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Thursday, March 08, 2012

Feynmann, John von Neumann, and Mental Models

Posted by Bill Benzon on 03/08/12 at 02:22 PM

Since George Dyson’s recent history of modern computing, Turing’s Cathedral: The Origins of the Digital Universe, was written in part to restore John von Neumann to prominence, I thought I’d republish a double review, lightly edited, I wrote some years ago: “A Tale of Two Geniuses,” Journal of Social and Evolutionary Systems, 17(2): 227-230, 1994. Richard Feynman was one of the geniuses and John von Neumann was the other.


* * * * * *

Genius:  The Life and Science of Richard Feynman, by James Gleick, New York:  Pantheon Books, 1992, 532 pp.

John von Neumann, by Norman Macrae, New York, Pantheon Books, 1992, 405 pp.

Students of cognitive evolution and of twentieth century thought are fortunate in the simultaneous appearance of these two biographies.  No doubt the simultaneity is mostly coincidence.  The physicist Richard Feynman is most widely known, alas, for two autobiographical collections of anecdotes which reveal him to be a waggish and riggish anti-establishment sort; he is most deeply known for his contributions to quantum electrodynamics.  John von Neumann was a thoroughly establishment sort - soldiers guarded his hospital room as he lay dying of brain cancer just in case he let out defense secrets in his sleep - and is most widely know as the name which appears in phrases like “computers using the von Neumann architecture.” The two men crossed paths in Los Alamos, where they worked on the atomic bomb.  That crossing is a reasonable place to begin our review.

Los Alamos

Feynman was recruited to Los Alamos while still a graduate student.  He was in charge of group T-4, Diffusion Problems.  The problem was to figure out how neutrons, which drive the fission reaction, diffuse through the explosive core.  Knowing the rate and pattern of diffusion was essential to determining the mass and configuration of fissile material.  Since the late 30s von Neumann had been working on similar problems in connection with shock waves and explosions in general and so was able to help the Los Alamos effort between 1943 and 1945.

The difficulty was that the relevant equations could not be solved analytically.  Rather, it was necessarily to simulate neutron diffusion numerically by calculating the step-by-step motion of individual neutrons.  That requires lots of calculations, which were performed by a group of people operating calculators.  The problems would be broken into components; each person would be responsible for one component, with each problem being passed from person to person as individual components where calculated.

Computing and von Neumann

That, of course, is the general way computers solve problems, with the computational plan being an algorithm.  But, they did not have computers at Los Alamos.  Computers came after the war and von Neumann was central to the effort.  He understood that the computer is essentially a logical device and clarified that logic with the concepts of the stored program (Macrae, pp. 282-284), the fetch-execute cycle (pp. 287), and conditional transfer (see Bernstein 1963, 1964, pp. 60 ff.).  That is to say, von Neumann clearly differentiated between the physical structures and connections of the devices from which the computer is constructed and the logical requirements which those devices have to fulfill.  For that he is the progenitor of the computer. 

Later on von Neumann initiated the use of computers in weather modeling.  This, plus his earlier work on shock waves and the atomic bomb, makes him one of the founders of numerical analysis, a loose collection of techniques important in many scientific and technical fields.  While pursuing the conceptual foundations of life, he worked out the concept of the cellular automaton, a highly parallel kind of computational device which is much favored by current theorists of chaos and dynamical systems.  His work on game theory created a new field of economic and strategic analysis.  Before the war von Neumann did important work on the mathematical foundations of quantum mechanics.

Feynmann and Quantum Mechanics

And so we segue to Feynman, whose most important work was that which he in the late 1940s on quantum electrodynamics.  The quantum world is notorious as the domain where the fundamental stuff of the universe acts sometimes like a wave, sometimes like a particle.  Particles and waves are readily visualized.  But how can you visualize something which is both and neither?  And, if you can’t visualize it, then how do you get the physical intuition which is, for many, so important to scientific thinking (cf. Miller, 1986)?  It was Feynman’s genius to create diagrammatic conventions for quantum interactions which made physical intuition much easier and facilitated calculation as well.  The so-called Feynman diagrams became ubiquitous once Feynman introduced them and, in 1949, Freeman Dyson [father of George Dyson] proved the diagrams to be equivalent to the more rigorously mathematical, and less intuitive, axiomatic approach of Julian Schwinger and Sin-Itoro Tomonaga.

