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Stephen Wolfram
Wolfram Research
Boston, MA

Registered: Oct 2003
Posts: 13

John von Neumann's 100th Birthday

Today (December 28, 2003) would have been John von Neumann's 100th birthday---if he had not died at age 54 in 1957, and that gives me an excuse today to write a little about him. (I would have liked to spend longer on this, but December 28 only lasts so long, and I have many other things to do.)

I've been interested in von Neumann for many years---not least because his work touched on some of my most favorite topics. He is mentioned in 12 separate places in my book---second in number only to Alan Turing, who appears 19 times. (See www.wolframscience.com/nks/index/names/t-z.html.)

I always feel that one can appreciate peoples' work better if one understands the people themselves better. And from talking to many people who knew him, I think I've gradually built up a decent picture of John von Neumann as a man.

He would have been fun to meet. He knew a lot, was very quick, always impressed people, and was lively, social and funny.

One video clip of him has survived. In 1955 he was on a television show called "Youth Wants to Know", which today seems painfully hokey. Surrounded by teenage kids, he is introduced as a commissioner of the Atomic Energy Commission---which in those days was a big deal. He is asked about an exhibit of equipment. He says very seriously that it's mostly radiation detectors. But then a twinkle comes into his eye, and he points to another item, and says deadpan "except this, which is a carrying case". And that's the end of the only video record of John von Neumann that exists.

Some scientists (such as myself) spend most of their lives pursuing their own grand programs, ultimately in a fairly isolated way. John von Neumann was instead someone who always liked to interact with the latest popular issues---and the people around them---and then contribute to them in his own characteristic way.

He worked hard, often on many projects at once, and always seemed to have fun. In retrospect, he chose most of his topics remarkably well. He studied each of them with a definite practical mathematical style. And partly by being the first person to try applying serious mathematical methods in various areas, he was able to make important and unique contributions.

But I've been told that he was never completely happy with his acheivements because he thought he missed some great discoveries. And indeed he was close to a remarkable number of important mathematics-related discoveries of the twentieth century: Godel's theorem, Bell's inequalities, information theory, Turing machines, computer languages---as well as my own more recent favorite core NKS discovery of complexity from simple rules.

But somehow he never quite made the conceptual shifts that were needed for any of these discoveries.

There were, I think, two basic reasons for this. First, he was so good at getting new results by the mathematical methods he knew that he was always going off to get more results, and never had a reason to pause and see whether some different conceptual framework should be considered. And second, he was not particularly one to buck the system: he liked the social milieu of science and always seemed to take both intellectual and other authority seriously.

By all reports, von Neumann was something of a prodigy, publishing his first paper (on zeros of polynomials) at the age of 19. By his early twenties, he was established as a promising young professional mathematician---working mainly in the then-popular fields of set theory and foundations of math. (One achievement was alternate axioms for set theory---see the NKS book, page 1155.)

Like many good mathematicians in Germany at the time, he worked on David Hilbert's program for formalizing mathematics, and for example wrote papers aimed at finding a proof of consistency for the axioms of arithmetic. But he did not guess the deeper point that Kurt Godel discovered in 1931: that actually such a proof is fundamentally impossible. I've been told that von Neumann was always disappointed that he had missed Godel's theorem. He certainly knew all the methods needed to establish it (and understood it remarkably quickly once he heard it from Godel). But somehow he did not have the brashness to disbelieve Hilbert, and go looking for a counterexample to Hilbert's ideas.

In the mid-1920s formalization was all the rage in mathematics, and quantum mechanics was all the rage in physics. And in 1927 von Neumann set out to bring these together---by axiomatizing quantum mechanics. A fair bit of the formalism that von Neumann built has become the standard framework for any more mathematical exposition of quantum mechanics. But I must say that I have always thought that it gave too much of an air of mathematical definiteness to ideas (particularly about quantum measurement) that are in reality depend on all sorts of physical details. And indeed some of von Neumann's specific axioms turned out to be too restrictive for ordinary quantum mechanics---obscuring for a number of years the phenomenon of entanglement, and later of criteria such as Bell's inequalities.

