John von Neumann and Scientific Method

John von Neumann and Scientific Method

Salim Rashid

Journal of the History of Ideas, Volume 68, Number 3, July 2007, pp.501-527 (Article)

Published by University of Pennsylvania PressDOI: 10.1353/jhi.2007.0025

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John von Neumann and Scientific Method

Salim Rashid

Philosophy is ‘‘the systematic misuse of words designed precisely for thatpurpose. . . .’’

Von Neumann, as quoted by Michael Suppe, ‘‘Becoming Michael,’’ inPhilosophers who Believe, 151.

I. INTRODUCTION

The topic of this paper is von Neumann’s methodological approach tomathematics, to science and, to economics. Von Neumann himself de-scribed it well when he spoke of the mathematician as being ‘‘opportunis-tic.’’ Everything that he did seems consistent with this utilitarian,handyman approach. While only two of his papers deal directly with meth-odology, there is much that can be gleaned, and sometimes inferred, fromvon Neumann’s writings, his correspondence and, in some sense, his life.Nonetheless, since few direct pronouncements are available, much that Isay will in some sense be an imputation, if I may borrow a game-theoreticterm and give it an everyday meaning. I should add that the imputation

I am very grateful to J. Ferreiros, R. James, R. Tieszen, G. Moore, J. Dauben, M. Redei,T. Saif, Paul Saylor, Marina Whitman, and especially to W. Aspray, Ed McPhail, andNicola Giocoli for all their help and advice. A special acknowledgment is due to KennethArrow. Earlier versions of this paper were presented at meetings of The History of Eco-nomics Society in Durham, North Carolina, and at that of the International Society forthe History and Philosophy of Mathematics in Seville, Spain.

Copyright � by Journal of the History of Ideas, Volume 68, Number 3 (July 2007)

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JOURNAL OF THE HISTORY OF IDEAS ✦ JULY 2007

seems quite straightforward to me. In effect, von Neumann said, ‘‘One maybe unable to prove that mathematics is consistent, but mathematics is indis-pensable for physics. Physics works; hence mathematics can happily con-tinue.’’ He recognized that this was a lowbrow defense. He made this pointonce in 1947 and again in 1954. There is no reason to doubt what hemeant. It is harder to account for the substantial variety of opinions sur-rounding von Neumann’s activities.

I will assume that the world-views of individuals possess a general co-herence. While this may be false in many a case, everything we know aboutvon Neumann’s life suggests that this is the proper starting point for‘‘Johnny,’’ as he was popularly known among friends. Von Neumannwanted an empirically-based mathematics; he came to this conclusion whileengaging in extensive work on hydrodynamics, especially turbulence. Thecentral question he posed was: how can mathematics be sufficiently realisticto be practically useful? The methods he used followed from this goal. Itwas the inability to find sufficiently realistic axioms for economics that con-stantly frustrated von Neumann, and, in my opinion, accounts for the ca-sual manner with which he devoted time to economics. I will try to justifymy interpretation by moving back in time to von Neumann’s first mathe-matical contributions, on logic and set theory, and then show that nothinghe did subsequent to his work on Turbulence contradicts this interpreta-tion. To establish science and knowledge as the servants of mankind is abroadly humanistic goal. It can require both the need to control our sur-roundings, as well as social and political involvement—if that is what cir-cumstances dictate. Hence this form of interpretation requires moreattention to biography than is usual for mathematicians. Steven Heims’saccount of von Neumann’s life is remarkable for the understanding it pro-vides of the Hungary in which von Neumann came to maturity and how itinfluenced the young man. Yet Heims’s generally informative and highlyreadable study is marred by the author’s inability to empathize with vonNeumann’s experiences after the Second World War, and ends up present-ing an overly simplistic view of von Neumann’s religious beliefs. NormanMcRae’s biography of von Neumann provides a suitable corrective, which,though occasionally gushing with admiration for its subject, contains muchexcellent material. Miklos Redei’s recently published selection of von Neu-mann’s letters has been very helpful in providing corroborating evidence.1

