Von Neumann Quotes

If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.

Young man, in mathematics you don't understand things. You just get used to them.

There's no sense in being precise when you don't even know what you're talking about

It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature.

In mathematics you don’t understand things. You just get used to them. —John von Neumann

Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.

Running overtime is the one unforgivable error a lecturer can make. After fifty minutes (one microcentury as von Neumann used to say) everybody's attention will turn elsewhere.

Von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of Von Neumann’s advice. It’s made me a very happy man ever since. But it was Von Neumann who put the seed in that grew into my active irresponsibility!

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn’t say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

Kurt Gödel's achievement in modern logic is singular and monumental – indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement." —John von Neumann

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work - that is correctly to describe phenomena from a reasonably wide area. Furthermore, it must satisfy certain esthetic criteria - that is, in relation to how much it describes, it must be rather simple.

Se la gente non crede che la matematica è semplice, è solo perché non capisce quanto è complicata la vita.

Can we survive technology?

It is only proper to realize that language is largely a historical accident.

...where were answers to the truly deep questions? Religion promised those, though always in vague terms, while retreating from one line in the sand to the next. Don't look past this boundary, they told Galileo, then Hutton, Darwin, Von Neumann, and Crick, always retreating with great dignity before the latest scientific advance, then drawing the next holy perimeter at the shadowy rim of knowledge.

A large part of mathematics which becomes useful developed with absolutely no desire to be useful, and in a situation where nobody could possibly know in what area it would become useful; and there were no general indications that it ever would be so.

The spectacular thing about Johnny [von Neumann] was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast. And like the modern computer, which no longer bothers to retrieve the logarithm of 11 from its memory (but, instead, computes the logarithm of 11 each time it is needed), Johnny didn't bother to remember things. He computed them. You asked him a question, and if he didn't know the answer, he thought for three seconds and would produce and answer.

An element which stimulates itself will hold a stimulus indefinitely.

The thing that Von Neumann had, which I’ve noticed that other geniuses have, is the ability to pick out, in a particular problem, the one crucial thing that’s important.

Von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in. So I have developed a very powerful sense of social irresponsibility as a result of Von Neumann’s advice. It’s made me a very happy man ever since.

The decisions we make in our lives—in business, saving and spending, health and lifestyle choices, raising our children, and relationships—easily fit von Neumann’s definition of “real games.” They involve uncertainty, risk, and occasional deception, prominent elements in poker. Trouble follows when we treat life decisions as if they were chess decisions.

In this sense, an object is of the highest degree of complexity if it can do very difficult and involved things.

That’s not what I am looking for. John Louis von Neumann said, “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” Mathematics may well be simple, but the complexities of race and culture are often irreducible. They cannot be wholly addressed in a single essay or book or television show or movie.

PROFESSOR EMERITUS WOTAN Ulm, of the University of Oxford East 5, author of the bestselling if controversial memoir Peer Reviewers and Other Idiots: A Life In Academia, had consented to give a recorded lecture on von Neumann replicators to be carried as briefing material on the US Navy twain USS Brian Cowley.

In the Game of Life, as in our world, self-reproducing patterns are complex objects. One estimate, based on the earlier work of mathematician John von Neumann, places the minimum size of a self-replicating pattern in the Game of Life at ten trillion squares—roughly the number of molecules in a single human cell.

Von Neumann told Shannon to call his measure entropy, since "no one knows what entropy is, so in a debate you will always have the advantage.

Most people avoid thinking if they can, some of us are addicted to thinking, but Von Neumann actually enjoyed thinking, maybe even to the exclusion of everything else.

The term bit (the contraction, by 40 bits, of “binary digit”) was coined by statistician John W. Tukey shortly after he joined von Neumann’s project in November of 1945.

Von Neumann was in many ways a traditional mathematician, who (like Turing) believed he needed to turn to partial differential equations in describing natural systems.

When part of this ecosystem was lacking, such as for John Atanasoff at Iowa State or Charles Babbage in the shed behind his London home, great concepts ended up being consigned to history’s basement. And when great teams lacked passionate visionaries, such as Penn after Mauchly and Eckert left, Princeton after von Neumann, or Bell Labs after Shockley, innovation slowly withered.

It is easy to imagine an advanced civilization sending out von Neumann probes to explore the galaxy. On arrival in a star system, one such machine would mine raw materials from asteroids or comets in order to replicate.

THE VASTNESS OF OUR MEMORY Holography also explains how our brains can store so many memories in so little space. The brilliant Hungarian-born physicist and mathematician John von Neumann once calculated that over the course of the average human lifetime, the brain stores something on the order of 2. 8 x 1020 (280, 000, 000, 000, 000, 000, 000) bits of information. This is a staggering amount of information, and brain researchers have long struggled to come up with a mechanism that explains such a vast capability. Interestingly, holograms also possess a fantastic capacity for information storage. By changing the angle at which the two lasers strike a piece of photographic film, it is possible to record many different images on the same surface. Any image thus recorded can be retrieved simply by illuminating the film with a laser beam possessing the same angle as the original two beams. By employing this method researchers have calculated that a one-inch-square of film can store the same amount of information contained in fifty Bibles!

