ‘Über die Lage der Nullstellen gewisser Minimumpolynome’, Jahresbericht der Deutschen Mathematiker-Vereinigung, 31 (1922).

5. Timothy Gowers (ed.), 2008, The Princeton Companion to Mathematics, Princeton University Press, Princeton.

6. Freeman Dyson, ‘A Walk through Johnny von Neumann’s Garden’, Notices of the American Mathematical Society, 60(2) (2013), pp. 154–61.

7. Andrew Janos, 1982, The Politics of Backwardness in Hungary, Princeton University Press, Princeton.

8. Vonneuman, John von Neumann as Seen by His Brother.

9. Hearings (1955), United States: US Government Printing Office.

10. Pál Prónay, 1963, A hatarban a halal kaszal: Fejezetek Prónay Pal feljegyzeseibol, ed. Agnes Szabo and Ervin Pamlenyi, Kossuth Könyvkiadó, Budapest.

11. Eugene P. Wigner, ‘Two Kinds of Reality’, The Monist, 48(2) (1964), pp. 248–64.

12. Quoted in the definitive work on mathematical modernism: Jeremy Gray, 2008, Plato’s Ghost: The Modernist Transformation of Mathematics, Princeton University Press, Princeton.

13. Ibid.

14. P. Stäckel, 1913, Wolfgang und Johann Bolyai: Geometrische Untersuchungen, Leben und Schriften der beiden Bolyai, Teubner, Leipzig, quoted in ibid.

15. Much of what follows is based on the accounts in Constance Reid, 1986, Hilbert-Courant, Springer, New York; and Gray, Plato’s Ghost.

16. Reid, Hilbert-Courant.

17. Bertrand Russell, 1967, The Autobiography of Bertrand Russell: 1872–1914 (2000 edn), Routledge, New York.

18. Ibid.

19. Reid, Hilbert-Courant.

20. John von Neumann, ‘Zur Einführung der transfiniten Zahlen’, Acta Scientiarum Mathematicarum (Szeged), 1(4) (1923), pp. 199–208.

21. Von Kármán, The Wind and Beyond. In 2008, the world would learn that, to the contrary, mathematics can make lots of money – and lose it too.

22. Wigner’s dream of pursuing theoretical physics after school had ended less happily than von Neumann’s. ‘And exactly how many jobs are there in Hungary for theoretical physicists?’ his father asked him. ‘Four,’ Wigner replied. He was promptly packed off to study chemical engineering in Berlin. Fellner would initially study the subject for similar reasons. All three would drop it quite soon after finishing their degrees to pursue their true passions.

23. Quoted in Stanisław Ulam, ‘John von Neumann 1903–1957’, Bulletin of the American Mathematical Society, 64 (1958), pp. 1–49.

24. John von Neumann, ‘Eine Axiomatisierung der Mengenlehre’, Journal für die reine und angewandte Mathematik, 154 (1925), pp. 219–40.

25. John von Neumann, ‘Die Axiomatisierung der Mengenlehre’, Mathematische Zeitschrift, 27 (1928), pp. 669–752.

26. Quoted in Dyson, Turing’s Cathedral.

CHAPTER 3: THE QUANTUM EVANGELIST

1. Einstein to Heinrich Zangger, 12 May 1912, The Collected Papers of Albert Einstein, Princeton University Press, Princeton, vol. 5, p. 299, quoted in Manjit Kumar, 2008, Quantum: Einstein, Bohr and the Great Debate about the Nature of Reality, Icon Books, London.

2. Werner Heisenberg, ‘Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen’, Zeitschrift für Physik, 33(1) (1925), pp. 879–93.

3. Well, not quite instantaneously it turns out. The results of some remarkable experiments published in June 2019 show transitions take a small but finite amount of time (many microseconds): https://doi.org/10.1038/s41586-019-1287-z.

4. An electron in the energy state n = 6 might, for example, drop down to the ground state or arrive there via n = 2, effectively making two instantaneous mini transitions; one from n = 6 to n = 2, and a second from n=2 to the ground state.

5. If I roll a dice twice and want to know the probability that I roll a 3 (1 in 6 or 1/6) followed by a 4 (ditto), then I multiply the two probabilities together; the answer is 1 in 36 or 1/36.

6. Here is one of Heisenberg’s three-by-three arrays multiplied by itself. The first digit in the resulting array is a result of calculating 2×2+5×1+4×4 = 25; the second digit of the first row is 2×5+5×1+4×2 = 23 and so on.

Jeremy Bernstein explains this superbly in more detail in his paper ‘Max Born and the Quantum Theory’, American Journal of Physics, 73 (2005), pp. 999–1008.

7. For example, . Flip the order of multiplication and you get .

8. Kumar, Quantum.

9. Born to Einstein, 15 July 1925, in Max Born, 2005, The Born–Einstein Letters 1916–1955: Friendship, Politics and Physics in Uncertain Times, Macmillan, New York.

10. Born’s equation connecting the position of a particle, x, with its momentum, p:

On the right-hand side is h, Planck’s constant; I, the identity or unit matrix; and the imaginary number i (defined by the property that its square is −1). Unit matrices have 1s on their diagonal and 0s everywhere else, like this . The I in the equation allows both sides to be written as matrices. Imaginary numbers obey the same mathematical rules as real numbers and have proved incredibly useful in physics and engineering (for example in the very non-imaginary realm of circuit theory). In quantum mechanics, they are a convenience that makes the equations easier to solve. They make the maths work.

11. The uncertainty principle states that if ∆x and ∆p are the respective uncertainties in the measurements of position and momentum, their product will always be at least . That is, . The more accurately we know the position of a particle, the less accurately can we know its momentum. Physics students are still sometimes taught that this uncertainty arises because of disturbances introduced by the act of measurement. This is not true. While in practice the measuring process does introduce some additional uncertainty, Heisenberg’s uncertainty principle places a fundamental limit on the precision with which a particle’s position and momentum can ever be determined, no matter how delicately they are measured. The uncertainty is an inherent property of the particle in question.

12. Louis de Broglie, ‘XXXV. A Tentative Theory of Light Quanta’, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 47(278) (1924), pp. 446–58.

13. Johnny’s compatriot Leo Szilard, recovered by now from his ‘intolerably boring’ maths lessons in Budapest, would try to patent the idea of using electron waves to image small objects, and Ernst Ruska and Max Knoll would (independently) build the first prototype of an electron microscope in 1931.

14. If you’re wondering how a function can satisfy an equation, a relatively simple example is f (x) + f (y) = x + y, which is satisfied by, for example, the function f (x) = x.

15. ψ_n=a₁ψ₁+a₂ψ₂+a₃ψ₃+a₄ψ₄+ … where a_n is the fraction each wave function makes up of the total.

16. Kumar, Quantum, p. 225.

17. Ibid.

18. Imagine writing a list of numbers beginning 1, 2, 3 … One can imagine writing next to each number an element from a Heisenberg matrix.

19. A bounded particle, such as the electron of a hydrogen atom, can also be in an infinite superposition of states. But, as we have seen, these states are countable: it is possible to assign quantum numbers to them. But a physical theory like quantum mechanics has to be able to cope with unbounded particles whizzing freely through space too.

20. The coordinates x, y, z are real numbers, so can be any point along a number line (the sort used to teach primary school kids how to add up). The real numbers include negative numbers, fractions and the irrational numbers, like pi and the square root of 2, which can’t be expressed as a fraction (they go on for ever after the decimal point, without repeating).

21. John von Neumann, 2018, Mathematical Foundations of Quantum Mechanics, Princeton University Press, Princeton.

22. Ian McEwan, 2010, Solar, Random House, London.

23. Freeman Dyson, quoted in Graham Farmelo, 2009, The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius, Faber and Faber, London.

24. Paul A. M. Dirac, ‘The Fundamental Equations of Quantum Mechanics’, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 109(752) (1925), pp. 642–53.

25. Paul Dirac, 1930, The Principles of Quantum Mechanics, Oxford University Press, Oxford.

26. Dyson, ‘A Walk through Johnny von Neumann’s Garden’, p. 154.

27. Operators are expressed as partial derivatives in wave mechanics.

28. Max Jammer, 1974, The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective, Wiley, Hoboken.

29. So x₁²+x₂²+x₃²+x₄²+x₅² … etc. has to be less than infinity. Mathematicians say the series of numbers has to ‘converge’.

30. What does a Hilbert space of quantum states look like? As the magnitude of each wave function (squared) is 1, their vectors are all of length one. If Hilbert space was merely two-dimensional, the collection of all possible state vectors would describe a circle centred on the origin, and in three dimensions they would describe the surface of a sphere. Since Hilbert space is infinite-dimensional, the tips of the state vectors representing the wave functions touch the surface of an infinite-dimensional ball, known as a hypersphere.

31. For those in the know, the orthogonal functions in question are a Fourier series of sines and cosines.

32. The Chebyshev polynomials of von Neumann’s very first paper are orthogonal.

33. Like this: ψ = c₁f₁+c₂f₂+c₃f₃+c₄f₄+ … Here, the fs are the orthogonal functions and the cs the coefficients.

34. I.e. |c₁|²+|c₂|²+|c₃|²+|c₄|²+ … etc. = 1.

35. Von Neumann, Mathematical Foundations of Quantum Mechanics.

36. Frank, The Social Construction of Hungarian Genius.

37. Interview of Eugene Wigner by Charles Weiner and Jagdish Mehra, 30 November 1966, Niels Bohr Library and Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history-programs/niels-bohr-library/oral-histories/4964.

38. ‘With Nordheim and (nominally) Hilbert’: David Hilbert, John von Neumann and Lothar Nordheim, ‘Über die Grundlagen der Quantenmechanik’, Mathematische Annalen, 98 (1927), pp. 1–30; ‘later on his own’: J. von Neumann, ‘Mathematische Begründung der Quantenmechanik’, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1927), pp. 1–57, and John von Neumann, ‘Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren’, Mathematische Annalen, 102 (1929), pp. 49–131.

39. Physicist Erich Hückel is credited with writing the poem. Translated by Felix Bloch. See Elisabeth Oakes, 2000, Encyclopedia of World Scientists, Facts on File, New York.

