Summary of contributions

1. Mathematical foundations for quantum mechanics (aged 22)
2. Mathematics of ballistics & explosions (in the 30s when sensed war was looming)
3. Analysis of the structure of self-replication which preceded the discovery of the structure of DNA
4. Manhattan project
5. Founded Game theory
6. Building the ENIAC, the first digital computer
7. Cellular automata

Introduction

• “As a child, von Neumann absorbed Ancient Greek and Latin, and spoke French, German and English as well as his native Hungarian. He devoured a forty-five-volume history of the world and was able to recite whole chapters verbatim decades later.”
• Grew up in Budapest
• Today he is much less remembered than his Princeton colleagues Einstein & Godel

A teenager tackles a crisis in Mathematics

• PhD thesis at 17
• “The axiomatization of mathematics, on the model of Euclid’s Elements, had reached new levels of rigour and breadth at the end of the 19th century, particularly in arithmetic, thanks to the axiom schema of Richard Dedekind and Charles Sanders Peirce, and in geometry, thanks to Hilbert’s axioms.[102] But at the beginning of the 20th century, efforts to base mathematics on naive set theory suffered a setback due to Russell’s paradox (on the set of all sets that do not belong to themselves).[103] The problem of an adequate axiomatization of set theory was resolved implicitly about twenty years later by Ernst Zermelo and Abraham Fraenkel. Zermelo–Fraenkel set theory provided a series of principles that allowed for the construction of the sets used in the everyday practice of mathematics, but did not explicitly exclude the possibility of the existence of a set that belongs to itself. In his doctoral thesis of 1925, von Neumann demonstrated two techniques to exclude such sets—the axiom of foundation and the notion of class.”

• “Hilbert’s dreams of a perfect mathematics were soon to be crushed. Within a decade, some of the brightest minds in mathematics answered his call. They would show that mathematics was neither complete nor consistent nor decidable. Shortly after Hilbert’s death in 1943, however, his failed programme would bear unexpected fruit. Hilbert had driven mathematicians to think extraordinarily systematically about the nature of problems that were or were not solvable through step-by-step, mechanical procedures. Through von Neumann, that abstruse pursuit would help to birth a truly revolutionary machine: the modern computer.”
  • Godel incompleteness theorems
  • But Hilbert’s notations still useful for building computers

The Quantum evangelist

• “As well as the vibrant nightlife of Berlin, the scientific culture was second to none. German, not English, was the language of science in the 1920s. Practically all the founding papers of quantum mechanics were written in it. There was a flood of congresses and conferences for young researchers to attend. Academic talks would often spill over into coffee houses and bars. ‘The United States in those years was a bit like Russia: a large country without first-rate scientific training or research,’ Wigner told an interviewer in 1988. ‘Germany was then the greatest scientific nation on earth.’”
• “The existence of GPS, computer chips, lasers and electron microscopes attest that quantum theory works beautifully.”
  • “At bottom, the entire computer industry is built on quantum mechanics. Modern semiconductor-based electronics rely on the band structure of solid objects. This is fundamentally a quantum phenomenon, depending on the wave nature of electrons, and because we understand that wave nature, we can manipulate the electrical properties of silicon. Mixing in just a tiny fraction of the right other elements changes the band structure and thus the conductivity; we know exactly what to add and how much to use thanks to our detailed understanding of the quantum nature of matter.” (link)

• “What von Neumann then proceeds to prove is that dispersive ensembles in quantum mechanics are
  homogeneous. All of the particles in the ensemble are in the same quantum superposition of states until there’s a measurement. As he has already shown that hidden variables would mean that ensembles cannot, in general, be homogeneous, von Neumann can rule them out. Von Neumann’s proof electrified those who were leaning towards the Copenhagen interpretation. As word spread that the young genius had decisively rejected hidden variable theories, ‘Von Neumann was hailed by his followers and credited even by his opponents,’ says the historian Max Jammer.53 By this time, von Neumann was enjoying the comforts of a new life in the United States”
• “Everett first hit upon his solution ‘after a slosh or two of sherry’ with his flatmate Charles Misner and Aage Petersen, Bohr’s assistant, who was visiting Princeton at the time.76 Von Neumann’s approach had been to distil from the physics the bare mathematical principles required to explain quantum phenomena, then, using only these laws, infer whatever one could about the nature of the quantum realm. But Everett realized von Neumann had not done that in his treatment of measurement. Instead, von Neumann had noted that observers never see a quantum superposition of states, only a single classical state. Then, he had assumed that at some point, a transition from quantum to classical must take place. Nothing in the maths necessitated wave function collapse. What, Everett wondered, if we really follow the maths to its logical conclusion? What if there is no collapse at all? Everett was led to a startling result. With no artificial boundary to constrain it, there is quantumness everywhere. All the particles in the universe are entwined in a single massive superposition of all possible states. Everett called this the ‘universal wave function’. Why, then, does an observer perceive only one outcome from a measurement and not some quantum fuzz of possibilities? That is where the ‘many worlds’ come in. Everett proposed that every time a measurement is made, the universe ‘splits’ to create a crop of alternative realities, in which each of the possibilities play out. (So Schrödinger’s cat is alive in one universe and dead in another. Or in several.)”