John von Neumann draws attention to what seemed to him a contrast. He remarked that for simple mechanisms, it is often easier to describe how they work than what they do, while for more complicated mechanisms, it is usually the other way around.

I have known a great many intelligent people in my life. I knew Max Planck, Max von Laue, and Wemer Heisenberg. Paul Dirac was my brother-in-Iaw; Leo Szilard and Edward Teller have been among my closest friends; and Albert Einstein was a good friend, too. And I have known many of the brightest younger scientists. But none of them had a mind as quick and acute as Jancsi von Neumann. I have often remarked this in the presence of those men, and no one ever disputed me. [...] But Einstein's understanding was deeper than even Jancsi von Neumann's. His mind was both more penetrating and more original than von Neumann's. And that is a very remarkable statement. Einstein took an extraordinary pleasure in invention. Two of his greatest inventions are the Special and General Theories of Relativity; and for all of Jancsi's brilliance, he never produced anything so original.

And' and 'or' are the basic operations of logic. Together with 'no' (the logical operation of negation) they are a complete set of basic logical operations—all other logical operations, no matter how complex, can be obtained by suitable combinations of these.

There is a belief, current in many countries, which has been elevated to the rank of an official article of faith in the United States, that free competition is itself a homeostatic process: that in a free market the individual selfishness of the bargainers, each seeking to sell as high and buy as low as possible, will result in the end in a stable dynamics of prices, and with redound to the greatest common good. This is associated with the very comforting view that the individual entrepreneur, in seeking to forward his own interest, is in some manner a public benefactor and has thus earned the great rewards with which society has showered him. Unfortunately, the evidence, such as it is, is against this simpleminded theory. The market is a game, which has indeed received a simulacrum in the family game of Monopoly. It is thus strictly subject to the general theory of games, developed by von Neumann and Morgenstern. This theory is based on the assumption that each player, at every stage, in view of the information then available to him, plays in accordance with a completely intelligent policy, which will in the end assure him of the greatest possible expectation of reward.

In any conceivable method ever invented by man, an automaton which produces an object by copying a pattern, will go first from the pattern to a description to the object. It first abstracts what the thing is like, and then carries it out. It's therefore simpler not to extract from a real object its definition, but to start from the definition.

Von Neumann, by contrast, wore a three-piece suit at almost all times, including on a donkey ride down the Grand Canyon; even as a student he was so well dressed that, upon first meeting him, the mathematician David Hilbert reportedly had but one question: Who is his tailor?45

It has become commonplace to conclude that humans are simply irrational—more Homer Simpson than Mr. Spock, more Alfred E. Neuman than John von Neumann. And, the cynics continue, what else would you expect from descendants of hunter-gatherers whose minds were selected to avoid becoming lunch for leopards?

They had already put together an entire committee to choose the best targets, but it was actually von Neumann who convinced them that they shouldn’t detonate the devices at ground level, but higher up in the atmosphere, since that way the blast wave would cause incomparably larger damage. He even calculated the optimal height himself—six hundred meters, about two thousand feet. And that is exactly how high our bombs were when they exploded above the roofs of those quaint wooden houses in Hiroshima and Nagasaki.

A system of logical instructions that an automaton can carry out and which causes the automaton to perform some organized task is called a code.

Von Neumann gave me an interesting idea: that you don’t have to be responsible for the world that you’re in.

Von Neumann was another innovator who stood at the intersection of the humanities and sciences.

Si alguna vez surge una raza mentalmente sobrehumana —dijo en cierta ocasión Edward Teller, el creador de la bomba de hidrógeno—, sus miembros se parecerán a Johnny von Neumann.

When we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics.

Real life consists of bluffing, of little tactics of deception, of asking yourself what is the other man going to think I mean to do,” von Neumann wrote. “And that is what games are about in my theory.

the groundbreakers in many sciences were devout believers. Witness the accomplishments of Nicolaus Copernicus (a priest) in astronomy, Blaise Pascal (a lay apologist) in mathematics, Gregor Mendel (a monk) in genetics, Louis Pasteur in biology, Antoine Lavoisier in chemistry, John von Neumann in computer science, and Enrico Fermi and Erwin Schrodinger in physics. That’s a short list, and it includes only Roman Catholics; a long list could continue for pages. A roster that included other believers—Protestants, Jews, and unconventional theists like Albert Einstein, Fred Hoyle, and Paul Davies—could fill a book.

ever accelerating progress of technology and changes in the mode of human life,” von Neumann explained to Stan Ulam, “gives the appearance of approaching some essential singularity in the history of the race.

Si Von Neumann y su equipo hubieran seguido liderando las innovaciones y las hubiesen hecho de dominio público, ¿habría conducido ese modelo de desarrollo de código abierto a un progreso más rápido de los ordenadores?