40. Steven Weinberg, ‘The Trouble with Quantum Mechanics’, The New York Review of Books, 19 January 2017.

41. Niels Bohr, ‘Wirkungsquantum und Naturbeschreibung’, Naturwiss, 17 (1929), pp. 483–6. First translated into English in 1934 as The Quantum of Action and the Description of Nature, Cambridge University Press, Cambridge.

42. Einstein to Max Born, 3 March 1947, in Born, The Born–Einstein Letters 1916–1955.

43. In standard histories of quantum mechanics, the Copenhagen interpretation first emerges in 1927, after being presented by Bohr at the fifth Solvay Conference in Brussels. The story goes that Bohr’s influential acolytes then promoted Bohr’s views until they became orthodoxy. The truth, however, appears to be that Bohr’s views are not perfectly aligned with the Copenhagen interpretation, and his writings were not widely read at the time. In fact, the phrase ‘Copenhagen interpretation’ first turns up only in 1955, in an essay by Heisenberg, who appears to have set down the ideas that scientists often regard as the canonical view of quantum mechanics (see, for example, Don Howard, ‘Who Invented the “Copenhagen Interpretation?” A Study in Mythology’, Philosophy of Science, 71(5) (2004), pp. 669–82, doi:10.1086/425941). Nonetheless, different aspects of the Copenhagen interpretation appeared long before Heisenberg collected them together under a single banner, and von Neumann’s work on the measurement problem is one of the earliest contributions to that synthesis.

44. David N. Mermin, ‘Could Feynman Have Said This?’, Physics Today, 57(5) (2004), pp 10–12.

45. Wigner changed his mind later.

46. Abraham Pais, ‘Einstein and the Quantum Theory’, Reviews of Modern Physics, 51 (1979), pp. 863–914.

47. Mathematical Foundations of Quantum Mechanics was not translated into English until 1955.

48. Andrew Hodges, 2012, Alan Turing: The Enigma. The Centenary Edition, Princeton University Press, Princeton.

49. Erwin Schrödinger, ‘Die gegenwärtige Situation in der Quantenmechanik’, Naturwissenschaften, 23(48) (1935), pp. 807–12.

50. Einstein to Born, 4 December 1926, in Born, The Born–Einstein Letters 1916–1955.

51. If they were measurable, then quantum mechanics could easily be proved wrong

52. This sounds like a fudge but need not be. There are many examples of hidden variable-like theories in physics that have turned out to be useful. The ideal gas law, for example, relates the pressure, volume and temperature of a fixed amount of gas. But there is a deeper ‘hidden variables’ theory – the kinetic theory of gases – that can be used to derive the law by considering the behaviour of the atoms or molecules of the gas bouncing around inside a vessel. Put another way, the ideal gas law is a result of the ‘hidden’ motions of gas particles.

53. Jammer, The Philosophy of Quantum Mechanics.

54. Andrew Szanton, 1992, The Recollections of Eugene P. Wigner: As Told to Andrew Szanton, Springer, Berlin.

55. Accounts differ on this. Wigner says the von Neumanns arrived a day after him. Von Neumann’s biographer claims they arrived a week later (Norman Macrae, 1992, John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence and Much More, Pantheon Books, New York).

56. David N. Mermin, ‘Hidden Variables and the Two Theorems of John Bell’, Reviews of Modern Physics, 65 (1993), pp. 803–15.

57. This and much of what follows is from Elise Crull and Guido Bacciagaluppi (eds.), 2016, Grete Hermann: Between Physics and Philosophy, Springer, Berlin, Heidelberg, New York.

58. Werner Heisenberg, 1971, Physics and Beyond: Encounters and Conversations, Harper and Row, New York.

59. The additivity postulate states that the sum of the expectation (average) values of two operators (acting on a system) is equal to the expectation value of their sum. This is true in both quantum and classical physics. For example, the average energy of a particle (the sum of the kinetic and potential energy of the particle) is equal to its average kinetic energy plus its average potential energy).

60. The essay was never published but has recently been recovered from Dirac’s archives and translated into English. See Crull and Bacciagaluppi, Grete Hermann.

61. Grete Hermann, ‘Die naturphilosophischen Grundlagen der Quantenmechanik’, Abhandlugen der Fries’schen Schule, 6(2) (1935), pp. 75–152.

62. Grete Hermann, ‘Die naturphilosophischen Grundlagen der Quantenmechanik’, Die Naturwissenschaften, 23(42) (1935), pp. 718–21.

63. The philosophy Hermann eventually expounded was a version of what is now known as the relational interpretation of quantum mechanics. Carlo Rovelli is generally credited with first describing the idea in 1994. Hermann got to it over half a century earlier. In relational interpretations, roughly speaking, quantum theory only describes the state of a system relative to other systems or observers. There is no ‘objective’ observer-independent state of a system ‘out there’. Two observers may have differing views on a quantum event until they compare them. According to Hermann, any observer can reconstruct the sequence of events that led to a particular observation, so causality, of sorts, is restored. This is all the causality required for most Kantians.

64. John Stewart Bell, interview in Omni, May 1988.

65. Quoted in Nicholas Gisin, ‘Sundays in a Quantum Engineer’s Life’ (2001), arXiv:quant-ph/0104140, https://arxiv.org/abs/quant-ph/0104140.

66. John Stewart Bell, 1987, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge.

67. From ‘On the Impossible Pilot Wave’, republished in ibid.

68. John Stewart Bell, 1966, ‘On the Problem of Hidden Variables in Quantum Mechanics’, Reviews of Modern Physics, 38 (1966), pp. 447–52.

69. N. D. Mermin, ‘Hidden Variables and the Two Theorems of John Bell’, Reviews of Modern Physics, 65 (1993), pp. 803–15.

70. Jeffrey Bub, ‘Von Neumann’s “No Hidden Variables” Proof: A Re-Appraisal’, Foundations of Physics, 40 (2010), pp. 1333–40; D. Dieks, ‘Von Neumann’s Impossibility Proof: Mathematics in the Service of Rhetorics’, Studies in History and Philosophy of Modern Physics, 60 (2017), pp. 136–48.

71. Michael Stöltzner, 1999, ‘What John von Neumann Thought of the Bohm Interpretation’, in D. Greenberger et al. (eds.), Epistemological and Experimental Perspectives on Quantum Physics, Kluwer Academic Publishers, Dordrecht.

72. Einstein to Born, 12 May 1952, in Born, The Born–Einstein Letters 1916–1955.

73. Albert Einstein, Boris Podolsky and Nathan Rosen, 1935, ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’, Physical Review, 47(10) (1935), pp. 777–80.

74. Using entangled photons rather than hydrogen atoms is easier for such experiments.

75. This and what follows is from Stefano Osnaghi, Fábio Freitas and Olival Freire Jr, ‘The Origin of the Everettian Heresy’, Studies in History and Philosophy of Modern Physics, 40, pp. 97–123.

76. Peter Byrne, 2010, The Many Worlds of Hugh Everett III: Multiple Universes, Mutual Assured Destruction, and the Meltdown of a Nuclear Family, Oxford University Press, Oxford.

77. Philip Ball discusses the problems and advantages of this and other interpretations of quantum mechanics in his wonderful 2018 book Beyond Weird, The Bodley Head, London.

78. G. C. Ghirardi, A. Rimini and T. Weber (1986), ‘Unified Dynamics for Microscopic and Macroscopic Systems’, Physical Review D, 34(2) (1986), pp. 470–91.

79. Von Neumann, Mathematical Foundations of Quantum Mechanics.

80. P. A. M. Dirac, 1978, Directions in Physics, Wiley, New York.

81. Laurent-Moïse Schwartz tidied up the mathematics of Dirac’s ‘improper’ delta functions in 1945 and was awarded the prestigious Fields Medal for his efforts.

82. Jammer, The Philosophy of Quantum Mechanics.

83. Many thanks to Ulrich Pennig, without whom I would have nothing at all to say about von Neumann algebras.

84. Dyson, ‘A Walk through Johnny von Neumann’s Garden’.

85. Carlo Rovelli, 2018, The Order of Time, Allen Lane, London.

86. Von Neumann to O. Veblen, 19 June 1933, Library of Congress archives.

87. Fabian Waldinger, ‘Bombs, Brains, and Science: The Role of Human and Physical Capital for the Creation of Scientific Knowledge’, Review of Economics and Statistics, 98(5) (2016), pp. 811–31.

88. A. Fraenkel, 1967, Lebenskreise, translated and quoted by David. E. Rowe, 1986, ‘“Jewish Mathematics” at Gottingen in the Era of Felix Klein’, Isis, 77(3), pp. 422–49.

CHAPTER 4: PROJECT Y AND THE SUPER

1. Paul Halmos, ‘The Legend of John von Neumann’, The American Mathematical Monthly, 80(4) (1973), pp. 382–94.

2. Mariette would later be instrumental in establishing Brookhaven National Laboratory on Long Island, working there for some twenty-eight years as a senior administrator. See Marina von Neumann Whitman, The Martian’s Daughter, University of Michigan Press, Ann Arbor, and https://www.bnl.gov/60th/EarlyBNLers.asp.

3. Richard Feynman with Ralph Leighton, 1985, Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character, W. W. Norton, New York.

4. Ergodic theory crops up in many diverse areas of physics and mathematics. Mathematicians Terence Tao and Ben Green used the theorem in 2004 to prove a conjecture related to prime numbers dating from the 1770s. They showed that there are sequences of prime numbers of arbitrary length where the difference between adjacent terms is a constant. For instance, 3,5,7 is one such sequence of length three.

5. The dispute between the two is discussed in Joseph D. Zund, ‘George David Birkhoff and John von Neumann: A Question of Priority and the Ergodic Theorems, 1931–1932’, Historia Mathematica, 29 (2002), pp. 138–56.

6. Garrett Birkhoff, 1958, ‘Von Neumann and Lattice Theory’, Bulletin of the American Mathematical Society, 64 (1958), pp. 50–56.

7. Alan Turing, ‘On Computable Numbers, with an Application to the Entscheidungsproblem’, published in two parts 1936–7, Proceedings of the London Mathematical Society, 42(1) (1937), pp. 230–65.

8. Interview with Herman Goldstine conducted by Nancy Stern, 1980, https://conservancy.umn.edu/bitstream/handle/11299/107333/oh018hhg.pdf?sequence=1&isAllowed=y.