• “Today, we know that Dirac was almost certainly wrong, and the hopes of Einstein were misplaced. There may yet be a better theory than quantum mechanics, but thanks to Bell’s work and the experiments that followed, we know that non-locality will be part and parcel of it. Conversely, von Neumann’s cautious conservatism appears with hindsight the correct attitude. There is no proof yet for a deeper alternative to the quantum theory that von Neumann helped to forge more than a hundred years ago. All the experiments to date have revealed no hidden variables, nothing to suggest causality reasserts itself at some deeper level. As far as we know, it’s quantum all the way down.”

Project Y and the Super

• “Guns used by the Central and Allied powers routinely lobbed shells thousands of feet in the air and over distances of several miles. Germany’s infamous Paris gun had a staggering firing range of over 70 miles. But a projectile hurled high and long in this way flies through progressively thinner air as it gains altitude, and so experiences less resistance to its motion. A failure to adequately account for this meant that early efforts to calculate trajectories were wildly off, and shells flew far beyond their intended targets. Throw in some more complications – a moving target, boggy ground and so forth – and the equations of motion often become impossible to solve exactly (in mathematical terms they become ‘non-linear’), forcing mathematicians to approximate. That required arithmetic and lots of it: hundreds of multiplications for a single trajectory. What was needed, but not available (yet), was a device able to perform such calculations accurately at the rate of thousands per second. Some of the earliest room-sized computers would be built to solve exactly this problem.”

• War prediction — “Still fighting for his commission to the Army and increasingly busy with military work, von Neumann could only worry from afar. Klári’s parents and Johnny’s mother and brother, Michael, were safely dispatched (Nicholas appears to have emigrated earlier) but Klári stayed behind to resolve some family affairs. ‘For God’s sake do not go to Pest,’ von Neumann begged on 10 August, ‘and get out of Europe by the beginning of Sept! I mean it!’17 Klári sailed from Southampton aboard the SS Champlain on 30 August.18 Germany invaded Poland the next day.”

• “For twelve months from September 1941, he was a member of the National Defence Research Committee (NDRC), which together with its successor, the Office of Scientific Research and Development (OSRD) would coordinate nearly all scientific research related to war. The government agency was conceived and led by Vannevar Bush, an influential engineer. Under Bush, who reported directly to the president, the NDRC and, later, the OSRD, would support a range of projects including radar, guided missiles and the proximity fuze. The most important work the agency would support would be research on an atom bomb.”
  • Vannevar Bush who we know well from The Dream Machine!

• “The arm of the OSRD charged with nuclear work morphed into what became known as the Manhattan Project. The massive effort to build the atom bomb, codenamed Project Y, would cost the US $2 billion (more than $20 billion today) and at its height employ more than 100,000 people.24 In September 1942, the forty-six-year-old Army engineer Leslie Groves was appointed to lead it. The very next month, Groves chose Oppenheimer to head the top-secret laboratory that would develop the bomb.”

• “Oppenheimer would earn a reputation at Los Alamos for making the right call at the right time. His next decision would prove to be inspired. ‘We are in what can only be described as a desperate need of your help,’ he wrote to von Neumann in July. ‘We have a good many theoretical people working here, but I think that if your usual shrewdness is a guide to you about the probable nature of our problems you will see why even this staff is in some respects critically inadequate.’ He invited von Neumann to ‘come, if possible as a permanent, and let me assure you, honored member of our staff’, adding that ‘a visit will give you a better idea of this somewhat Buck Rogers project than any amount of correspondence’.29 Von Neumann accepted.”