1 Steve J. Heims, John von Neumann and Norbert Weiner: from Mathematics to the Tech-nologies of Life and Death (Cambridge Mass.: MIT Press, 1980); Norman Macrae, Johnvon Neumann (New York: Pantheon Books, 1992; and Providence, R. I.: American

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Rashid ✦ John von Neumann and Scientific Method

Von Neumann’s remarkable self-analytical lecture of 1947, ‘‘The Mathe-matician,’’ provides such a suitable beginning for a study of his methodol-ogy that I will use it as a scaffolding for my case. The early example of settheory and the later one of turbulence will then show that the imputationof views is consistent. A final section relates this to the nature of von Neu-mann as a person.

II. THE MATHEMATICIAN

The lecture on ‘‘The Mathematician’’ is for a general audience and beginswith a disclaimer about the limitations of the author and the vastness of thesubject. The guiding theme is set by his first substantive pronouncement.

The most vitally characteristic fact about mathematics is, in myopinion, its quite peculiar relationship to the natural sciences, ormore generally, to any science which interprets experience on ahigher than purely descriptive level. . . .

Most people, mathematicians and others, will agree thatmathematics is not an empirical science, or at least that it is prac-ticed in a manner which differs in several decisive respects fromthe techniques of the empirical sciences. And, yet, its developmentis very closely linked with the natural sciences. One of its mainbranches, geometry, actually started as a natural, empirical sci-ence.2

This is a significant characterization of the nature and scope of mathemat-ics. One would have liked to see this statement followed by examples illus-trating the sociological basis for this characterization of mathematics asdependent upon the material world. If one were to conjecture and ‘‘fill inthe gaps,’’ one might first argue something to the effect that such an ‘‘empir-ical’’ relationship is necessitated by the importance of the material world inguiding our intuition; and secondly, that the argument would underline theobvious fact that we consistently obtain institutional and financial supportonly by claiming to theorize about the material world.

Mathematical Society, 1999); John von Neumann: Selected Letters, ed. Miklos Redei(Providence, R.I.: American Mathematical Society, 2005).2 John von Neumann, Collected Works, 6 vols., ed. by A. H. Taub (New York: MacMil-lan, 1963), 1:1

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After describing each science on a scale determined by its successful useof mathematics, von Neumann goes on to an infelicitous summary of hisviews.

There is a quite peculiar duplicity in the nature of mathematics.One has to realize this duplicity, to accept and to assimilate it intoone’s thinking on the subject. This double face is the face of mathe-matics, and I do not believe that any simplified, Unitarian view ofthe thing is possible without sacrificing the essence.

I do not pretend to know English better than von Neumann, but neither‘‘duplicity’’ nor ‘‘Unitarian’’ expresses his true meaning. If I may rephrasehim, he meant to say that ‘‘mathematics has an inescapably dual nature andany monolithic picture of it is misleading.’’ This is the rendering of vonNeumann that I intend to illustrate from his lecture.

Von Neumann begins by minimizing the role of axiomatics as markingthe success of Greek geometry; instead, according to von Neumann, it iscontact with the empirical world that sanctified all geometric arguments.He realizes that he goes against the grain, so he suggests that our contraryviews are to be explained by centuries of schooling and not by the realityof mathematical practice. This is already a considerable challenge to theimportance of the a priori in mathematics. Then von Neumann goes on tothe calculus, where his argument is much stronger. The main discoveries inthe calculus were of explicitly physical origin and it was the practical suc-cess of the calculus that led to its diffusion—even though the logical foun-dations of the calculus3 remained inconsistent for over a century.

The third argument that von Neumann uses follows directly from thecheckered history of the calculus. Because mathematics has been supportedby physical arguments, it has succeeded despite considerable fluctuation inthe rigor of its arguments over the past three centuries. Von Neumanndraws the significant conclusion that ‘‘The variability of the concept ofrigor shows that something else beside mathematical abstraction must enterinto the makeup of mathematics.’’ Note that he uses the word ‘‘makeup’’and not ‘‘foundations’’ as he probably would if he took mathematics to bean axiomatic enterprise. This variability extends to matters of style andwhile recognizing that style is a secondary matter, von Neumann notes that

3 Judith Grabiner, The Origins of Cauchy’s Rigorous Calculus (Cambridge, Mass.: MITPress, 1981).

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such wide and consistent variations of style may well be reflective of a morepersistent reality.