If a mentally superhuman race ever develops, its members will resemble Johnny von Neumann,” Edward Teller once said. “If you enjoy thinking, your brain develops. And that is what von Neumann did. He enjoyed the functioning of his brain.

If each of them creates another thousand robots, then we have a million. Then a billion. Then a trillion. In just a few generations, we can have an expanding sphere containing quadrillions of these devices, which scientists call von Neumann machines.

There’s a cellular automaton called TVC. After Turing, von Neumann and Chiang. Chiang’s version was N-dimensional. That leaves plenty of room for data within easy reach. In two dimensions, the original von Neumann machine had to reach further and further - and wait longer and longer - for each successive bit of data. In a six-dimensional TVC automaton, you can have a three-dimensional grid of computers, which keeps on growing indefinitely - each with its own three-dimensional memory, which can also grow without bound. And when the simulated TVC universe being run on the physical computer is suddenly shut down, the best explanation for what I’ve witnessed will be a continuation of that universe - an extension made out of dust. Maria could almost see it: a vast lattice of computers, a seed of order in a sea of random noise, extending itself from moment to moment by sheer force of internal logic, “accreting” the necessary building blocks from the chaos of non-space-time by the very act of defining space and time.

Nash’s equilibrium, when it exists, is that point where neither player can do any better, or have no regrets, given what the opponent has done. Neither can have regrets because of how the other person played the game. It may not be the best option for either player, but it’s the best under the circumstances. There isn’t always an equilibrium in a game, or a Nash equilibrium in a game theory matrix. However, if it exists, in many cases the Nash equilibrium is a far better outcome for both players than the von Neumann saddle point. In the Kelley apartment cleaning game-theory matrices above, the Nash equilibrium is for them both to clean. Consider his payoffs. He does much better if he cleans no matter what she decides to do (because 5.7 is much greater than -2.2). Now consider her payoffs. She also does better if she cleans no matter what he does (because 8.5 is much greater than -6.6). So they have a stable Nash equilibrium at the joint strategy = (Male Cleans, Female Cleans). Then neither of them can have regrets about that choice because with that choice neither of them can do any better, regardless of what the partner does. With the Nash equilibrium their strategy is to maximize one’s own gains even if it means maximizing the partner’s gains (as well as one’s own).

En palabras de George Dyson: «El computador de programa almacenado, tal y como lo imaginó Alan Turing y lo plasmó John von Neumann, diluyó la distinción entre números que significan cosas y números que hacen cosas. Nuestro universo nunca volvería a ser el mismo».

Von Neumann makes two important observations here: acceleration and singularity. The first idea is that human progress is exponential (that is, it expands by repeatedly multiplying by a constant) rather than linear (that is, expanding by repeatedly adding a constant).

Out of the prospering but vulnerable Hungarian Jewish middle class came no fewer than seven of the twentieth century’s most exceptional scientists: in order of birth, Theodor von Kármán, George de Hevesy, Michael Polanyi, Leo Szilard, Eugene Wigner, John von Neumann and Edward Teller.

Twenty minutes later, Three Body’s Von Neumann architecture human-formation computer had begun full operations under the Qin 1.0 operating system. “Run solar orbit computation software ‘Three Body 1.0’!” Newton screamed at the top of his lungs. “Start the master computing module! Load the differential calculus module! Load the finite element analysis module! Load the spectral method module! Enter initial condition parameters … and begin calculation!” The motherboard sparkled as the display formation flashed with indicators in every color. The human-formation computer began the long computation.

He was "a magician, a magician in the sense that he took what was given and simply forced the conclusions logically out of it, whether it was algebra, geometry, or whatever. He had some way of forcing out the results that made him different from the rest of the people." Israel Halperin about von Neumann

singularity, a term that von Neumann coined and the futurist Ray Kurzweil and the science fiction writer Vernor Vinge popularized, which is sometimes used to describe the moment when computers are not only smarter than humans but also can design themselves to be even supersmarter, and will thus no longer need us mortals.

The other line of argument, which leads to the opposite conclusion, arises from looking at artificial automata. Everyone knows that a machine tool is more complicated than the elements which can be made with it, and that, generally speaking, an automaton A, which can make an automaton B, must contain a complete description of B, and also rules on how to behave while effecting the synthesis. So, one gets a very strong impression that complication, or productive potentiality in an organization, is degenerative , that an organization which synthesizes something is necessarily more complicated, of a higher order, than the organization it synthesizes. This conclusion, arrived at by considering artificial automaton, is clearly opposite to our early conclusion, arrived at by considering living organisms.

There is reason to suspect that our predilection for linear codes, which have a simple, almost temporal sequence, is chiefly a literary habit, corresponding to our not particularly high level of combinatorial cleverness, and that a very efficient language would probably depart from linearity,” von Neumann suggested in 1949.

The general opinion in theoretical physics had accepted the idea that the principle of continuity ("natura non facit saltus"), prevailing in the microsoptic world, is merely simulated by an averaging process in a world which in truth is discontinuous by its very nature. This simulation is such that a man generally percieves the sum of many billions of elementary processes simultaneously, so that the leveling law of large numbers completely obscures the real nature of the individual processes.