9. Ulam, Adventures of a Mathematician.

10. Quoted in Macrae, John von Neumann.

11. Von Neumann Whitman, The Martian’s Daughter.

12. ‘It’s basically Angry Birds,’ says historian Thomas Haigh. http://opentranscripts.org/transcript/working-on-eniac-lost-labors-information-age/.

13. Quoted in Dyson, Turing’s Cathedral.

14. Quoted in ibid.

15. Quoted in Macrae, John von Neumann.

16. Von Neumann Whitman, The Martian’s Daughter.

17. Quoted in Dyson, Turing’s Cathedral.

18. https://libertyellisfoundation.org/passenger-details/czoxMzoiOTAxMTk4OTg3MDU0MSI7/czo4OiJtYW5pZmVzdCI7.

19. Details of Meitner’s life drawn from Ruth Lewin Sime, 1996, Lise Meitner: A Life in Physics, University of California Press, Berkeley.

20. John von Neumann, 2005, John von Neumann: Selected Letters, ed. Miklós Rédei, American Mathematical Society, Providence, R.I.

21. Subrahmanyan Chandrasekhar and John von Neumann, 1942, ‘The Statistics of the Gravitational Field Arising from a Random Distribution of Stars. I. The Speed of Fluctuations’, Astrophysical Journal, 95 (1942), pp. 489–531.

22. Thomas Haigh and Mark Priestly have recently made the case that von Neumann was not much influenced by Turing when it came to computer design, based on the text of three lectures they discovered: ‘Von Neumann Thought Turing’s Universal Machine Was “Simple and Neat”. But That Didn’t Tell Him How to Design a Computer’, Communications of the ACM, 63(1) (2020), pp. 26–32.

23. The name of the committee was not an acronym. A member of the committee, John Cockcroft, had received a cryptic telegram from Lise Meitner via an English friend of hers: ‘MET NIELS AND MARGRETHE RECENTLY BOTH WELL BUT UNHAPPY ABOUT EVENTS PLEASE INFORM COCKCROFT AND MAUD RAY KENT’. Cockcroft puzzled over the contents of the telegram. With a ‘y’ replaced with an ‘i’, he reasoned that the last three words were an anagram for ‘radium taken’. Were the Germans stockpiling the radioactive substance for work on a reactor or bomb? The committee was suitably galvanized by Meitner’s cunning warning, taking for its name the first word of the mysterious triplet in her message: MAUD. Many years after Cockcroft and others puzzled over the import of her words, it would emerge that Meitner’s message was never meant to be a call to action but was addressed to the former governess of Bohr’s children. Her name was Maud Ray. She lived in Kent. Meitner, the committed pacifist, had inadvertently helped to launch and energize the pursuit of the atomic bomb in two countries.

24. Two great accounts of the development of the atom bomb are Richard Rhodes, 2012, The Making of the Atom Bomb, Simon & Schuster, London, and the more recent Jim Baggott, 2012, Atomic: The First War of Physics and the Secret History of the Atom Bomb: 1939–49, Icon Books, London.

25. Kenneth D. Nichols, quoted in Peter Goodchild, 1980, J. Robert Oppenheimer: Shatterer of Worlds. Houghton Mifflin, New York.

26. Quoted in Rhodes, The Making of the Atom Bomb.

27. The technical problems encountered by the Manhattan Project and some of von Neumann’s contribution to solving them are from Lillian Hoddeson, Paul W. Henriksen, Roger A. Meade and Catherine Westfall, 1993, Critical Assembly: A Technical History of Los Alamos during the Oppenheimer Years, 1943–1945, Cambridge University Press, Cambridge.

28. The discovery was kept secret until after the war.

29. Quoted in Hoddeson et al., Critical Assembly.

30. John von Neumann, 1963, Oblique Reflection of Shocks, in John von Neumann: Collected Works, ed. A. H, Taub, vol. 6: Theory of Games, Astrophysics, Hydrodynamics and Meteorology, Pergamon Press, Oxford.

31. Hoddeson et al., Critical Assembly.

32. Kistiakowsky claimed to have taught von Neumann how to play poker at Los Alamos, only stopping when ‘he began to play better than I’. Yet von Neumann was familiar enough with the game to mention poker in a paper on game theory he published in 1928 and by the time he met Kistiakowsky, he had finished Theory of Games and Economic Behavior, which includes various analyses of the game. Perhaps von Neumann, who was a notoriously poor player, was trying to lull his friend into a false sense of security. https://www.manhattanprojectvoices.org/oral-histories/george-kistiakowskys-interview.

33. In the months that followed, experiments confirmed that reactor-produced samples contained a high-proportion of plutonium-240, which decays faster than the plutonium-239 needed at Los Alamos.

34. Arjun Makhijani, ‘“Always” the Target?’, Bulletin of the Atomic Scientists, 51(3) (1995), pp. 23–7.

35. ‘Personal Justice Denied: Report of the Commission on Wartime Relocation and Internment of Civilians’, National Archives. Government Printing Office, Washington, D.C., December 1982, https://www.archives.gov/research/japanese-americans/justice-denied.

36. Von Neumann’s notes on the Target Committee’s agenda are taken from Macrae, John von Neumann.

37. http://www.dannen.com/decision/targets.html.

38. https://www.1945project.com/portfolio-item/shigeko-matsumoto/.

39. The Committee for the Compilation of Materials on Damage Caused by the Atomic Bombs in Hiroshima and Nagasaki, 1981, Hiroshima and Nagasaki: The Physical, Medical, and Social Effects of the Atomic Bombings, Basic Books, New York.

40. Freeman Dyson, 1979. Disturbing the Universe, Harper and Row, New York.

41. Quoted in von Neumann Whitman, The Martian’s Daughter.

42. Quoted in Dyson, Turing’s Cathedral.

43. Interview with the author. 14 January 2019.

44. Von Neumann Whitman, The Martian’s Daughter.

45. Interview with the author.

46. German A. Goncharov, ‘Thermonuclear Milestones: (1) The American Effort’, Physics Today, 49(11) (1996), pp. 45–8.

47. https://www.globalsecurity.org/wmd/intro/classical-super.htm.

48. Goncharov, ‘Thermonuclear Milestones’.

49. German A. Goncharov, ‘Main Events in the History of the Creation of the Hydrogen Bomb in the USSR and the USA’, Physics – Uspekhi, 166 (1996), pp. 1095–1104.

CHAPTER 5: THE CONVOLUTED BIRTH OF THE MODERN COMPUTER

1. From her unfinished memoirs, quoted extensively in Dyson, Turing’s Cathedral. This chapter is particularly indebted to Dyson; Thomas Haigh, Mark Priestley and Crispin, Rope, 2016, ENIAC in Action: Making and Remaking the Modern Computer, MIT Press, Cambridge, Mass., and William Aspray, 1990, John von Neumann and the Origins of Modern Computing, MIT Press, Cambridge, Mass.

2. A slightly longer excerpt is quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory Cambridge University Press, Cambridge.

3. See Macrae, John von Neumann.

4. Earl of Halsbury, ‘Ten Years of Computer Development’, Computer Journal, 1 (1959), pp. 153–9.

5. Brian Randell, 1972, On Alan Turing and the Origins of Digital Computers, University of Newcastle upon Tyne Computing Laboratory, Technical report series.

6. Quoted in Aspray, John von Neumann and the Origins of Modern Computing.

7. Interview with William Aspray, quoted in ibid.

8. Hermann H. Goldstine, 1972, The Computer from Pascal to von Neumann, Princeton University Press, Princeton.

9. Herman Goldstine, interview with Albert Tucker and Frederik Nebeker, 22 March 1985, https://web.math.princeton.edu/oral-history/c14.pdf.

10. Harry Reed, 18 February 1996, ACM History Track Panel, quoted in Thomas J. Bergin (ed.), 2000, 50 Years of Army Computing: From ENIAC to MSRC, Army Research Lab Aberdeen Proving Ground MD.

11. Their contribution was discovered by the historian Thomas Haigh and his colleagues. See Haigh et al., ENIAC in Action.

12. http://opentranscripts.org/transcript/working-on-eniac-lost-labors-information-age/.

13. From dissertation work by Anne Fitzpatrick. Quoted in Haigh et al., ENIAC in Action.

14. Quoted in ibid.

15. Ibid.

16. Originally circulated in 1945, a carefully edited version was published by Michael D. Godfrey in 1993. John von Neumann, ‘First Draft of a Report on the EDVAC’, IEEE Annals of the History of Computing, 15 (1993), pp. 27–75.

17. Wolfgang Coy, 2008, The Princeton Companion to Mathematics, Princeton University Press, Princeton.

18. The details of Gödel’s life and work here are drawn largely from John W. Dawson, 1997, Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley, Mass., and Rebecca Goldstein, 2005, Incompleteness: The Proof and Paradox of Kurt Gödel. W. W. Norton & Company, New York.

19. In a syllogism, like the one below, a pair of premises leads to a conclusion:

All men are mortal

Socrates is a man

Therefore, Socrates is mortal

In the symbols of first-order logic, this becomes:

∀x(M(x) → P(x))

M(a)

P(a)

… where

∀ means ‘every’, → means ‘implies’ or ‘is’

M(a), which is read as ‘M of a’, means that any a has the property M.

M = the property of being a man

P = the property of being mortal

a is Socrates.

20. David Hilbert and Wilhelm Ackermann, 1928, Grundzüge der theoretischen Logik, Julius Springer, Berlin (later translated as Principles of Mathematical Logic).

21. E.g. set M = the property of being an orange, P = the property of being the colour green, and insert into the syllogism above.

22. Andrew Wiles of the University of Oxford proved Fermat’s last theorem in 1994, more than 350 years after Pierre de Fermat stated the problem and asserted he had a ‘marvellous proof’ which his margin was too narrow to contain. Wiles’s proof ran to over 100 pages. Goldbach’s conjecture remains unproven.

23. ‘Gödel began the same way, assigning a unique number to each symbol in the Principia’: after that he used prime numbers to denote the position of a particular symbol (the first five primes are 2, 3, 5, 7 and 11). He then raised each prime to the power of the symbol number of the symbol at that position in the statement. Multiplying these together produces the Gödel number of that statement. So, for example, let’s define the following symbol numbers: M = 1, a = 2, ( = 3, and ) = 4. Then the Gödel number of the expression M(a) is 2¹ × 3³ × 5² × 7⁴ = 3,241,350. As you can see, Gödel numbers soon get very large indeed. The so-called ‘fundamental theorem of arithmetic’ states that every integer greater than 1 is either prime or is a unique product of primes. That means in turn every statement has a unique Gödel number, and by factorizing that number, we can always recover the encoded statement.