• “At S-10000, Bainbridge authorized the start of the twenty-minute countdown at 5.09.45 a.m. A few hundred scientists and VIPs had gathered at Compañia Hill, the designated observation point 20 miles northwest of the tower. Von Neumann was among them. The scientists wagered on the explosive yield of the bomb. Some still thought zero was the most likely figure. Downplaying the chance of success, Oppenheimer chose 300 tons TNT equivalent. Teller, an optimist where bombs were concerned, picked 45,000 tons and passed around a bottle of suntan lotion. The sight of scientists slapping on sunscreen in the pitch dark perturbed some of the assembled VIPs. At 5.29 a.m., just before sunrise, an electric pulse from 10,000 yards away detonates the thirty-two atomic lenses arrayed around the pit. The shock waves, initially convex, become concave as they travel through the carefully configured layers of explosives, merging, finally, into a rapidly contracting spherical front. The force of the implosion compresses the core to less than half its original size. The shock wave then reaches the centre of the pit, crushing the initiator and blending polonium with beryllium. In the following ten billionths of a second, nine or ten neutrons are liberated. It is enough. About a kilogram of the liquefied plutonium and some of the uranium tamper, fissions. A gram of matter is converted to energy. Then everything vaporises.”

• “‘That was at least 5,000 tons and probably a lot more,’ von Neumann said quietly. Fermi had ripped a sheet of paper into pieces, letting them fall when the air blast hit. They were blown about 8 feet. Consulting a table he had prepared earlier, he declared the blast equivalent to 10,000 tons of TNT. They were both off. The best estimates of Trinity’s power put the figure somewhere between 20,000 and 22,000 tons. Oppenheimer reached for poetry, recalling a verse from ancient Hindu scripture, the Bhagavad Gita, which he had read in the original Sanskrit. ‘Now I am become Death,’ he said, ‘the destroyer of worlds.’ Bainbridge was pithier. ‘Now we are all sons of bitches,’ he told Oppenheimer. ‘The war is over,’ Groves’ deputy, Thomas Farrell, declared at Base Camp, 5 miles south of S-10000. ‘Yes,’ Groves replied, ‘after we drop two bombs on Japan.’”

• “Von Neumann’s votes in the end were for a) Kyoto, b) Hiroshima, c) Yokohama and d) Kokura arsenal – exactly the recommendations the committee as a whole were to make later.37 Stimson was completely opposed to bombing Kyoto, the cultural heart of Japan, for eleven centuries its capital, and a city he had visited during his honeymoon. The port of Nagasaki was chosen instead. By August, Yokohama had been bombed so thoroughly that it was removed from the list, leaving only Hiroshima, Kokura arsenal and Nagasaki.”
• (They dropped the bombs quite high up in the air due to von Neumann – and others – calculations about greatest impact)

The convoluted birth of the first modern computer

• “One evening in the summer of 1944, on the platform of Aberdeen train station, Goldstine saw someone he recognized – someone whose lectures he had once attended and who was by now the most famous scientist in America after Einstein. He introduced himself to von Neumann, and the two began to chat while waiting for their train. Goldstine explained that his role involved liaising with the Moore School in Philadelphia and mentioned the project they were working on together: an electronic computer capable of more than 300 multiplications per second. At that, Goldstine says, ‘the whole atmosphere of our conversation changed from one of relaxed good humour to one more like the oral examination for the doctor’s degree in mathematics.’”

• “Now we think of a personal computer as one which you carry around with you,’ says mathematician Harry Reed, who joined the project in 1950. ‘The ENIAC was actually one that you kind of lived inside.’10 The ENIAC was the brainchild of John W. Mauchly, a former physics teacher whose dreams of a research career had been dashed by the Great Depression. He had won a scholarship to study at Johns Hopkins University in Baltimore, Maryland and without bothering to finish his undergraduate degree he earned a PhD in physics from there in 1932. He worked briefly as a research assistant but had the misfortune of beginning his hunt for a university post during one of the longest periods of economic decline in recent history. Mauchly’s academic job hunt stalled. Instead, he had to settle for employment at Ursinus, a small liberal arts college in Pennsylvania, where he became the head and, indeed, the sole member of the physics department. He was still there when war broke out.”