The key to innovation—at Bell Labs and in the digital age in general—was realizing that there was no conflict between nurturing individual geniuses and promoting collaborative teamwork. It was not either-or. Indeed, throughout the digital age, the two approaches went together. Creative geniuses (John Mauchly, William Shockley, Steve Jobs) generated innovative ideas. Practical engineers (Presper Eckert, Walter Brattain, Steve Wozniak) partnered closely with them to turn concepts into contraptions. And collaborative teams of technicians and entrepreneurs worked to turn the invention into a practical product. When part of this ecosystem was lacking, such as for John Atanasoff at Iowa State or Charles Babbage in the shed behind his London home, great concepts ended up being consigned to history’s basement. And when great teams lacked passionate visionaries, such as Penn after Mauchly and Eckert left, Princeton after von Neumann, or Bell Labs after Shockley, innovation slowly withered.

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work.

According to one story, Von Neumann was asked to assist with the design of a new supercomputer, required to solve a new and important mathematical problem which was beyond the capacities of existing supercomputers. He asked to have the problem explained to him, solved it in moments with pen and paper and then turned down the request. Von

Anybody who looks at living organisms knows perfectly well that they can produce other organisms like themselves. This is their normal function, they wouldn't exist if they didn't do this, and it's not plausible that this is the reason why they abound in the world. In other words, living organisms are very complicated aggregations of elementary parts, and by any reasonable theory of probability or thermodynamics highly improbable. That they should occur in the world at all is a miracle of the first magnitude; the only thing which removes, or mitigates, this miracle is that they reproduce themselves. Therefore, if by any peculiar accident there should ever be one of them, from there on the rules of probability do not apply, and there will be many of them, at least if the milieu is reasonable. But a reasonable milieu is already a thermodynamically much less improbable thing. So, the operations of probability somehow leave a loophole at this point, and it is by the process of self-reproduction that they are pierced.

Vinge compares it to the Cold War strategy called MAD—mutually assured destruction. Coined by acronym-loving John von Neumann (also the creator of an early computer with the winning initials, MANIAC), MAD maintained Cold War peace through the promise of mutual obliteration. Like MAD, superintelligence boasts a lot of researchers secretly working to develop technologies with catastrophic potential. But it’s like mutually assured destruction without any commonsense brakes. No one will know who is ahead, so everyone will assume someone else is. And as we’ve seen, the winner won’t take all. The winner in the AI arms race will win the dubious distinction of being the first to confront the Busy Child.

Furthermore, it's equally evident that what goes on is actually one degree better than self-reproduction, for organisms appear to have gotten more elaborate in the course of time. Today's organisms are phylogenetically descended from others which were vastly simpler than they are, so much simpler, in fact, that it's inconceivable, how any kind of description of the latter, complex organism could have existed in the earlier one. It's not easy to imagine in what sense a gene, which is probably a low order affair, can contain a description of the human being which will come from it. But in this case you can say that since the gene has its effect only within another human organism, it probably need not contain a complete description of what is to happen, but only a few cues for a few alternatives. However, this is not so in phylogenetic evolution. That starts from simple entities, surrounded by an unliving amorphous milieu, and produce, something more complicated. Evidently, these organisms have the ability to produce something more complicated than themselves.

Natura non facis Saltus.

When we talk mathematics, we may be discussing a secondary language built on the primary language truly used by the central nervous system.

As a thought experiment, von Neumann's analysis was simplicity itself. He was saying that the genetic material of any self-reproducing system, whether natural or artificial, must function very much like a stored program in a computer: on the one hand, it had to serve as live, executable machine code, a kind of algorithm that could be carried out to guide the construction of the system's offspring; on the other hand, it had to serve as passive data, a description that could be duplicated and passed along to the offspring. As a scientific prediction, that same analysis was breathtaking: in 1953, when James Watson and Francis Crick finally determined the molecular structure of DNA, it would fulfill von Neumann's two requirements exactly. As a genetic program, DNA encodes the instructions for making all the enzymes and structural proteins that the cell needs in order to function. And as a repository of genetic data, the DNA double helix unwinds and makes a copy of itself every time the cell divides in two. Nature thus built the dual role of the genetic material into the structure of the DNA molecule itself.

In the seventy years since von Neumann effectively placed his “Draft Report” on the EDVAC into the public domain, the trend for computers has been, with a few notable exceptions, toward a more proprietary approach. In 2011 a milestone was reached: Apple and Google spent more on lawsuits and payments involving patents than they did on research and development of new products.64

Von Neumann at six joked with his father in classical Greek and had a truly photographic memory: he could recite entire chapters of books he had read.392 Edward Teller, like Einstein before him, was exceptionally late in learning—or choosing—to talk.393 His grandfather warned his parents that he might be retarded, but when Teller finally spoke, at three, he spoke in complete sentences.