24. ‘The Gödelian strange loop that arises in formal systems in mathematics … is a loop that allows such a system to “perceive itself”, to talk about itself, to become “self-aware”, and in a sense it would not be going too far to say that by virtue of having such a loop, a formal system acquires a self.’ Douglas R. Hofstadter, 1979, Gödel, Escher, Bach: An Eternal Golden Braid, Basic Books, New York. Quote from the preface to the twentieth-anniversary edition.

25. Not that he knew or cared much about that.

26. Minutes of the Institute for Advanced Study Electronic Computing Project, Meeting 1, 12 November 1945, IAS, quoted in Dyson, Turing’s Cathedral.

27. Martin Davis, 2000, The Universal Computer: The Road from Leibniz to Turing, W. W. Norton & Company, New York.

28. Kurt Gödel, ‘Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I’, Monatshefte für Mathematik und Physik, 38 (1931), pp. 173–98.

29. Von Neumann, Selected Letters.

30. Any details of Turing’s life are drawn from Andrew Hodges, 2012, Alan Turing: The Enigma. The Centenary Edition, Princeton University Press, Princeton. My brief description of Turing’s paper is abridged from Charles Petzold, 2008, The Annotated Turing: A Guided Tour Through Alan Turing’s Historic Paper on Computability and the Turing Machine, Wiley, Hoboken, and ‘Computable Numbers: A Guide’, in Jack B. Copeland (ed.), 2004, The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and Artificial Life plus The Secrets of Enigma, Oxford University Press, Oxford.

31. Alonzo Church, ‘A Note on the Entscheidungsproblem’, Journal of Symbolic Logic, 1(1) (1936), pp. 40–41.

32. Turing, ‘On Computable Numbers’.

33. He summarizes the behaviour of the machine in a table that looks something like this:

where P0 means ‘print a 0’, P1 means ‘print a 1’ and R means move right one square.

34. Charles Petzold describes Turing machines that can, for example, add and multiply two binary numbers together. See Petzold, The Annotated Turing.

35. Supplied with a copy of the standard description, all the universal machine needs to start work is the initial state of the Turing machine it is imitating and the first symbol that machine is supposed to read. It begins by scanning the standard description on its tape for the symbol and initial state that it has been given. Once it has found this configuration, the universal machine can read what it should do next: the coded m-description supplies the symbol to be printed, how its head should move and the next state. With this information to hand, the universal machine winds its tape back to where it started, prints the next symbol and the new state of the Turing machine it is simulating. Now the universal machine has a new symbol and state to look for in the standard description on its tape. This cyclical ‘search and match’, followed by print symbol and switching to next state, can continue for ever – or until the ‘program’ on its tape terminates. The output of the universal machine is not identical to that of the machine it is emulating. The universal machine, for instance, requires extra space for printing states. Nonetheless, it does reproduce the expected sequence exactly – even if there are gaps in between for ‘rough working’.

36. John von Neumann, 1963, ‘The General and Logical Theory of Automata’, lecture given at the Hixon Symposium on Cerebral Mechanisms in Behaviour, 20 September 1948, reproduced in Collected Works, vol. 5: Design of Computers, Theory of Automata and Numerical Analysis, Pergamon Press, Oxford.

37. Briefly, Turing’s strategy is this: he begins by considering whether it is possible for some Turing machine to tell from another machine’s standard description whether it will print digits for ever or halt. A little confusingly, he calls machines that print for ever ‘circle free’ and those that eventually stop as ‘circular’. Turing then proves that no machine exists that is capable of deciding if another, arbitrary machine is circle free. Any such machine, he shows, gets stuck in a loop when trying to assess its own standard description. Next, he likewise demonstrates that no machine can be concocted that can decide whether another machine ever prints a given symbol – ‘0’, for example. He does so by showing that this would also imply the existence of a machine that could determine whether a machine is circle free. Since he has already shown no machine can determine if another prints for ever by analysing its standard description, there can also be no machine that determines if another prints ‘0’. Finally, Turing constructs a rather complex statement in first-order logic that, in effect, says, ‘0 appears somewhere on the tape of machine M’. He calls this formula Un(M), for ‘undecidable’. Now, imagine a machine that can determine whether any statement in first-order logic can be proved using first-order logic. The philosopher Jack Copeland calls this machine ‘Hilbert’s dream’. Alas, when Hilbert’s dream attempts to chunter its way through Un(M) to decide if it is provable, the wheels come off. For Turing has already demonstrated that no machine can ever determine this. Hilbert’s dream disappears in a puff of logic.

38. From Klára von Neumann’s papers. Quoted in Dyson, Turing’s Cathedral.

39. Haigh quashes some of the wilder claims made for Turing in Thomas Haigh, ‘Actually, Turing Did Not Invent the Computer’, Communications of the ACM, 57(1) (2014), pp. 36–41.

40. Copeland, The Essential Turing.

41. First Draft of a Report on the EDVAC. A carefully proofread and edited version was published by Michael D. Godfrey, IEEE Annals of the History of Computing, 15(4) (1993), pp. 27–75.

42. W. S. McCulloch and W. Pitts, 1943, ‘A Logical Calculus of the Ideas Immanent in Nervous Activity’, Bulletin of Mathematical Biophysics, 5 (1943), pp. 115–33.

43. Goldstine to von Neumann, 15 May, 1945, quoted in Haigh et al., ENIAC in Action.

44. From ibid.

45. The computer’s ‘main memory’ or ‘primary storage’ loses data as soon as the computer powers down. The auxiliary memory, provided by a hard drive or flash memory, holds data until overwritten.

46. John W. Mauchly, ‘Letter to the Editor’, Datamation, 25(11) (1979), https://sites.google.com/a/opgate.com/eniac/Home/john-mauchly.

47. J. Presper Eckert, oral history interview by Nancy B. Stern, 28 October 1977, http://purl.umn.edu/107275.

48. From an unpublished book, quoted in Haigh et al., ENIAC in Action. This remains a controversial statement in computer history. In their thorough, recent work on the ENIAC, Haigh and his colleagues conclude: ‘Our best interpretation of the evidence is that, by editing, assembling, and extending ideas discussed at the joint meetings with the ENIAC team, von Neumann established, for the first time, the EDVAC architecture as a unified whole. See ibid.

49. Von Neumann to Aaron Townshend, 6 June 1946, quoted in Aspray, John von Neumann and the Origins of Modern Computing.

50. John von Neumann, deposition concerning EDVAC report, n.d. [1947], IAS, quoted in Dyson, Turing’s Cathedral.

51. Von Neumann to Frankel, 29 October 1946, quoted in Aspray, John von Neumann and the Origins of Modern Computing.

52. I. J. Good, 1970, ‘Some Future Social Repercussions of Computers’, International Journal of Environmental Studies, 1 (1970), pp. 67–79.

53. Eckert and Mauchly left the Moore School in March 1946 after refusing to sign new agreements to hand over their inventions to the University of Pennsylvania. They founded their own computer company, producing the UNIVAC I, the first of a series of machines for business and military applications, which was sold at a loss to the US Census Bureau. The Eckert–Mauchly Computer Corporation soon ran into severe financial difficulties, which were exacerbated when Mauchly was falsely accused of having communist sympathies. Some lucrative defence contracts were cancelled, and the company was put up for sale. The firm was acquired in 1950 by Remington Rand, which after a merger became Sperry Rand. As the new owners of Eckert and Mauchly’s patents, Sperry Rand would defend them, unsuccessfully, in court. Despite the devastating conclusion of that battle, Sperry Rand lives on today: after a takeover in 1986 by Burroughs, a business machines maker, the combined company became the computer giant Unisys.

54. Norbert Wiener to John von Neumann, 24 March 1945. Quoted in both Macrae, John von Neumann, and Dyson, Turing’s Cathedral.

55. Minutes of the Regular Meeting of the Board of Trustees, Institute for Advanced Study, 19 October 1945, quoted in Goldstine, The Computer from Pascal to von Neumann.

56. Dyson, Disturbing the Universe.

57. Klára von Neumann, Johnny, quoted in Dyson, Turing’s Cathedral.

58. Minutes of the Regular Meeting of the Board of Trustees, Institute for Advanced Study, 19 October 1945, quoted in Goldstine, The Computer from Pascal to von Neumann.

59. John von Neumann to Lewis Strauss, October 1945, quoted in Andrew Robinson (ed.), 2013, Exceptional Creativity in Science and Technology: Individuals, Institutions, and Innovations, Templeton Press, West Conshohocken, Pennsylvania.

60. All details of the ENIAC’s conversion and programming are from Haigh et al., ENIAC in Action.

61. In conversation with N. Cooper: see N. G. Cooper et al. (eds.), 1989, From Cardinals to Chaos: Reflection on the Life and Legacy of Stanislaw Ulam, Cambridge University Press, Cambridge.

62. Quoted in Dyson, Turing’s Cathedral.

63. Klára von Neumann, c.1963, quoted in Dyson, Turing’s Cathedral.

64. Ulam, quoted in Roger Eckhardt, 1987, Stan Ulam, John von Neumann, and the Monte Carlo Method, Los Alamos Science, 15, Special Issue (1987), pp. 131–7, https://permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-UR-88-9068.

65. Ulam was playing Canfield (US) or Demon (UK) solitaire, which is notoriously difficult to win.

66. Ulam testimony during ENIAC patent trial, quoted in Dyson, Turing’s Cathedral.

67. Haigh et al. were not able to find the program code for the first Monte Carlo run, but code listings for later runs were written in Klári’s hand.

68. ‘Eniac, the only electronic computer among the four “mathematical brains” now in use, is being converted so that it can handle without resetting all types of mathematical problems to which it is adapted.’ The changes would, the Times says, give ENIAC ‘a substantial part of the efficiency which is being built into the Edvac’ (Will Lissner, ‘“Brain” Speeded Up for War Problems’, New York Times, 13 December 1947), and ‘Under changes being made, it is hoped to boost Eniac’s actual weekly output from the equivalent of 10,000 man-hours to 30,000’ (Will Lissner, ‘Mechanical “Brain” Has Its Troubles’, New York Times, 14 December 1947).