• “Curiously, von Neumann was mentally prepared for this cutting-edge contribution to computing by his involvement in the foundational crisis that had riven mathematics in the early twentieth century. An unlikely turn of history would entangle the intellectual roots of the modern computer with Hilbert’s challenge to prove that mathematics was complete, consistent and decidable. Soon after Hilbert issued his challenge, the intellectually dynamic but psychologically frail Austrian mathematician Kurt Gödel would demonstrate that it is impossible to prove that mathematics is either complete or consistent. Five years after Gödel’s breakthrough, a twenty-three-year-old Turing would attack Hilbert’s ‘decision problem’ (Entscheidungsproblem) in a way completely unanticipated by any other logician, conjuring up an imaginary machine to show that mathematics is not decidable. The formalisms of these two logicians would help von Neumann crystallize the structure of the modern computer.”

• “The result of his musings, First Draft of a Report on the EDVAC, would become the most influential document in the history of computing.16 ‘Today,’ says computer scientist Wolfgang Coy, ‘it is considered the birth certificate of modern computers.’17 Just over a decade after Hilbert’s quest for a perfectible mathematics had run aground, his programme would unexpectedly bear spectacular fruit.”

• “Von Neumann kept thinking about Gödel’s proof after the Königsberg conference. On 20 November, he wrote excitedly to Gödel. ‘Using the methods you employed so successfully … I achieved a result that seems to me to be remarkable, namely,’ von Neumann continued with a flourish, ‘I was able to show that the consistency of mathematics is unprovable.’ Von Neumann promised to send him his proof, which he said would soon be ready for publication. But it was too late. Gödel, probably sensing that von Neumann was hot on his heels after their conversation in Königsberg, had already sent his paper to a journal.28 He now sent a copy to von Neumann. Crestfallen, von Neumann wrote back, thanking him. ‘As you have established the theorem on the unprovability of consistency as a natural continuation and deepening of your earlier results,’ he added, ‘I clearly won’t publish on this subject.’ So saying, von Neumann quietly passed up the opportunity to stake a claim on the most remarkable result in mathematical history.”

• EDVIAC inventors (Eckert-Mauchly Computer Corporation) vs von Neumann — “What had become the longest trial in the history of the federal court system concluded with the ruling that the most valuable invention of the twentieth century could not be patented. The open source movement, born a decade or so later, would soon shun corporate secrecy, lauding the benefits of freely sharing information to drive forward innovation. Thanks to von Neumann those principles were baked into computing from the very beginning.”

A theory of games

• “Von Neumann founded the field of game theory as a mathematical discipline.^[270] He proved his minimax theorem in 1928. It establishes that in zero-sum games with perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses. When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy that will result in the minimization of his maximum loss” (wikipedia)
• “Game theory quickly proved itself worthy of the accolade in spectacular style. The same year the three laureates would receive their gold medals from the king of Sweden, the US government was preparing to auction bands of the radio spectrum to telecoms firms. Thousands of licences worth billions of dollars were at stake. Many past sell-offs had flopped.72 In New Zealand, a botched ‘second-price’ auction, in which the winner only pays the second-highest bid, resulted in a firm that bid NZ$7 million paying NZ$5,000; and a university student picking up a licence to run a television network for a small city for nothing – because no one else had bid.”

Misc

• “In the eleven months that he was hospitalized, von Neumann received a stream of visitors – family, friends, collaborators and the military men with whom he had spent so much time in the latter years of his life. Strauss recalled ‘the extraordinary picture, of sitting beside the bed of this man … who had been an immigrant, and there surrounding him, were the Secretary of Defense, the Deputy Secretary of Defense, the Secretaries of Air, Army, Navy, and the Chiefs of Staff’.100 He left hospital briefly in a wheelchair to accept the Medal of Freedom from President Eisenhower. ‘I wish I could be around long enough to deserve this honour,’ he told the president. ‘You will be with us for a long time,’ Eisenhower reassured him, ‘we need you.’”

• “But then Gödel, never very tactful, moves on to more important matters. ‘Since you now, as I hear, are feeling stronger, I would like to allow myself to write you about a mathematical problem, of which your opinion would very much interest me …’ He then launches into the description of a Turing machine, which if ever shown to actually exist, ‘would have consequences of the greatest importance’. Namely, despite Turing’s negative answer to the Entscheidungsproblem, Gödel said, it would imply that the discovery of some mathematical proofs can be automated. It is not known what von Neumann made of Gödel’s problem or if he even saw the letter. Klári was by now answering most correspondence on his behalf. What Gödel was describing is now known as the P versus NP problem and would only be rigorously stated in 1971. Today, it is one of the most important unsolved problems in mathematics.”
• Von Neumann died age 53. Some think in part due to radiation from the Manhattan project tests.

Also read/watch

• https://mathshistory.st-andrews.ac.uk/Biographies/Von_Neumann/