Web 2.0 is our code word for the analog increasingly supervening upon the digital—reversing how digital logic was embedded in analog components, sixty years ago. Search engines and social networks are just the beginning—the Precambrian phase. “If the only demerit of the digital expansion system were its greater logical complexity, nature would not, for this reason alone, have rejected it,” von Neumann admitted in 1948.

In the first case it emerges that the evidence that might refute a theory can often be unearthed only with the help of an incompatible alternative: the advice (which goes back to Newton and which is still popular today) to use alternatives only when refutations have already discredited the orthodox theory puts the cart before the horse. Also, some of the most important formal properties of a theory are found by contrast, and not by analysis. A scientist who wishes to maximize the empirical content of the views he holds and who wants to understand them as clearly as he possibly can must therefore introduce other views; that is, he must adopt a pluralistic methodology. He must compare ideas with other ideas rather than with 'experience' and he must try to improve rather than discard the views that have failed in the competition. Proceeding in this way he will retain the theories of man and cosmos that are found in Genesis, or in the Pimander, he will elaborate them and use them to measure the success of evolution and other 'modern' views. He may then discover that the theory of evolution is not as good as is generally assumed and that it must be supplemented, or entirely replaced, by an improved version of Genesis. Knowledge so conceived is not a series of self-consistent theories that converges towards an ideal view; it is not a gradual approach to truth. It is rather an ever increasing ocean of mutually incompatible alternatives, each single theory, each fairy-tale, each myth that is part of the collection forcing the others in greater articulation and all of them contributing, via this process of competition, to the development of our consciousness. Nothing is ever settled, no view can ever be omitted from a comprehensive account. Plutarch or Diogenes Laertius, and not Dirac or von Neumann, are the models for presenting a knowledge of this kind in which the history of a science becomes an inseparable part of the science itself - it is essential for its further development as well as for giving content to the theories it contains at any particular moment. Experts and laymen, professionals and dilettani, truth-freaks and liars - they all are invited to participate in the contest and to make their contribution to the enrichment of our culture. The task of the scientist, however, is no longer 'to search for the truth', or 'to praise god', or 'to synthesize observations', or 'to improve predictions'. These are but side effects of an activity to which his attention is now mainly directed and which is 'to make the weaker case the stronger' as the sophists said, and thereby to sustain the motion of the whole.

> In effect, though Wiener didn't quite express it this way, cybernetics was offering an alternative to the Skinnerian worldview, in which human beings were just stimulus-response machines to be manipulated and conditioned for their own good. It was likewise offering an alternative to von Neumann's worldview, wherein human beings were unrealistically rational technocrats capable of anticipating, controlling, and managing their society with perfect confidence. Instead, cybernetics held out a vision of humans as neither gods nor clay but rather "machines" of the new kind, embodying purpose—and thus, autonomy. No, we were not the absolute masters of our universe; we lived in a world that was complex, confusing, and largely uncontrollable. But neither were we helpless. We were embedded in our world, in constant communication with our environment and one another. We had the power to act, to observe, to learn from our mistakes, and to grow. "From the point of view of cybernetics, the world is an organism," Wiener declared in his autobiography. "In such a world, knowledge is in its essence the process of knowing. . . . Knowledge is an aspect of life which must be interpreted while we are living, if it is to be interpreted at all. Life is the continual interplay between the individual and his environment rather than a way of existing under the form of eternity.

Your thoughts on numerology are most interesting,” Waterhouse says loudly, running Mr. Drkh off the rhetorical road. “I myself studied with Drs. Turing and von Neumann at the Institute for Advanced Studies in Princeton.” Father John snaps awake, and Mr. Drkh looks as if he’s just taken a fifty-caliber round in the small of his back. Clearly, Mr. Drkh has had a long career of being the weirdest person in any given room, but he’s about to go down in flames.

Did von Neumann understand the potential of the machines he helped to invent? Yes, he did. In reflective mood in 1955, he noted that the ‘over-all capacity’ of computers had ‘nearly doubled every year’ since 1945 and often implied in conversation that he expected that trend to continue. His observations prefigure ‘Moore’s law’, named after Intel’s cofounder Gordon Moore, who predicted in 1965 that the number of components on an integrated circuit would double every year.

This behavior can be more easily captured by continuous, analog networks than it can be defined by digital, algorithmic codes. These analog networks may be composed of digital processors, but it is in the analog domain that the interesting computation is being performed. “The purely ‘digital’ procedure is probably more circumstantial and clumsy than necessary,” von Neumann warned in 1951. “Better, and better integrated, mixed procedures may exist.”49 Analog is back, and here to stay.

What we are creating now is a monster whose influence is going to change history, provided there is any history left,” he said, in Klári’s account, “yet it would be impossible not to see it through, not only for the military reasons, but it would also be unethical from the point of view of the scientists not to do what they know is feasible, no matter what terrible consequences it may have. And this is only the beginning!” The concerns von Neumann voiced that night were less about nuclear weapons, and more about the growing powers of machines.