Feynman went on to do important work in superfluids, weak nuclear force and, while on sabbatical, did some creditable molecular biology.  In the wake of the Challenger disaster Feynman received a great deal of attention by performing a simple demonstration with ice water and a rubber ring.  That simple demonstration unmasked the self-serving bureaucratic disregard for reality which led to the Challenger disaster.  He also served on the board of directors of Thinking Machines, Inc., whose massively parallel computers are often used to implement models based on von Neumann’s concept of the cellular automaton.

Mental Math, Computing, and Late 20th Century Thought

With this return of von Neumann we get to the point of this review:  When taken together, what do Feynman and von Neumann give us?  Their intellectual lives crossed paths only once, albeit in a caldron whose intellectual fecundity may, in the long run, outlive the bomb which was its immediate purpose.  For the most part, they worked in separate arenas.  But taken together those various arenas encompass much of the deepest and most rigorous scientific and technical thinking of our century.  Perhaps by looking at their work we can gain some insight into the basis of that thinking.

Mental mathematics is a motif which crops up in both books.  Feynman and von Neumann worked in fields were calculational facility was widespread and both were among the very best at mental mathematics.  In itself such skill has no deep intellectual significance.  Doing it depends on knowing a vast collection of unremarkable calculational facts and techniques and knowing one’s way around in this vast collection.  Before the proliferation of electronic calculators the lore of mental math used to be collected into books on mental calculation.  Virtuosity here may have gotten you mentioned in “Ripley’s Believe it or Not” or a spot on a TV show, but it wasn’t a vehicle for profound insight into the workings of the universe. 

Yet, this kind of skill was so widespread in the scientific and engineering world that one has to wonder whether there is some connection between mental calculation, which has largely been replaced by electronic calculators and computers, and the conceptual style, which isn’t going to be replaced by computers anytime soon.  Perhaps the domain of mental calculations served as a matrix in which the conceptual style of Feynman, von Neumann, (and their peers and colleagues) was nurtured.

Enter, Piaget

Piaget (1976, pp. 320 ff.) talks of higher mental processes which operate on lower level processes; perhaps the world of calculations is the lower level world over which all these thinkers built their higher level processes.  In the terms David Hays and I introduced in our account of cognitive evolution (Benzon & Hays, 1990), these higher level processes implement models.  Earlier science had been based on theories, while philosophy is grounded in rationalization, with modeling, theorizing and rationalizing understood as distinct types of abstract conceptualization.  I’m suggesting that these conceptual models, which are central to 20th century thought, were originally constructed in a mental space richly populated with the tricks and procedures of mental calculation. It is in such a mental space that von Neumann and Feynman made their contributions to our thought.

If so, does the advent of the electronic calculator and computer mean that we have just thrown away the possibility of such deep thought by raising a generation of thinkers who routinely turn to calculators and computers for tasks that Feynman, von Neumann, and their many colleagues would peform in their heads?  It is possible.  But, another possibility is more interesting, and not nearly so depressing in its implications.

Yes, the models of Feynman and von Neumann are built on a foundation of calculational wizardry.  But most of that wizardry was irrelevant to that genius.  It turns out that the relevant component is not only required in programming computers, but is used there in a form unadulterated by a vast collection of mere facts and details.  Thus programming, with its own collection of tricks and procedures, provides a much more effective basis for creating the conceptual matrix required to understand particle physics, game theory, and, of course, computing itself. A student who has learned to program need not be a Feynman or a von Neumann to grasp matters which, not so long ago, only Feynman and von Neumann and very few others, could grasp.  Thus, far from destroying the necessary conceptual matrix, computers may make that matrix more readily available.

This is, of course, only a speculation; not even that, it is that most fragile of conceptual objects, a Mere Speculation.  Disreputable as they are, Mere Speculations are nonetheless unavoidable, for they are starting points.  One can only wonder at how many Mere Speculations von Neuman and Feynman must have considered as they worked their way toward their more substantial contributions.

References

Benzon, W. L.  & Hays, D.G. (1990) “The Evolution of Cognition.” Journal of Social and Biological Structures, 13, 297-320.

Bernstein, J. (1963, 1964) The Analytical Engine.  New York:  Random House.

Miller, A. I. (1986) Imagery in Scientific Thought.  Cambridge, MA:  The MIT Press.

Piaget, J. (1976) The Grasp of Consciousness.  Cambridge, MA:  Harvard University Press.


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