69. Nicholas Metropolis, Jack Howlett and Gian-Carlo Rota (eds.), 1980, A History of Computing in the Twentieth Century, Academic Press, New York.

70. John von Neumann, 1951, ‘Various Techniques Used in Connection with Random Digits’, https://mcnp.lanl.gov/pdf_files/nbs_vonneumann.pdf.

71. S. Ulam to von Neumann, 12 May 1948, Putnam, New York quoted in Dyson, Turing’s Cathedral.

72. Von Neumann to S. Ulam, 11 May 1948, quoted in Haigh et al., ENIAC in Action.

73. Klári von Neumann to S. Ulam, 12 June 1948, quoted in ibid.

74. See Haigh et al., ENIAC in Action.

75. Von Neumann to S. Ulam, 18 November 1948, quoted in ibid.

76. Klári von Neumann to the Ulams, quoted in Dyson, Turing’s Cathedral.

77. The full name of MANIAC I was Mathematical Analyzer Numerical Integrator and Computer Model I.

78. See Haigh et al., ENIAC in Action.

79. Julian Bigelow interview with Nancy Stern, quoted in Dyson, Turing’s Cathedral.

80. Dyson, Turing’s Cathedral.

81. Interview with Richard R. Mertz, quoted in ibid.

82. Interview with Nancy Stern, quoted in ibid.

83. Stanley A. Blumberg and Gwinn Owens, 1976, Energy and Conflict: The Life and Times of Edward Teller,.

84. John von Neumann, ‘Defense in Atomic War’, Scientific Bases of Weapons, Journal of American Ordnance Association, 6(38) (1955), pp. 21–3, reprinted in Collected Works, vol. 6.

85. Julian Bigelow, ‘Computer Development at the Institute for Advanced Study’, in Metropolis et al. (eds.), A History of Computing in the Twentieth Century, pp. 291–310.

CHAPTER 6: A THEORY OF GAMES

1. Von Neumann Whitman, The Martian’s Daughter.

2. Ibid.

3. Ibid.

4. Jacob Bronowski, The Ascent of Man, Little, Brown, 1975.

5. This chapter is indebted to Robert Leonard, Von Neumann, Morgenstern and the Creation of Game Theory: From Chess to Social Science, 1900–1960, Cambridge University Press, Cambridge and William Poundstone, Prisoner’s Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, Doubleday, New York. For a brief non-mathematical introduction to game theory, try Ken Binmore, Game Theory: A Very Short Introduction, Oxford University Press, Oxford.

6. ‘I was in one of those moods where danger is attractive,’ Lasker wrote after a match against Dutch champion Abraham Speijer. ‘Hence I plunged from the start into a combination the outcome of which was exceedingly doubtful.’ Needless to say, Lasker came out on top.

7. Quoted in Leonard, Von Neumann, Morgenstern and the Creation of Game Theory.

8. Emanuel Lasker, [1906/7], Kampf, Lasker’s Publishing Co., New York, reprinted in 2001 by Berlin-Brandenburg, Potsdam, quoted in ibid.

9. Emanuel Lasker, Lasker’s Manual of Chess, New York: Dover, 1976 (original: Lehrbuch des Schachspiels, 1926; first English translation, 1927)], quoted in ibid.

10. John von Neumann, ‘Zur Theorie der Gesellschaftsspiele’, Mathematische Annalen, 100 (1928), pp. 295–320. Translation by Sonya Bargmann, ‘On the Theory of Games of Strategy’, Contributions to the Theory of Games, 4 (1959), pp. 13–42.

11. Von Neumann Whitman, The Martian’s Daughter.

12. Von Neumann, ‘Zur Theorie der Gesellschaftsspiele’.

13. That no real player is actually perfectly rational does not matter here. Perfectly straight lines do not exist either, but geometry has still proved quite useful in the real world.

14. On von Neumann’s 1928 proof and the priority dispute with Borel see Tinne Hoff Kjeldsen, ‘John von Neumann’s Conception of the Minimax Theorem: A Journey Through Different Mathematical Contexts’, Archive for History of Exact Sciences, 56 (2001), pp. 39–6.

15. Maurice Fréchet, ‘Emile Borel, Initiator of the Theory of Psychological Games and Its Application’, Econometrica, 21 (1953), pp. 95–6.

16. Maurice Fréchet, ‘Commentary on the Three Notes of Emile Borel’, Econometrica, 21 (1953), pp. 118–24.

17. John von Neumann, ‘Communication on the Borel Notes’, Econometrica 21 (1953), pp. 124–5.

18. Halperin Interview, The Princeton Mathematics Community in the 1930s, Transcript Number 18 (PMC18), quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

19. Péter Rózsa, Játék a Végtelennel, 1945. Translated by Z. P. Dienes, Playing with Infinity: Mathematical Explorations and Excursions, Dover Publications, New York.

20. John von Neumann, ‘Über ein ökonomisches Gleichungssystem und eine Verallgemeinerung des Brouwerschen Fixpunktsatzes’, Ergebnisse eines Mathematische Kolloquiums, 8 (1937), ed. Karl Menger, pp. 73–83,

21. Translated as ‘A Model of General Economic Equilibrium’, Review of Economic Studies, 13 (1945), pp. 1–9.

22. Macrae, John von Neumann.

23. The function f(x) = x is all fixed points, for example, while f(x) = 7 has one at x = 7.

24. John von Neumann, ‘The Impact of Recent Development in Science on the Economy and Economics’, speech delivered, published in Looking Ahead, 4 (1956), also in A. Bródy and T. Vámos (eds.), 1995, The Neumann Compendium, World Scientific, London.

25. John von Neumann to Oskar Morgenstern, 8 October 1947, quoted in Oskar Morgenstern, ‘The Collaboration Between Oskar Morgenstern and John von Neumann on the Theory of Games’, Journal of Economic Literature, 14(3) (1976), pp. 805–16.

26. See Macrae, John von Neumann.

27. E. Roy Weintraub, ‘On the Existence of a Competitive Equilibrium: 1930–1954’, Journal of Economic Literature, 21(1) (1983), pp. 1–39.

28. Sylvia Nasar, 1998, A Beautiful Mind, Simon & Schuster, New York.

29. Quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

30. Morgenstern diary entries quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

31. Oskar Morgenstern, 1928, Wirtschaftprognose: Eine Untersuchung ihrer Voraussetzungen und Möglichkeiten, Julius Springer, Vienna. Translation quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

32. Morgenstern, ‘The Collaboration’.

33. Oskar Morgenstern, Diary, 18 November 1938, quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

34. Morgenstern, Diary, 15 February 1939, quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

35. Einstein to Queen Elizabeth of Belgium, 20 November 1933, quoted in Jagdish Mehra, 1975, The Solvay Conferences on Physics: Aspects of the Development of Physics since 1911, D. Reidel, Dordrecht.

36. Morgenstern, ‘The Collaboration’.

37. Morgenstern, Diary, 26 October 1940.

38. Ibid., 22 January 1941.

39. Ibid.

40. Israel Halperin, 1990, ‘The Extraordinary Inspiration of John von Neumann’, in Proceedings of Symposia in Pure Mathematics, vol. 50: The Legacy of John von Neumann, ed. James Glimm, John Impagliazzo and Isadore Singer, American Mathematical Society, Providence, R.I., pp. 15–17.

41. Morgenstern, Diary, 12 July 1941.

42. Klára von Neumann, unpublished papers, quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

43. Morgenstern, Diary, 7 August 1941.

44. Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

45. Morgenstern, Diary, 14 April 1942.

46. Morgenstern, ‘The Collaboration’.

47. This illustration is my version of the explanation given in Binmore, Game Theory.

48. The scale need not go from 0 to 100. A utility score on one scale can be converted to a score on the other in the same way that a temperature measured in degrees Celsius can be converted into degrees Fahrenheit.

49. John von Neumann and Oskar Morgenstern, 1944, Theory of Games and Economic Behavior. Princeton University Press, Princeton.

50. Daniel Kahneman, 2011, Thinking, Fast and Slow, Farrar, Straus and Giroux, New York.

51. German logician Ernst Zermelo proved in 1912 that from a winning position either black or white can force a win. Unlike von Neumann, he allowed for games with infinitely many moves by ignoring the standard stopping rules of chess and did not use the method of backward induction for his proof. See U. Schwalbe and P. Walker, ‘Zermelo and the Early History of Game Theory’, Games and Economic Behavior, 34 (2001), pp. 123–37.

52. In this scenario if Holmes goes to Dover, Moriarty’s average payout is 60 per cent of 100 (60) if he goes to Dover plus 40 per cent of -50 (-20) if stopped at Canterbury. If Holmes disembarks early, on the other hand, then Moriarty’s strategy yields 40 per cent of 100. So in both cases Moriarty’s strategy yields 40 utils.

53. Binmore, Game Theory.

54. There are 33 ‘cards’ higher than hers and 66 of equal or lower value.

55. The threshold is given by .

56. The optimal frequency of bluff bids is .

57. Binmore, Game Theory.

58. I.e. Players A, B and C can form the coalitions (A,B), (B,C) and (A,C).

59. Imagine two players form a coalition and divide the payoff equally, each getting ½. The third player, who now faces receiving a payout of -1, quickly offers one of the others ¾ utils if they team up with him. He would then pocket ¼ utils. But then there would be nothing to stop the newly jilted player (now facing a payout of -1) from offering him more than ¼ utils to form another alliance and so on …

60. W Barnaby, ‘Do Nations Go to War Over Water?’, Nature, 458 (2009), pp. 282–3.

61. Von Neumann calls these ‘imputations’, a term still used in game theory today.

62. Michael Bacharach, 1989, ‘Zero-sum Games’, in John Eatwell, Murray Milgate and Peter Newman (eds.), Game Theory, The New Palgrave, Palgrave Macmillan, London, pp. 253–7.

63. Von Neumann knew Walras’s work – his own model of General Economic Equilibrium was an attempt to rectify some of the flaws he saw there.

64. John McDonald, 1950, Strategy in Poker, Business and War, W. W. Norton, New York.

65. Ibid.

66. Jacob Marschak, ‘Von Neumann and Morgenstern’s New Approach to Static Economics’, Journal of Political Economy, 54 (1946), pp. 97–115.