Computers were built in the late 1940s because mathematicians like John von Neumann thought that if you had a computer—a machine to handle a lot of variables simultaneously—you would be able to predict the weather. Weather would finally fall to human understanding. And men believed that dream for the next forty years. They believed that prediction was just a function of keeping track of things. If you knew enough, you could predict anything. That’s been a cherished scientific belief since Newton.” “And?” “Chaos theory throws it right out the window. It says that you can never predict certain phenomena at all. You can never predict the weather more than a few days away. All the money that has been spent on long-range forecasting—about half a billion dollars in the last few decades—is money wasted. It’s a fool’s errand. It’s as pointless as trying to turn lead into gold. We look back at the alchemists and laugh at what they were trying to do, but future generations will laugh at us the same way. We’ve tried the impossible—and spent a lot of money doing it. Because in fact there are great categories of phenomena that are inherently unpredictable.

Complex networks—of molecules, people, or ideas—constitute their own simplest behavioral descriptions. This behavior can be more easily captured by continuous, analog networks than it can be defined by digital, algorithmic codes. These analog networks may be composed of digital processors, but it is in the analog domain that the interesting computation is being performed. “The purely ‘digital’ procedure is probably more circumstantial and clumsy than necessary,” von Neumann warned in 1951. “Better, and better integrated, mixed procedures may exist.”49 Analog is back, and here to stay.

The truth was that von Neumann had been unhappy at the IAS for several years before his death. ‘Von Neumann, when I was there at Princeton, was under extreme pressure,’ says Benoît Mandelbrot, who had come to the IAS in 1953 at von Neumann’s invitation, ‘from mathematicians, who were despising him for no longer being a mathematician; by the physicists, who were despising him for never having been a real physicist; and by everybody for having brought to Princeton this collection of low-class individuals called “programmers”’. ‘Von Neumann,’ Mandelbrot continues, ‘was simply being shunned.

on his first visit to the Los Alamos Tech Area, when he had encountered von Neumann discussing theory with dark, intense Edward Teller, the “tremendously long formulae on the blackboard” had scared him. “Seeing all these complications of analysis, I was dumbfounded, fearing I would never be able to contribute anything.” But the equations stayed on the board from day to day, which meant to Ulam that the pace of invention was relatively slow, and he soon regained confidence.1340 “I found out that the main ability to have was a visual, and also an almost tactile, way to imagine the physical situations, rather than a merely logical picture of the problems.

though the ENIAC had been designed by men, this gruelling, fiddly job of actually building it was almost exclusively the work of women, who laboured nights and weekends until it was complete.11 Buried in the project’s payroll records are the names of nearly fifty women and perhaps many more who were only listed by their initials.

Von Neumann saw great promise in a redesign of the ENIAC computer’s memory. He believed there was a way to turn the computer into an “electronic brain” capable of storing not just data and instructions, as was the case with ENIAC, but additional information that would allow the computer to perform a myriad of computational functions on its own. This was called a stored-program computer, and it “broke the distinction between numbers that mean things and numbers that do things,” writes von Neumann’s biographer George Dyson, adding, “Our universe would never be the same.” These “instructions” that von Neumann imagined were the prototype of what the world now knows as software.

Ohm found that the results could be summed up in such a simple law that he who runs may read it, and a schoolboy now can predict what a Faraday then could only guess at roughly. By Ohm's discovery a large part of the domain of electricity became annexed by Coulomb's discovery of the law of inverse squares, and completely annexed by Green's investigations. Poisson attacked the difficult problem of induced magnetisation, and his results, though differently expressed, are still the theory, as a most important first approximation. Ampere brought a multitude of phenomena into theory by his investigations of the mechanical forces between conductors supporting currents and magnets. Then there were the remarkable researches of Faraday, the prince of experimentalists, on electrostatics and electrodynamics and the induction of currents. These were rather long in being brought from the crude experimental state to a compact system, expressing the real essence. Unfortunately, in my opinion, Faraday was not a mathematician. It can scarcely be doubted that had he been one, he would have anticipated much later work. He would, for instance, knowing Ampere's theory, by his own results have readily been led to Neumann's theory, and the connected work of Helmholtz and Thomson. But it is perhaps too much to expect a man to be both the prince of experimentalists and a competent mathematician.

The four had come to an exciting decision" during the six months of the blockade threatened by the authorities, they would make the ruins a laboratory, a demonstration of how well and happily men could live with virtually no machines. They saw now the common man's wisdom in wrecking practically everything. That was the way to do it, and the hell with moderation! "All right, so we'll heat our water and cook our food and light and warm our homes with wood fires," said Lasher. "And walk wherever we're going," said Finnerty. "And read books instead of watching television," said von Neumann. "The Renaissance comes to upstate New York! We'll rediscover the two greatest wonders of the world, the human mind and hand.

An important viewpoint in classifying games is this: Is the sum of all payments received by all players (at the end of the game) always zero; or is this not the case? If it is zero, then one can say that the players pay only to each other, and that no production or destruction of goods is involved. All games which are actually played for entertainment are of this type. But the economically significant schemes are most essentially not such. There the sum of all payments, the total social product, will in general not be zero, and not even constant. I.e., it will depend on the behavior of the players—the participants in the social economy. This distinction was already mentioned in 4.2.1., particularly in footnote 2, p. 34. We shall call games of the first-mentioned type zero-sum games, and those of the latter type non-zero-sum games.