Finally, von Neumann clinches the argument by referring to the recenthistory of set theory and the foundations of mathematics. He notes how thearguments of Russell, Weyl, and Brouwer seemed acceptable and correct tomost mathematicians. And yet, when faced with the thought that the scopeof mathematics would be severely restricted if they stood by their principles,most mathematicians continued to conduct mathematics the old easy andloose way.4 The autobiographical ending is noteworthy.

I have told the story of this controversy in such detail, because Ithink that it constitutes the best caution against taking the immov-able rigor of mathematics too much for granted. This happened inour own lifetime, and I know myself how humiliatingly easily myown views regarding the absolute mathematical truth changedduring this episode, and how they changed three times in succes-sion.5

Immediately after, von Neumann goes on to consider an undeniable socialfact: why does the psychological feeling of most mathematicians militateagainst this seeming dependence upon the real world? Whatever the origi-nal stimulus may have been, in the actual practice of mathematics, in theformulation of the problem and in selecting criteria for success, the mathe-matician is guided by internal, aesthetic criteria. Von Neumann relates thisto the approach of recent figures in the philosophy of science (one wouldthink of Henri Poincare and Pierre Duhem):

. . . the criterion of success [for a theory in physics] is simplywhether it can, by a simple and elegant classifying and correlatingscheme, cover very many phenomena, which without this schemewould seem complicated and heterogeneous, . . . Now this crite-rion, as set forth here, is clearly to a great extent of an aestheticalnature. For this reason it is very closely akin to the mathematicalcriteria of success, which, as you shall see, are almost entirely aes-thetical.6

4 And some who gave up the easy ways were persuaded to return! Van der Waardenquoted in Gregory H. Moore, Zermelo’s Axiom of Choice: its Origins, Development,and Influence (New York: Springer-Verlag, 1982).5 von Neumann, Collected Works,1:6.6 Ibid., 1:9.

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Mathematicians themselves do not feel the guidance of the physicalworld in their arguments and proofs and this independence from ‘‘realworld sensations’’ is a feeling so real and so primary that von Neumannhad to find a way to express its force by noting its high aesthetic properties.But in the conclusion, von Neumann again returns to the dominance ofpractice. While the actual work of a mathematician may be subjectivelyand internally guided by a professional or personal aesthetic, the subjectof mathematics itself cannot and should not stray from the constrainedconnections with the real world. He begins this marvelous passage by re-minding us that only the initial inspiration and not the method of attackare taken from the real world.

I think that it is a relatively good approximation to truth—whichis much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the geneal-ogy is sometimes long and obscure. But, once they are soconceived, the subject begins to live a peculiar life of its own andis better compared to a creative one, governed by almost entirelyaesthetical motivations, than to anything else and, in particular, toan empirical science.

However, von Neumann cannot allow the internal aesthetic to becomedominant and so he returns to stress the importance, even the primacy, ofthe real world.

There is, however, a further point which, I believe, needs stressing.As a mathematical discipline travels far from its empirical source,or still more, if it is a second and third generation only indirectlyinspired by ideas coming from ‘‘reality,’’ it is beset with very gravedangers. It becomes more and more purely aestheticizing, moreand more purely l’art pour l’art. This need not be bad, if the fieldis surrounded by correlated subjects, which still have closer empir-ical connections, or if the discipline is under the influence of menwith an exceptionally well-developed taste. But there is a gravedanger that the subject will develop along the line of least resis-tance, that the stream, so far from its source, will separate intoa multitude of insignificant branches, and that the discipline willbecome a disorganized mass of details and complexities. In otherwords, at a great distance from its empirical source, or after much

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‘‘abstract’’ inbreeding, a mathematical subject is in danger of de-generation. At the inception the style is usually classical; when itshows signs of becoming baroque, then the danger signal is up. Itwould be easy to give examples, to trace specific evolutions intothe baroque and the very high baroque, but this, again, would betoo technical.7

The characterization of ‘‘abstract’’ mathematics as ‘‘baroque’’ has a pejora-tive tinge and von Neumann drives home the conclusion.