But von Neumann's work on quantum mechanics had a variety of fertile mathematical spinoffs, and particularly what are now called von Neumann algebras have recently become popular in mathematics and mathematical physics.

Interestingly, von Neumann's approach to quantum mechanics was at first very much aligned with traditional calculus-based mathematics---investigating properties of Hilbert spaces, continuous operators, etc. But gradually it became more focused on discrete concepts, particularly early versions of "quantum logic". In a sense von Neumann's quantum logic ideas were an early attempt at defining a computational model of physics. But he did not pursue this, and did not go in the directions that have for example led to modern ideas of quantum computing.

By the 1930s von Neumann was publishing several papers a year, on a variety of popular topics in mainstream mathematics, often in collaboration with contemporaries of significant later reputation (Wigner, Koopman, Jordan, Veblen, Birkhoff, Kuratowski, Halmos, Chandrasekhar, etc.). Von Neumann's work was unquestionably good and innovative, though very much in the flow of development of the mathematics of its time.

Despite von Neumann's early interest in logic and the foundations of math, he (like most of the math community) moved away from this by the mid-1930s. In Cambridge and then in Princeton he encountered the young Alan Turing---even offering him a job as an assistant in 1938. But he apparently paid little attention to Turing's classic 1936 paper on Turing machines and the concept of universal computation, writing in a recommendation letter on June 1, 1937 that "[Turing] has done good work on ... theory of almost periodic functions and theory of continuous groups".

As it did for many scientists, von Neumann's work on the Manhattan Project appears to have broadened his horizons, and seems to have spurred his efforts to apply his mathematical prowess to problems of all sorts---not just in traditional mathematics. His pure mathematical colleagues seem to have viewed such activities as a peculiar and somewhat suspect hobby, but one that could generally be tolerated in view of his respectable mathematical credentials.

At the Institute for Advanced Study in Princeton, where von Neumann worked, there was strain, however, when he started a project to build an actual computer there. Indeed, even when I worked at the Institute in the early 1980s, there were still pained memories of the project. The pure mathematicians at the Institute had never been keen on it, and the story was that when von Neumann died, they had been happy to accept Thomas Watson of IBM's offer to send a truck to take away all of von Neumann's equipment. (Amusingly, the 6-inch on-off switch for the computer was left behind, bolted to the wall of the building, and has recently become a prized possession of a computer industry acquaintance of mine.)

I had some small interaction with von Neumann's heritage at the Institute in 1982 when the then-director (Harry Woolf) was recruiting me. (Harry's original concept was to get me to start a School of Computation at the Institute, to go along with the existing School of Natural Sciences and School of Mathematics. But for various reasons, this was not what happened.) I was concerned about intellectual property issues, having just had some difficulty with them at Caltech. Harry's response---that he attributed to the chairman of their board of trustees was: "Look, von Neumann developed the computer here, but we insisted on giving it away; after that, why should we worry about any intellectual property rights?". (The practical result was a letter disclaiming any rights to any intellectual property that I produced at the Institute.)

Among several of von Neumann's interests outside of mainstream pure mathematics was his attempt to develop a mathematical theory of biology and life (see the NKS book, page 876). In the mid-1940s there had begun to be---particularly from wartime work on electronic control systems---quite a bit of discussion about analogies between "natural and artificial automata", and "cybernetics". And von Neumann decided to apply his mathematical methods to this. I've been told he was particularly impressed by the work of McCullough and Pitts on formal models of the analogy between brains and electronics (see the NKS book, page 1099). (There were undoubtedly other influences too: John McCarthy told me that around 1948 he visited von Neumann, and told him about applying information theory ideas to thinking about the brain as an automaton; von Neumann's main response at the time was just "write it up!".)

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. I've been told that at Los Alamos von Neumann was very taken with electrically stimulated jellyfish, which he appears to have viewed as doing some kind of continuous analog of the information processing of an electronic circuit. In any case, by about 1947, he had conceived the idea of using partial differential equations to model a kind of factory that could reproduce itself, like a living organism.