Nevertheless, Oppenheimer strongly believed it was essential that the Institute remain a home to both science and the humanities. In his speeches about the Institute, Oppenheimer continually emphasized that science needed the humanities to better understand its own character and consequences. Only a few of the senior resident mathematicians agreed with him, but their support was critical. Johnny von Neumann was almost as interested in ancient Roman history as he was in his own field. Others shared Oppenheimer’s interest in poetry. He hoped that he could make the Institute a haven for scientists, social scientists and humanists interested in a multidisciplinary understanding of the whole human condition. It was an irresistible opportunity, a chance to bring together the two worlds, science and the humanities, that had engaged him equally as a young man.

That seems reasonable,” I agreed. “My own personal theory is that extraterrestrial life could be here already … and how would we necessarily know? If there is life in the universe, the form of life that will prove to be most successful at propagating itself will be digital life; it will adopt a form that is independent of the local chemistry, and migrate from one place to another as an electromagnetic signal, as long as there’s a digital world—a civilization that has discovered the Universal Turing Machine—for it to colonize when it gets there. And that’s why von Neumann and you other Martians got us to build all these computers, to create a home for this kind of life.” There was a long, drawn-out pause. “Look,” Teller finally said, lowering his voice to a raspy whisper, “may I suggest that instead of explaining this, which would be hard … you write a science-fiction book about it.” “Probably someone has,” I said. “Probably,” answered Teller, “someone has not.

The key to innovation-at Bell Labs and in the digital age in general-was realizing that there was no conflict between nurturing individual geniuses and promoting collaborative teamwork. It was not either-or. Indeed, throughout the digital age, the two approaches went together. Creative geniuses (John Mauchly, William Shockley, Steve Jobs) generated innovative ideas. Practical engineers (Presper Eckert, Walter Brattain, Steve Wozniak) partnered closely with them to turn concepts into contraptions. And collaborative teams of technicians and entrepreneurs worked to turn the invention into a practical product. When part of this ecosystem was lacking, such as for John Atanasoff at Iowa State or Charles Babbage in the shed behind his London home, great concepts ended up being consigned to history's basement. And when great teams lacked passionate visionaries, such as Penn after Mauchly and Eckert left, Princeton after von Neumann, or Bell Labs after Shockley, innovation slowly withered.

It is a well known phenomenon in many branches of the exact and physical sciences that very great numbers are often easier to handle than those of medium size. An almost exact theory of a gas, containing about 1025 freely moving particles, is incomparably easier than that of the solar system, made up of 9 major bodies; and still more than that of a multiple star of three or four objects of about the same size. This is, of course, due to the excellent possibility of applying the laws of statistics and probabilities in the first case.

Von Neumann, in his thought experiment about self-replication, had written that he had avoided the “most intriguing, exciting, and important question of why the molecules or aggregates that in nature really occur … are the sorts of thing they are, why they are essentially very large molecules in some cases but large aggregations in other cases.”20 Pattee suggested that it is the very size of the molecules that ties the quantum and classical worlds together: “Enzymes are small enough to take advantage of quantum coherence to attain the enormous catalytic power on which life depends, but large enough to attain high specificity and arbitrariness in producing effectively decoherent products that can function as classical structures.”21 Quantum coherence basically means that subatomic particles sync together to “cooperate” to produce decoherent products, which are particles that do not have quantum properties. Pattee notes that there is now research that supports his proposal that enzymes require quantum effects22 and that life would be impossible in a strictly quantum world.23 Both are needed: a quantum layer and a classical physical layer.

a harbinger of a third wave of computing, one that blurred the line between augmented human intelligence and artificial intelligence. “The first generation of computers were machines that counted and tabulated,” Rometty says, harking back to IBM’s roots in Herman Hollerith’s punch-card tabulators used for the 1890 census. “The second generation involved programmable machines that used the von Neumann architecture. You had to tell them what to do.” Beginning with Ada Lovelace, people wrote algorithms that instructed these computers, step by step, how to perform tasks. “Because of the proliferation of data,” Rometty adds, “there is no choice but to have a third generation, which are systems that are not programmed, they learn.”27 But even as this occurs, the process could remain one of partnership and symbiosis with humans rather than one designed to relegate humans to the dustbin of history. Larry Norton, a breast cancer specialist at New York’s Memorial Sloan-Kettering Cancer Center, was part of the team that worked with Watson. “Computer science is going to evolve rapidly, and medicine will evolve with it,” he said. “This is coevolution. We’ll help each other.”28 This belief that machines and humans will get smarter together is a process that Doug Engelbart called “bootstrapping” and “coevolution.”29 It raises an interesting prospect: perhaps no matter how fast computers progress, artificial intelligence may never outstrip the intelligence of the human-machine partnership. Let us assume, for example, that a machine someday exhibits all of the mental capabilities of a human: giving the outward appearance of recognizing patterns, perceiving emotions, appreciating beauty, creating art, having desires, forming moral values, and pursuing goals. Such a machine might be able to pass a Turing Test. It might even pass what we could call the Ada Test, which is that it could appear to “originate” its own thoughts that go beyond what we humans program it to do. There would, however, be still another hurdle before we could say that artificial intelligence has triumphed over augmented intelligence. We can call it the Licklider Test. It would go beyond asking whether a machine could replicate all the components of human intelligence to ask whether the machine accomplishes these tasks better when whirring away completely on its own or when working in conjunction with humans. In other words, is it possible that humans and machines working in partnership will be indefinitely more powerful than an artificial intelligence machine working alone?