67. Robert J. Leonard, ‘Reading Cournot, Reading Nash: The Creation and Stabilisation of the Nash Equilibrium’, Economic Journal, 104(424) (1994), pp. 492–511.

68. Ibid.

69. William F. Lucas, ‘A Game with No Solution’, Bulletin of the American Mathematics Society, 74 (1968), pp. 237–9.

70. Gerald L. Thompson, 1989, ‘John von Neumann’, in Eatwell et al. (eds.), Game Theory, pp. 242–52.

71. The story of the controversy over the award is told in Nasar, A Beautiful Mind.

72. See John McMillan, ‘Selling Spectrum Rights’, Journal of Economic Perspectives, 8(3) (1994), pp. 145–62. Early competitions to assign public assets to companies were known as ‘beauty contests’. Each firm submitted a lengthy document explaining why their bid was the best, and a panel of officials decided the winner. Unfortunately, the officials charged with the task could have no real idea what the asset was really worth to the competing businesses – and the businesses had no incentive to tell them. That method was replaced with lotteries, which were also gamed. One winner in 1989 sold their licence to operate mobile phones on Cape Cod to Southwestern Bell for $41 million.

73. Paul Milgrom, ‘Putting Auction Theory to Work: The Simultaneous Ascending Auction’, Journal of Political Economy, 108(2) (2000), pp. 245–72. In a simultaneous ascending auction, bidding progresses in rounds, with the current standings revealed between rounds. The auction is over when there are no new bids on any licence. There are penalties for withdrawing bids to help keep bidders honest. Auctioning off licences in batches rather than sequentially prevents a firm from driving up prices of licences sold earlier so that competitors cannot afford to bid for those sold later. The setup is good for bidders too – they can adjust their bids to collect bundles of licences that complement each other.

74. See e.g. Thomas Hazlett, 2009, ‘U.S. Wireless License Auctions: 1994–2009’, https://www.accc.gov.au/system/files/Hazlett%2C%20Thomas%20%28Auctions%20Paper%29.pdf, and a complete list of FCC auctions here: https://capcp.la.psu.edu/data-and-software/fcc-spectrum-auction-data.

75. Ostrom was the only woman to win the prize until Esther Duflo in 2019.

76. Derek Wall, 2014, The Sustainable Economics of Elinor Ostrom: Commons, Contestation and Craft, Routledge, London.

77. Elinor Ostrom, ‘Design Principles of Robust Property Rights Institutions: What Have We Learned?’, in Gregory K. Ingram and Yu-Hung Hong (eds.), 2009, Property Rights and Land Policies, Lincoln Institute of Land Policy, Columbia University Press, New York.

78. See Kahneman, Thinking, Fast and Slow.

79. I am indebted to Michael Ostrovsky for his help with present-day applications of game theory and his magisterial round-up of thirty years of work in the field.

80. For any particular search term, advertisers submit bids stating the maximum they are willing to pay each time someone clicks through to their web page. The highest bidder has their ad shown most prominently on the search results page – but pays the second-highest bid. The second-highest bidder secures the second-most prominent position on the results page – and pays the third-highest bid, and so on. Designing and perfecting these ‘generalized second price’ auctions brought the first wave of game theorists into tech firms. For a brief history and review of such auctions, see one of the most cited economics papers of this century: Benjamin Edelman, Michael Ostrovsky and Michael Schwarz ‘Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords’, American Economic Review, 97(1) (2007), pp. 242–59.

81. W. D. Hamilton, 1996, Narrow Roads of Gene Land, vol. 1: Evolution of Social Behaviour, Oxford University Press, Oxford.

82. For the remarkable story of Price’s quest to understand altruism, see Oren Harman, 2010, The Price of Altruism: George Price and the Search for the Origins of Kindness, Bodley Head, London.

83. Elinor Ostrom, 2012, ‘Coevolving Relationships between Political Science and Economics’, Rationality, Markets and Morals, 3 (2012), pp. 51–65.

CHAPTER 7: THE THINK TANK BY THE SEA

1. Details of the history of the RAND Corporation are mainly from Fred Kaplan, 1983, The Wizards of Armageddon, Stanford University Press, Stanford, and David Jardini, 2013, Thinking Through the Cold War: RAND, National Security and Domestic Policy, 1945–1975, Smashwords, as well as Poundstone, Prisoner’s Dilemma and Alex Abella, 2008, Soldiers of Reason: The RAND Corporation and the Rise of the American Empire, Harcourt, San Diego, Calif. Daniel Bessner, 2018, Democracy in Exile: Hans Speier and the Rise of the Defense Intellectual, Cornell University Press, Ithaca, offers another perspective on RAND. The definitive intellectual history of game theory is Paul Erickson, 2015, The World the Game Theorists Made, The University of Chicago Press, Chicago.

2. ‘The RAND Hymn’, words and music by Malvina Reynolds, copyright 1961 Schroder Music Company, renewed 1989.

3. Kaplan, The Wizards of Armageddon.

4. Quoted in Jacob Neufeld, 1990, The Development of Ballistic Missiles in the United States Air Force, 1945–1960, United States Government Printing Office, Washington, D.C.

5. Kaplan, Wizards of Armageddon.

6. Ibid.

7. Ibid.

8. H. H. Arnold, 1949, Global Mission, Harper & Brothers, New York.

9. Full report available: https://www.governmentattic.org/TwardNewHorizons.html See also https://apps.dtic.mil/dtic/tr/fulltext/u2/a954527.pdf.

10. Interview with Collbohm by Martin Collins and Joseph Tatarewicz, 28 July 1987, https://www.si.edu/media/NASM/NASM-NASM_AudioIt-000006640DOCS.pdf.

11. Quoted in Abella, Soldiers of Reason.

12. Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

13. Quoted in Larry Owens, 1989, Mathematicians at War: Warren Weaver and the Applied Mathematics Panel, 1942–45, ed. David Rower and John McCleary, Academic Press, Boston.

14. Bernard Lovell, 1988, ‘Blackett in War and Peace’, The Journal of the Operational Research Society, 39(3) (1988), pp. 221–33.

15. All quotations here are from Erickson, The World the Game Theorists Made.

16. Quoted in Abella, Soldiers of Reason.

17. Kaplan, The Wizards of Armageddon.

18. John Williams to von Neumann 16 December 1947, quoted in Erickson, The World the Game Theorists Made.

19. Poundstone, Prisoner’s Dilemma.

20. Ibid.

21. See George B. Dantzig, ‘The Diet Problem’, Interfaces, 20(4) (1990), pp. 43–7. Dantzig later tried using the algorithm to lose weight, promising his wife that he would follow whatever optimal diet RAND’s computer suggested. Unfortunately, owing to an oversight during optimization of the program, the computer recommended that Dantzig drink 500 gallons of vinegar.

22. https://apps.dtic.mil/dtic/tr/fulltext/u2/a157659.pdf. George B. Dantzig, 1985, Impact of Linear Programming on Computer Development, Department of Operations Research, Stanford University.

23. Interview with Robert Leonard, 27 February 1990, quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

24. Willis H. Ware, 2008, RAND and the Information Evolution: A History in Essays and Vignettes, RAND Corporation.

25. Clay Blair Jr, ‘Passing of a Great Mind’, Fortune, 25 February 1957, p. 89.

26. Quoted in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

27. Much of this work is collected and summarized in Melvin Dresher, 1961, Games of Strategy: Theory and Applications, available as RAND Corporation document number CB-149-1 (2007).

28. This episode is related in Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

29. Hans Speier interview with Martin Collins, 5 April 1988, https://www.si.edu/media/NASM/NASM-NASM_AudioIt-000003181DOCS.pdf.

30. Mathematician Joseph Malkevitch gives an excellent example of how Shapley values work by considering the case of three (fictional) towns ordered to clean up their sewage outflow. See http://www.ams.org/publicoutreach/feature-column/fc-2016-09.

31. Quoted in Alvin E. Roth, ‘Lloyd Shapley (1923–2016)’, Nature, 532 (2016), p. 178.

32. D. Gale and L. S. Shapley, ‘College Admissions and the Stability of Marriage’, American Mathematical Monthly, 69 (1962), pp. 9–15.

33. Sylvia Nasar, 1998, A Beautiful Mind, Simon & Schuster, New York.

34. Ibid.

35. J. F. Nash, ‘The Bargaining Problem’, Econometrica, 28 (1950), pp. 155–62.

36. See Leonard, Reading Cournot, Reading Nash: ‘Above all, as Shubik (1991) reports, von Neumann “hated it!”, clearly finding it foreign to his whole conception of game theory.’

37. Email from John Nash to Robert Leonard, 20 February 1993, quoted in Nasar, A Beautiful Mind.

38. See, for example, Herman Goldstine interview with Albert Tucker and Frederik Nebeker, 22 March 1985, https://web.math.princeton.edu/oral-history/c14.pdf. ‘I don’t think Johnny suffered brilliant people easily,’ says Goldstine. Tucker cites the example of graduate student Harold Kuhn pointing out a possible problem with von Neumann’s expanding economy paper. ‘Johnny actually got angry,’ says Tucker, ‘and I really think that he was thinking very fast to get himself out of it.’

39. John F. Nash Jr, ‘Equilibrium Points in N-Person Games’, Proceedings of the National Academy of Sciences of the United States of America, 36 (1) (1950), pp. 48–9.

40. Nasar, A Beautiful Mind.

41. Sylvia Nasar claims Nash was not gay: https://www.theguardian.com/books/2002/mar/26/biography.highereducation.

42. Leonard, Von Neumann, Morgenstern, and the Creation of Game Theory.

43. Steve J. Heims, 1982, John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, MIT Press, Cambridge, Mass.

44. Quoted in Dyson, Turing’s Cathedral.

45. Quoted in ibid.

46. See Leonard, ‘Reading Cournot, Reading Nash’.

47. Merrill M. Flood, 1952, Some Experimental Games, RAND Research Memorandum RM-789-1.

48. From Poundstone, Prisoner’s Dilemma.

49. Flood, Some Experimental Games.

50. Ibid.

51. This is the method of ‘backward induction’, also used earlier by von Neumann in Theory of Games.

52. Poundstone, Prisoner’s Dilemma.

53. Evidence of cooperation in the one-shot Prisoner’s Dilemma has been found in many studies, e.g. R. Cooper, D. V. DeJong, R. Forsythe and T. W. Ross, ‘Cooperation Without Reputation: Experimental Evidence from Prisoner’s Dilemma Games’, Games and Economic Behavior, 12(2) (1996), pp. 187–218, and J. Andreoni and J. H. Miller (1993). ‘Rational Cooperation in the Finitely Repeated Prisoner’s Dilemma, Experimental Evidence’, The Economic Journal, 103(418), pp. 570–85.