Von Neumann always seems to have been very taken with children, and I am told that it was in playing with an erector set owned by the son of his game theory collaborator Oskar Morgenstern that von Neumann realized that his self-reproducing factory could actually be built out of discrete robotic-like parts. (There was already something of a tradition of building computers out of Meccano---and indeed for example some of Hartree's early articles on analog computers appeared in Meccano Magazine.)

An electrical engineer named Julian Bigelow, who worked on von Neumann's IAS computer project, pointed out that 3D parts were not necesary, and that 2D would work just as well. (When I was at the Institute in the early 1980s Bigelow was still there, though unfortunately viewed as a slightly peculiar relic of von Neumann's project.)

Stan Ulam told me that he had independently thought about making mathematical models of biology, but in any case, around 1951 he appears to have suggested to von Neumann that one should be able to use a simplified, essentially combinatorial model---based on something like the infinite matrices that Ulam had encountered in the so-called Polish Book of math problems to which he had contributed.

The result of all this was a model that was formally a two-dimensional cellular automaton. Systems equivalent to two-dimensional cellular automata were arising in several other contexts around the same time (see the NKS book, page 876). von Neumann seems to have viewed his version as a convenient framework in which to construct a mathematical system that could emulate engineered computer systems---especially the EDVAC on which von Neumann worked.

In the period 1952-3 von Neumann sketched an outline of a proof that it was possible for a formal system to support self reproduction. Whenever he needed a different kind of component (wire, oscillator, logic element, etc.) he just added it as a new state of his cellular automaton, with new rules. He ended up with a 29-state system, and a 200,000 cell configuration that could reproduce itself. (von Neumann himself did not complete the construction. This was done in the early 1960s by a former assistant of von Neumann's named Arthur Burks, who had left the IAS computer project to concentrate on his interests in philosophy, though who maintains even today an interest in cellular automata.)

From the point of view of NKS, von Neumann's system now seems almost grotesquely complicated. But von Neumann's intuition told him that one could not expect a simpler system to show something as sophisticated and biological as self reproduction. What he said was that he thought that below a certain level of complexity, systems would always be "degenerative", and always generate what amounts to behavior simpler than their rules. But then, from seeing the example of biology, and of systems like Turing machines, he believed that above some level, there should be an "explosive" increase in complexity, with systems able to generate other systems more complex than themselves. But he said that he thought the threshold for this would be systems with millions of parts.

Twenty-five years ago I might not have disagreed too strongly with that. And certainly for me it took several years of computer experimentation to understand that in fact it takes only very simple rules to produce even the most complex behavior. So I do not think it surprising---or unimpressive---that von Neumann failed to realize that simple rules were enough.

Of course, as one often realizes in retrospect, he did have some other clues. He knew about the idea of generating pseudorandom numbers from simple rules, even suggesting the "middle square method" (see NKS page 975.) He had the beginnings of the idea of doing computer experiments in areas like number theory. He analysed the first 2000 digits of pi and e, computed on the ENIAC, finding that they seemed random---though making no comment on it (see the NKS book, page 912). (He also looked at ContinuedFraction[2^(1/3)]; see the NKS book, page 914.)

I have asked many people who knew him why von Neumann never considered simpler rules. Marvin Minsky told me that he actually asked von Neumann about this directly, but that von Neumann had been somewhat confused by the question. It would have been much more Ulam's style than von Neumann's to have come up with simpler rules, and Ulam indeed did try making a one-dimensional analog of 2D cellular automata, but came up not with 1D cellular automata, but with a curious number-theoretical system (see the NKS book, page 908).

In the last ten years of his life, von Neumann got involved in an impressive array of issues. Some of his colleagues seem to have felt that he spent too little time on each one, but still his contributions were usually substantial---sometimes directly in terms of content, and usually at least in terms of lending his credibility to emerging areas.