What is life? Well, what do living things do? One answer is, they reproduce. Life makes more life. However, logic told him that “what goes on is actually one degree better than self-reproduction, for organisms appear to have gotten more elaborate in the course of time.”9 Life did not just make more life. Life could increase in complexity; it could evolve. Von Neumann became increasingly interested in what an evolvable, autonomous, self-replicating machine (“an automaton”) would logically require when placed in an environment with which it could interact. His string of logic led him to the conclusion that the automaton needed a description of how to copy itself and a description of how to copy that description so it could hand it off to the next, freshly minted automaton. The original automaton also needed a mechanism to do the actual construction and copy job. It needed information and construction. However, this would cover only replication. Von Neumann reasoned that he had to add something in order for the automaton to be able to evolve, to increase in complexity. He concluded that it needed a symbolic self-description, a genotype, a physical structure independent of the structure it was describing, the phenotype. Linking the symbolic description with what it refers to would require a code, and now his automatons would be able to evolve.

Turing was offered a choice: imprisonment or probation contingent on receiving hormone treatments via injections of a synthetic estrogen designed to curb his sexual desires, as if he were a chemically controlled machine. He chose the latter, which he endured for a year. Turing at first seemed to take it all in stride, but on June 7, 1954, he committed suicide by biting into an apple he had laced with cyanide. His friends noted that he had always been fascinated by the scene in Snow White in which the Wicked Queen dips an apple into a poisonous brew. He was found in his bed with froth around his mouth, cyanide in his system, and a half-eaten apple by his side. Was that something a machine would have done? I. Stirling’s formula, which approximates the value of the factorial of a number. II. The display and explanations of the Mark I at Harvard’s science center made no mention of Grace Hopper nor pictured any women until 2014, when the display was revised to highlight her role and that of the programmers. III. Von Neumann was successful in this. The plutonium implosion design would result in the first detonation of an atomic device, the Trinity test, in July 1945 near Alamogordo, New Mexico, and it would be used for the bomb that was dropped on Nagasaki on August 9, 1945, three days after the uranium bomb was used on Hiroshima. With his hatred of both the Nazis and the Russian-backed communists, von Neumann became a vocal proponent of atomic weaponry.

I work in theoretical computer science: a field that doesn’t itself win Fields Medals (at least not yet), but that has occasions to use parts of math that have won Fields Medals. Of course, the stuff we use cutting-edge math for might itself be dismissed as “ivory tower self-indulgence.” Except then the cryptographers building the successors to Bitcoin, or the big-data or machine-learning people, turn out to want the stuff we were talking about at conferences 15 years ago—and we discover to our surprise that, just as the mathematicians gave us a higher platform to stand on, so we seem to have built a higher platform for the practitioners. The long road from Hilbert to Gödel to Turing and von Neumann to Eckert and Mauchly to Gates and Jobs is still open for traffic today. Yes, there’s plenty of math that strikes even me as boutique scholasticism: a way to signal the brilliance of the people doing it, by solving problems that require years just to understand their statements, and whose “motivations” are about 5,000 steps removed from anything Caplan or Bostrom would recognize as motivation. But where I part ways is that there’s also math that looked to me like boutique scholasticism, until Greg Kuperberg or Ketan Mulmuley or someone else finally managed to explain it to me, and I said: “ah, so that’s why Mumford or Connes or Witten cared so much about this. It seems … almost like an ordinary applied engineering question, albeit one from the year 2130 or something, being impatiently studied by people a few moves ahead of everyone else in humanity’s chess game against reality. It will be pretty sweet once the rest of the world catches up to this.

Civilization Number 184 was destroyed by the stacked gravitational attractions of a tri-solar syzygy. This civilization had advanced to the Scientific Revolution and the Industrial Revolution. In this civilization, Newton established nonrelativistic classical mechanics. At the same time, due to the invention of calculus and the Von Neumann architecture computer, the foundation was set for the quantitative mathematical analysis of the motion of three bodies. After a long time, life and civilization will begin once more, and progress through the unpredictable world of Three Body. We invite you to log on again.

Authentic humans don’t show the perfect, chessmaster appreciation of consequences that von Neumann’s theory demands. Instead, decision makers resort to heuristics, or mental shortcuts, to arrive at quick, intuitive choices.