54. Quoted in von Neumann Whitman, The Martian’s Daughter.

55. Clay Blair Jr, ‘Passing of a Great Mind’, Life Magazine, 25 February 1957.

56. E.g. Alexander Field, ‘Schelling, von Neumann, and the Event That Didn’t Occur’, Games, 5(1) (2014), pp. 53–89.

57. Ulam, ‘John von Neumann 1903–1957’.

58. Eugene Wigner, 1957. ‘John von Neumann (1903–1957)’, Yearbook of the American Philosophical Society, later in 1967, Symmetries and Reflections: Scientific Essays of Eugene P. Wigner, Indiana University Press, Bloomington.

59. See e.g. R. Buhite and W. Hamel, ‘War for Peace: The Question of an American Preventive War against the Soviet Union, 1945–1955’, Diplomatic History, 14(3) (1990), pp. 367–84.

60. William L. Laurence, ‘How Soon Will Russia Have the A-Bomb?’, Saturday Evening Post, 6 November 1948, p. 182.

61. Buhite and Hamel ‘War for Peace’.

62. Quoted in Poundstone, Prisoner’s Dilemma.

63. Russell spent much of the fifties denying he had ever advocated bombing the Soviets if they did not give up their nuclear ambitions. He became the first president of the Campaign for Nuclear Disarmament in 1958 and in 1961, at age eighty-nine, he was jailed for seven days for organizing an anti-bomb protests in London. Russell finally came clean during an interview with the BBC in 1959. Asked if it was true that he had advocated a preventive war against Soviet Russia, he answered, ‘It’s entirely true, and I don’t repent of it. It was not inconsistent with what I think now. What I thought all along was that a nuclear war in which both sides had nuclear weapons would be an utter and absolute disaster.’

64. John von Neumann to Klára von Neumann, 8 September, 1954, quoted in Dyson, Turing’s Cathedral.

65. Bernard Brodie, 1959, Strategy in the Missile Age, available as RAND Corporation document number CB-137-1 (2007).

66. https://www.manhattanprojectvoices.org/oral-histories/george-kistiakowskys-interview.

67. Oppenheimer was indubitably a supporter of the Communist Party. See Ray Monk, 2012, Robert Oppenheimer: A Life Inside the Center, Doubleday, New York and Toronto.

68. Quoted in Macrae, John von Neumann. A charitable view of contemporary Britain, which did not make Alan Turing an earl. He was charged with gross indecency and his security clearance was revoked.

69. J. Robert Oppenheimer Personnel Hearings Transcripts, volume XII, https://www.osti.gov/includes/opennet/includes/Oppenheimer%20hearings/Vol%20XII%20Oppenheimer.pdf.

70. John Earl Haynes, Harvey Klehr and Alexander Vassiliev, 2009, ‘Enormous: The KGB Attack on the Anglo-American Atomic Project’, in Spies: The Rise and Fall of the KGB in America, translations by Philip Redko and Steven Shabad, Yale University Press, New Haven.

71. Quoted in Dyson, Turing’s Cathedral.

72. Erickson, The World the Game Theorists Made.

73. Roberta Wohlstetter, 1962, Pearl Harbor: Warning and Decision, Stanford University Press, Stanford.

74. Albert Wohlstetter, Fred Hoffman, R. J. Lutz and Henry S. Rowen, 1954, Selection and Use of Strategic Air Bases, RAND Corporation, Santa Monica.

75. Albert Wohlstetter, ‘The Delicate Balance of Terror’, Foreign Affairs, 37 (January 1959); an earlier and more complete version dated December 1958 is available as RAND Paper P-1472, https://www.rand.org/pubs/papers/P1472.html.

76. Kaplan, The Wizards of Armageddon.

77. The coinage is Alex Abella’s.

78. Kaplan, The Wizards of Armageddon.

79. Ibid.

80. Herman Kahn and Irwin Mann, 1957, Game Theory, https://www.rand.org/pubs/papers/P1166.html.

81. Sharon Ghamari-Tabrizi, 2005, The Worlds of Herman Kahn: The Intuitive Science of Thermonuclear War, Harvard University Press, Cambridge, Mass.

82. Herman Kahn, 1960, On Thermonuclear War, Princeton University Press, Princeton.

83. The director borrowed so much from On Thermonuclear War that Kahn demanded royalties, to which Kubrick replied, ‘That’s not how it works, Herman!’

84. Dr Strangelove, or: How I Stopped Worrying and Learned to Love the Bomb, directed by Stanley Kubrick (1964).

85. James R. Newman, ‘Two Discussions of Thermonuclear War’, Scientific American, March 1961.

86. Kahn joked he gained ten pounds to disprove Newman’s doubts about his existence.

87. Thomas C. Schelling, 1958, ‘The Strategy of Conflict: Prospectus for a Reorientation of Game Theory’, Journal of Conflict Resolution, 2(3), pp. 203–64.

88. Thomas C. Schelling, 1960, The Strategy of Conflict, Harvard University Press, Cambridge, Mass.

89. Thomas C. Schelling, ‘Bargaining, Communication, and Limited War’, Conflict Resolution, 1(1) (1957), pp. 20, 34.

90. Von Neumann, ‘Defense in Atomic War’.

91. Von Neumann was probably referring to one of the recent hydrogen bomb tests, rather than the atom bombs dropped on Hiroshima and Nagasaki, which were a thousand times less powerful.

92. For a summary of US thinking on deterrence see Marc Trachtenberg, ‘Strategic Thought in America, 1952–1966’, Political Science Quarterly, 104(2) (1989), pp. 301–34.

93. https://fas.org/irp/doddir/dod/jp3_72.pdf.

CHAPTER 8: THE RISE OF THE REPLICATORS

1. https://www.youtube.com/watch?v=3KJbrb0P8jQ&feature=emb_title.

2. Alex Ellery, ‘Are Self-Replicating Machines Feasible?’, Journal of Spacecraft and Rockets, 53(2) (2016), pp. 317–27.

3. Correspondence with author.

4. What could possibly go wrong?

5. Von Neumann, ‘The General and Logical Theory of Automata’.

6. Ibid.

7. John von Neumann, ‘The General and Logical Theory of Automata’, originally published in Lloyd A. Jeffress (ed.), 1951, Cerebral Mechanisms in Behavior: The Hixon Symposium, Wiley, New York.

8. Robert A. Freitas Jr and Ralph C. Merkle, 2004, Kinematic Self-Replicating Machines, Landes Bioscience, Georgetown, Texas, http://www.MolecularAssembler.com/KSRM.htm.

9. Von Neumann, ‘The General and Logical Theory of Automata’.

10. Erwin Schrödinger, 1944, What Is Life?, Cambridge University Press, Cambridge.

11. For a critical appreciation of Schrödinger’s What Is Life?, see Philip Ball, 2018, ‘Schrödinger’s Cat among Biology’s Pigeons: 75 Years of What Is Life?’, Nature, 560 (2018), pp. 548–50.

12. Sydney Brenner, 1984, ‘John von Neumann and the History of DNA and Self-replication’, https://www.webofstories.com/play/sydney.brenner/45.

13. Dyson, Disturbing the Universe.

14. Arthur W. Burks, 1966, Theory of Self-reproducing Automata, University of Illinois Press, Urbana.

15. John G. Kemeny, ‘Man Viewed as a Machine’, Scientific American, 192(4) (1955), pp. 58–67.

16. Philip K. Dick, ‘Autofac’, Galaxy, November 1955.

17. See Lawrence Sutin, [1989], Divine Invasions: A Life of Philip K. Dick, Harmony Books, New York.

18. Von Neumann called the struts ‘rigid members’, an expression that I avoid for reasons that I hope are obvious.

19. He chooses a transmission state with its output on the southern side.

20. Umberto Pesavento, 1995, ‘An Implementation of Von Neumann’s Self-reproducing Machine’, Artificial Life, 2(4) (1995), pp. 337–54.

21. For a look at the history of that field, see Steven Levy, 1993, Artificial Life: A Report from the Frontier Where Computers Meet Biology, Vintage, New York.

22. Arthur W. Burks, 1966, Theory of Self-reproducing Automata, University of Illinois Press, Urbana.

23. Details are from Conway’s biography: Siobhan Roberts, 2015, Genius at Play: The Curious Mind of John Horton Conway, Bloomsbury, London.

24. Ibid.

25. Ibid.

26. Ibid.

27. Martin Gardner, 2013, Undiluted Hocus-Pocus: The Autobiography of Martin Gardner, Princeton University Press, Princeton.

28. Quoted in Levy, Artificial Life.

29. These are the AND, OR and NOT operations of Boolean algebra.

30. In 2001, Paul Rendell implemented a Turing machine in Life and, later, a universal version: http://www.rendell-attic.org/gol/tm.htm

31. Quoted in Fred Hapgood, 1987, ‘Let There Be Life’, Omni, 9(7) (1987), http://www.housevampyr.com/training/library/books/omni/OMNI_1987_04.pdf.

32. E. O. Wilson, 1975, Sociobiology: The New Synthesis, Harvard University Press, Cambridge, Mass.

33. Levy, Artificial Life.

34. Quoted in ibid.

35. Published as Tommaso Toffoli, 1977, ‘Computation and Construction Universality of Reversible Cellular Automata’, Journal of Computer and System Sciences, 15(2), pp. 213–31.

36. Quoted in Levy, Artificial Life.

37. Ibid.

38. Fredkin made the case for this in Edward Fredkin, 1990, ‘Digital Mechanics: An Informational Process based on Reversible Universal Cellular Automata’, Physica D, 45 (1990), pp. 254–70.

39. Steven Levy, ‘Stephen Wolfram Invites You to Solve Physics’, Wired (2020), https://www.wired.com/story/stephen-wolfram-invites-you-to-solve-physics/.