He made mistakes, of course. He thought that each logical step in computation would necessarily dissipate a certain amount of heat, whereas in fact reversible computation is in principle possible. He thought that the unreliability of components would be a major issue in building large computer systems; he apparently did not have an idea like error-correcting codes. He is reputed to have said that no computer program would ever be more than a few thousand lines long. He was probably thinking about proofs of theorems---but did not think about subroutines, the analog of lemmas.

Von Neumann was a great believer in the efficacy of mathematical methods and models, perhaps implemented by computers. In 1950 he was optimistic that accurate numerical weather forecasting would soon be possible (see the NKS book page 1132). In addition, he believed that with methods like game theory it should be possible to understand much of economics and other forms of human behavior (see the NKS book page 1135).

Von Neumann was always quite a believer in using the latest methods and tools (I'm sure he would have been a big Mathematica user today). He typically worked directly with one or two collaborators, sometimes peers, sometimes assistants, though he maintained contact with a large network of scientists. (A typical communication was a letter he wrote to Alan Turing in 1949, in which he asks "What are the problems on which you are working now, and what is your program for the immediate future?".) In his later years he often operated as a distinguished consultant, brought in by the government, or other large organizations. His work was then often presented as a report, that was accorded particular weight because of his distinguished consultant status. (It was also often a good and clear piece of work.) He was often viewed a little ambivalently as an outsider in the fields he entered---positively because he brought his distinction to the field, negatively because he was not in the clique of experts in the field.

Particularly in the early 1950s, von Neumann became deeply involved in military consulting, and indeed I wonder how much of the intellectual style of cold war U.S. military strategic thinking actually originated with him. He seems to have been quite flattered that he called upon to do this consulting, and he certainly treated the government with considerably more respect than many other scientists of his day. Except sometimes in his exuberence to demonstrate his mathematical and calculational prowess, he seems to have always been quite mature and diplomatic. The transcript of his testimony at the Oppenheimer security hearing certainly for example bears this out.

Nevertheless, von Neumann's military consulting involvements left some factions quite negative about him. It's sometimes said, for example, that von Neumann might have been the model for the sinister Dr. Strangelove character in Stanley Kubrick's movie of that name (and indeed von Neumann was in a wheelchair for the last year of his life). And vague negative feelings about von Neumann surface for example in a typical statement I heard recently from a science historian of the period---that "somehow I don't like von Neumann, though I can't remember exactly why".

I recently met von Neumann's only child---his daughter Marina, who herself has had a distinguished career, mostly at General Motors. She reinforced my impression that until his unpleasant final illness, John von Neumann was a happy and energetic man, working long hours on mathematical topics, and always having fun. She told me that when he died, he left a box that he directed should be opened fifty years after his death. What does it contain? His last sober predictions of a future we have now seen? Or a joke---like a funny party hat of the type he liked to wear? It'll be most interesting in 2007 to find out.

Last edited by Stephen Wolfram on 12-28-2003 at 09:23 PM

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Old Post 12-28-2003 09:01 PM
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Gunnar Tomasson


Registered: Oct 2003
Posts: 69

Stephen:

As long-time student of the epistemological aspects of theoretical science in general and economics in particular, I would submit that John von Neumann’s contribution in the field of general-equilibrium economics – a branch of “economics” which John Maynard Keynes judged in 1934 to be “little better than nonsense” – offers a striking example of what might, charitably, be termed von Neumann’s epistemological innocence.

For, as shown in the following extract from my working note of many years ago on related issues, Neumann’s contribution paid no heed to Newton’s warning at the outset of Book III of ‘Principia’ that, when pushing the boundaries of science, “We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising”.