40. On the priority dispute between Fredkin and Wolfram, see Levy, Artificial Life, and Keay Davidson, ‘Cosmic Computer – New Philosophy to Explain the Universe’, San Francisco Chronicle, 1 July 2002, https://www.stephenwolfram.com/media/cosmic-computer-new-philosophy-explain-universe/.

41. Steven Wolfram, 2002, A New Kind of Science, Wolfram Media, Champagne, Ill.

42. https://www.wolframscience.com/reference/notes/876b.

43. See Wolfram, A New Kind of Science.

44. Matthew Cook, ‘Universality in Elementary Cellular Automata’, Complex Systems, 15 (2004), pp. 1–40.

45. Steven Wolfram, 1984, ‘Universality and Complexity in Cellular Automata’, Physica D, 10(1–2), pp. 1–35.

46. Steven Wolfram, 2002, A New Kind of Science, Wolfram Media, Champagne, Ill. https://www.wolframscience.com/nks/.

47. Steven Levy, 2002, ‘The Man Who Cracked the Code to Everything …’, Wired, 1 June 2002, https://www.wired.com/2002/06/wolfram/.

48. Steven Levy, ‘Great Minds, Great Ideas’, Newsweek, 27 May 2002, p. 59, https://www.newsweek.com/great-minds-great-ideas-145749.

49. https://writings.stephenwolfram.com/2020/04/finally-we-may-have-a-path-to-the-fundamental-theory-of-physics-and-its-beautiful/.

50. See Wolfram’s Registry of Notable Universes, https://www.wolframphysics.org/universes/.

51. Adam Becker, ‘Physicists Criticize Stephen Wolfram’s “Theory of Everything”’, Scientific American, https://www.scientificamerican.com/article/physicists-criticize-stephen-wolframs-theory-of-everything/.

52. Franz L. Alt, 1972, ‘Archaeology of Computers Reminiscences, 1945–1947’, Communications of the ACM, 15(7) (1972), pp. 693–4, doi:https://doi.org/10.1145/361454.361528.

53. For more on Barricelli see Dyson, Turing’s Cathedral, chapter 12; Robert Hackett, ‘Meet the Father of Digital Life’, Nautilus, 12 June 2014, https://nautil.us/issue/14/mutation/meet-the-father-of-digital-life; and Alexander, R. Galloway, Creative Evolution, http://cultureandcommunication.org/galloway/pdf/Galloway-Creative_Evolution-Cabinet_Magazine.pdf.

54. The experiments using the IAS computer are described in Nils Aall Barricelli, ‘Numerical Testing of Evolution Theories. Part I: Theoretical Introduction and Basic Tests’, Acta Biotheoretica, 16 (1963), pp. 69–98.

55. Ibid.

56. Nils Aall Barricelli to John von Neumann, 22 October 1953, member file on Barricelli, IAS School of Mathematics, members, Ba–Bi, 1933–1977, IAS Archives.

57. Jixing Xia et al., ‘Whitefly Hijacks a Plant Detoxification Gene That Neutralizes Plant Toxins’, Cell, 25 March 2021, https://doi.org/10.1016/j.cell.2021.02.014.

58. Quoted in Levy, Artificial Life.

59. Christopher G. Langton, ‘Self-reproduction in Cellular Automata’, Physica 10D (1984), pp. 135–44.

60. Quoted in Levy, Artificial Life.

61. Quoted in ibid.

62. Christopher G. Langton, 1990, ‘Computation at the Edge of Chaos: Phase Transitions and Emergent Computation’, Physica D, 42 (1990), pp. 12–37.

63. Christopher G. Langton (ed.), 1989, Artificial Life, Santa Fe Institute Studies in the Sciences of Complexity, vol. 6, Addison-Wesley, Reading, Mass.

64. D. G. Gibson et al., ‘Creation of a Bacterial Cell Controlled by a Chemically Synthesized Genome’, Science, 329 (2010), pp. 52–6.

65. See, for example, Nicholas Wade, ‘Researchers Say They Created a “Synthetic Cell”’, New York Times, 20 May 2010.

66. Clyde A. Hutchison III et al., ‘Design and Synthesis of a Minimal Bacterial Genome’, Science, 351 (2016), aad625.

67. Marian Breuer et al., 2019, ‘Essential Metabolism for a Minimal Cell’, eLife, 8 (2019), doi:10.7554/eLife.36842.

68. Kendall Powell, 2018, ‘How Biologists Are Creating Life-like Cells from Scratch’, Nature, 563 (2018), pp. 172–5.

69. Eric Drexler, 1986, Engines of Creation, Doubleday, New York.

70. Lionel S. Penrose, ‘Self-Reproducing Machines’, Scientific American, 200(6) (1959), pp. 105–14. Videos of Penrose’s models in action can be viewed at: https://www.youtube.com/watch?v=2_9ohFWR0Vs and https://www.youtube.com/watch?v=1sIph9VrmpM.

71. Homer Jacobson, ‘On Models of Reproduction’, American Scientist, 46(3) (1958), pp. 255–84.

72. Edward F. Moore, 1956, ‘Artificial Living Plants’, Scientific American, 195(4) (1956), pp. 118–26.

73. Dyson, Disturbing the Universe.

74. Freitas and Merkle, Kinematic Self-Replicating Machines.

75. Robert A. Freitas Jr, 1980, ‘A Self-Reproducing Interstellar Probe’, Journal of the British Interplanetary Society, 33 (1980), pp. 251–64.

76. R. A. Freitas and W. P. Gilbreath (eds.), 1982, Advanced Automation for Space Missions, NASA Conference Publications CP-2255 (N83-15348), https://en.wikisource.org/wiki/Advanced_Automation_for_Space_Missions.

77. Richard Laing, ‘Automaton Models of Reproduction by Self-inspection’, Journal of Theoretical Biology, 66(3) (1977), pp. 437–56.

78. Olivia Brogue and Andreas M. Hein, ‘Near-term Self-replicating Probes – A Concept Design’, Acta Astronautica, published online 2 April 2021, https://doi.org/10.1016/j.actaastro.2021.03.004.

79. R. T. Fraley et al., ‘Expression of Bacterial Genes in Plant Cells’, Proceedings of the National Academy of Sciences, USA, 80(15) (1983), pp. 4803–7.

80. Drexler, Engines of Creation.

81. Jim Giles, 2004, ‘Nanotech takes small step towards burying ‘grey goo’’, Nature, 429, pp. 591.

82. Drexler, Engines of Creation.

83. ‘Nanotechnology: Drexler and Smalley Make the Case For and Against “Molecular Assemblers”’, Chemical and Engineering News, 81(48) (2003), pp. 37–42.

84. Salma Kassem et al., ‘Steroedivergent Synthesis with a Programmable Molecular Machine’, Nature, 549(7672) (2017), pp. 374–8.

85. A. H. J. Engwerda and S. P. Fletcher, ‘A Molecular Assembler That Produces Polymers’, Nature Communications, 11 (2020), https://doi.org/10.1038/s41467-020-17814-0.

86. Thomas C. Schelling, ‘Dynamic Models of Segregation’, Journal of Mathematical Sociology, 1 (1971), pp. 143–86.

87. ‘Parable of the Polygons’ is a game based on Schelling’s model: https://ncase.me/polygons/.

88. Schelling, ‘Dynamic Models of Segregation’.

89. Ulam, Adventures of a Mathematician.

90. Goldstine, The Computer from Pascal to von Neumann.

91. Robert Jastrow, 1981, The Enchanted Loom: Mind in the Universe, Simon and Schuster, New York.

92. https://www.gsmaintelligence.com/data/.

93. Jeremy Bernstein, ‘John von Neumann and Klaus Fuchs: An Unlikely Collaboration’, Physics in Perspective, 12 (2010), pp. 36–50.

94. Philip J. Hilts, 1982, Scientific Temperaments: Three Lives in Contemporary Science, Simon and Schuster, New York.

95. Dyson, Turing’s Cathedral.

96. John von Neumann, 2012 (first published 1958), The Computer and the Brain, Yale University Press, New Haven.

97. Von Neumann, The Computer and the Brain.

98. Robert Epstein, ‘The Empty Brain’, Aeon, 18 May 2016, https://aeon.co/essays/your-brain-does-not-process-information-and-it-is-not-a-computer.

99. Stan isław Ulam,’John von Neumann 1903–1957’, Bulletin of the American Mathematical Society, 64(3) (1958), pp. 1–49.

100. John von Neumann, Documentary Mathematical Association of America, 1966. Many thanks to David Hoffman, the film’s producer, for sending me the DVD in 2019. Now available to watch here: https://archive.org/details/JohnVonNeumannY2jiQXI6nrE.

101. Macrae, John von Neumann.

102. ‘Benoît Mandelbrot – Post-doctoral Studies: Weiner and Von Neumann (36/144)’, Web of Stories – Life Stories of Remarkable People, https://www.youtube.com/watch?v=U9kw6Reml6s.

103. https://rjlipton.wpcomstaging.com/the-gdel-letter/. Also see Richard J. Lipton, 2010, The P=NP Question and Gödel’s Lost Letter, Springer, New York.

104. John von Neumann to Marina von Neumann, 19 April 1955, quoted in von Neumann Whitman, The Martian’s Daughter.

105. Quoted in Dyson, Turing’s Cathedral.

106. Quoted in ibid.

107. Email to author.

108. John von Neumann, Documentary Mathematical Association of America, 1966.

109. Ulam, Adventures of a Mathematician.

110. Quoted in Dyson, Turing’s Cathedral.

111. Quoted in von Neumann Whitman, The Martian’s Daughter.

EPILOGUE: THE MAN FROM WHICH FUTURE?

1. Frank, J. Tipler, ‘Extraterrestrial Beings Do Not Exist’, Quarterly Journal of the Royal Astronomical Society, 21(267) (1981).

2. Ulam, ‘John von Neumann 1903–1957’.

3. Heims, John von Neumann and Norbert Wiener.

4. Von Neumann to Rudolf Ortvay, 29 March 1939, in von Neumann, Selected Letters.

5. ‘Benoît Mandelbrot – A Touching Gesture by Von Neumann’, Web of Stories – Life Stories of Remarkable People, https://www.youtube.com/watch?v=wu6vGDk5kzY.

6. Interview with author, 14 January, 2019.

7. John von Neumann, ‘Can We Survive Technology?’, Fortune, June 1955.