Extract begins:

The mathematical genius, John von Neumann, is recognized as the foremost pioneer in the construction of general equilibrium models in the twentieth century. In a recent book, his model was summarized briefly as follows:

"In a new [1937] breakthrough John von Neumann first formulated a balanced and steady-state growth of a general economic equilibrium and proved the existence of a solution. The breakthrough did not lie in the subject matter, which was still allocation and relative price in general equilibrium using maxima and minima. Indeed, all economists can appreciate the simple beauty, yet high degree of generality, characterizing the von Neumann model. There is substitution in both production and consumption. The model "can handle capital goods without fuss and bother," as Dorfman-Samuelson-Solow put it. There is explicit optimization in the model: the solution weeds out all but the most profitable process or processes. There are free and economic goods, indeed the solution tells us which will be free and which economic." [106]

Economists in the modern mainstream macroeconomics tradition have analyzed von Neumann's model from the point of view of mathematics, neglecting to test it for analytical coherence and consistency along the following lines:

1. von Neumann assumed "that the natural factors of production, including labour, can be expanded in unlimited quantities." [107]

2. He also defined a "free good" to be one whose supply exceeds the need for it.

3. Thus, "free goods" within von Neumann's model may be "free" one day and "non-free" the next, although he did not make that point.

4. Since all factors of production "can be expanded in unlimited quantities" by assumption, von Neumann's model has built into its premises the conclusion that all goods may in time become "free" goods.

5. Since time is not an essential feature of any general equilibrium model, why should only one or some rather than all goods be held to be "free" and not "economic"?

6. Speaking of a mathematical equation relating to the subject matter, von Neumann said: "[Its] meaning is: it is impossible to consume more of a good G in the total process than is being produced. If, however, less is consumed, i.e., if there is excess production of G, G becomes a free good and its price y = 0." [108]

7. von Neumann recognized that there was mathematically nothing to preclude all goods from being available in infinite supply so that all goods would be "free."

8. von Neumann declared this mathematical possibility to be "meaningless." [109]

9. In principle, a mathematical model can only yield conclusions, which are already implied by its premises.

10. Therefore, when von Neumann found it "meaningless" that all goods could be "free goods," he effectively declared the premises of his model to be "meaningless."

In von Neumann's case, therefore, a "necessary element [was] omitted to be taken into account: and thus the only effect of the operation [was] to mislead," as Bentham had cautioned…. [As in: "I have by me a large quarto of mathematics, written by a mathematician and politician of deserved eminence, in which the utility of numbers, as a security for good judicature, is assured. The conclusions of mathematicians, though always mathematically just, are not unfrequently physically false: that is, they would be true if things were not as they are. Some necessary element is omitted to be taken into account: and thus the only effect of the operation is to mislead."]

End of extract.

The point at issue concerns that which Hermann Weyl termed “Realistic Mathematics” – that is to say, mathematics which relate to “evidence of experiments” and not to “dreams and vain fictions of our own devising.”

A point which awaits further exploration insofar as NKS is concerned.

Gunnar

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Old Post 12-29-2003 01:16 AM
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Charlie Stromeyer jr.


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There was a brief tribute to von Neumann in the April 8 edition of BusinessWeek [1]. Gunnar is correct in that traditional applications of game theory (and equilibrium) theory to economics ala Nash or von Neumann were overly naive but, as the BusinessWeek piece mentions, "The mathematical game theory he published in 1928 became the defining framework for viewing the Cold War as a "zero-sum" relationship between the U.S. and the Soviet Union."

I am guessing now that Thomas Carlyle dubbed economics the "dismal science" because of its inherent lack of the potential to be falsifiable via well controlled experiments. Smith and Kanneman won the Nobel prize in economics by showing how well controlled psychology(-like) experiments with individuals can teach us about broader economic ideas.

However, the main problem with behavioal economics or finance in my opinion is that for about the last 6,000 years or so of human history the vast majority of wealth within any society has been controlled or dominated by a relatively small minority, and many of these big money institutions and individuals are aware of the results from behavioral finance studies and they may not on average behave like the subjects of these studies when considering how to deploy large amounts of capital.

Regarding von Neumann and the atomic bomb [1], BusinessWeek also has a tribute to Enrico Fermi. As we all know, since the development of the A bomb there has not been another major and violent world war, and this may be their greatest gift to us.

Given that von Neumann, Watson of IBM and others knew what universal Turing machines were then why did they incorrectly assume that PCs would never amount to anything significant? It was von Neumann who envisioned the "electronic brain", and we now know that a biological brain is a network of cells which is a network of molecules, so by analogy why not guess that there could be a network of TMs?


[1] THE GREAT INNOVATORS

The MANIAC's Father
Inventing the computer that helped the U.S. win the Cold War was only one of John von Neumann's many accomplishments

In early 1956, Marina von Neumann, daughter of world-renowned mathematician John von Neumann, brought her fiancé, Robert Whitman, home to Princeton, N.J., to meet her famous father. The eminent computer pioneer decided to show his future son-in-law the MANIAC (mathematical analyzer, numeral integrator, and computer) -- the legendary machine von Neumann had built in the six years after World War II. The most powerful and accurate computer ever designed to that point, MANIAC helped the U.S. beat the Soviet Union in the race for the hydrogen bomb in 1952 and was a forerunner of the modern computer age.


Outside the door to the Electronic Computer Project building at Princeton's Institute for Advanced Study (IAS), where the computer was housed, von Neumann fumbled for the right key. "He went through all his keys," recalls Marina Whitman, today a professor of business administration and public policy at the University of Michigan. "He said, 'Here's my house keys, here's the key to the Swiss Institute of Technology from 1929, here's this one and that one.' Of course we never got it open, and Bob never did get to see the computer."

Von Neumann's absent-mindedness -- like that of his tutor, IAS colleague, and friend Albert Einstein -- may have been the only glitch in a singular intellect that changed the course of history. Born in Hungary in 1903, von Neumann was a child prodigy whose impressive tricks were signs of genius. It's said he could multiply eight-digit numbers in his head as a youngster and regularly beat adults at kriegspiel (a sister game to chess). By the time he was 23, he had earned a doctorate in mathematics from the University of Budapest, as well a degree in chemical engineering from the Swiss Federal Institute of Technology.

COMPLEX COMPUTATIONS._ Ironically, chemical engineering was one of the few subjects that held no interest for von Neumann throughout his life. From his perch at the Universities of Berlin and Hamburg and then, after 1933, at the IAS, he broke ground in three major fields -- mathematics, quantum physics, and computer science.

He had not only an unusual range of interests but a remarkable gift for producing work of almost immediate practical value. The mathematical "game theory" he published in 1928 became the defining framework for viewing the Cold War as a "zero-sum" relationship between the U.S. and the Soviet Union. As a member of the Manhattan Project, his computations regarding the implosion rate of plutonium headed off several false starts by the nuclear scientists at Los Alamos who developed the atom bomb.

Then there was his real legacy -- the recognition that computing power could be used for jobs beyond solving simple linear problems such as adding or subtracting, that it could in fact tackle complex mathematical problems involving several variables. The nascent field of computer science commanded little respect in 1946, and when von Neumann announced that he wanted to build his computer at the IAS, his colleagues turned up their noses -- the theorists at Princeton's landmark think tank didn't build things.

BIG BLUE LEGACY._ Eventually, though, von Neumann secured funding, mostly from the U.S. military. That was only fitting, since his experience at Los Alamos had given him the computational background he needed to envision the potential power of an "electronic brain," as the concept was often referred to at the time.

IBM (IBM ) subsequently hired von Neumann as a part-time consultant, and, using his concepts, developed the first mass-produced computer in 1953: the IBM 650 magnetic drum calculator. Big Blue sold 450 of the machines that year, and by 1961 its share of the computer market stood at more than 80%.

Von Neumann died of cancer in 1957, when Marina was just 21. Today, while she sometimes thinks of what might have been had he lived longer, she feels lucky that her parents made a pact when they divorced: Marina would go live with her father when it was time for high school. As she puts it: "Basically, my mother felt that anyone lucky enough to be John von Neumann's daughter really ought to get the chance to know him."


By Mike Brewster in New York

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Old Post 04-21-2004 06:57 PM
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Catherine Boucher
Stephen Wolfram Science Group

Registered: Aug 2003
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John von Neumann's box

See also
http://forum.wolframscience.com/sho...=&threadid=1382

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Old Post 02-08-2007 06:48 PM
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