In any event, whenever this stage is reached, the only remedyseems to me to be the rejuvenating return to the source: the reinjec-tion of more or less directly empirical ideas. I am convinced thatthis was a necessary condition to conserve the freshness and thevitality of the subject and that this will remain equally true in thefuture.

Von Neumann’s message may be summarized thus: ‘‘Seek those problemsthe objective world presents you and then solve them by your own criteria.’’What von Neumann does not address is why these internal mathematicalcriteria will also allow us to solve the practical problem according to thepractical man’s criteria. It was a friend and fellow Hungarian, Eugene Wig-ner, who later asked the famous question about ‘‘[t]he unreasonable effec-tiveness of mathematics. . . .’’8

III. LOGIC AND SET THEORY

Von Neumann’s earliest work focused upon logic, set theory, and quantummechanics. His later attitude is already apparent in his earliest public dis-cussion on the foundations of mathematics at the famous conference atKonigsberg. Von Neumann begins his presentation of the formalist positionby describing it as logically and historically subsequent to the logicist posi-tion, which he identifies most with Russell, and the intuitionist position,which he identifies with Brouwer. The formalist position of Hilbert and

7 Ibid., 1:9.8 Eugene Wigner, ‘‘The Unreasonable Effectiveness of Mathematics in the Natural Sci-ences,’’ in Communications in Pure and Applied Mathematics13, no. 1 (February 1960):1–9

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his school is thus a continuation and a clarification of the other two. ‘‘Anunderstanding of them is, of course, a necessary prerequisite for an under-standing of the utility, tendency, and modus procedendi of Hilbert’s theoryof proof. We turn instead directly to the theory of proof.’’9

Von Neumann’s presentation is throughout that of someone adoptinga minimalist defense. The crucial issue was that even though the content ofmathematical truth could not be agreed upon, the methods of communicat-ing it could.

The leading idea of Hilbert’s theory of proof is that, even if thestatements of classical mathematics should turn out to be false asto content, nevertheless, classical mathematics involves an inter-nally closed procedure which operates according to fixed rulesknown to all mathematicians and which consists basically in con-structing successively certain combinations of primitive symbolswhich are considered ‘‘correct’’ or ‘‘proved.’’ This construction-procedure, moreover, is ‘‘finitary’’ and directly constructive.10

An elementary proposition of logic is that inconsistency can be finitelychecked. This is noted by von Neumann at the end of his paper and is ofsignificance in leading to his faith in applications. He appears to argue thateven though the global structure of mathematics may not have a consis-tency proof—the results needed for individual applications can be‘‘checked’’ and relied upon.

The actual production of such a class [of axioms for all of Mathe-matics] at this time is unthinkable, however, for it poses difficultiescomparable to those raised by the decision problem. But the fol-lowing remark leads from this problem to a much simpler one: Ifour system were inconsistent, then there would exist a proof of 1� 2 in which only a finite number of axioms are used. Let the setof these axioms be called M. Then the axiom system M is alreadyinconsistent. Hence the axiom system of classical mathematics iscertainly consistent if every finite subsystem thereof is consistent.And this is surely the case if, for every finite set of axioms M, wecan give a class of formulas RM which has the following properties:

9 Johann von Neumann, ‘‘The Formalist Foundations of Mathematics,’’ in Paul Bennacer-raf and Hilary Putnam, eds., Philosophy of Mathematics: Selected Readings, 2nd edition(Cambridge and New York: Cambridge University Press, 1983), 61.10 Ibid., 61–62.

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(�) Every axiom of M belongs to RM.(�) If a and a V b belong to RM, then b also belongs to RM.(�) 1 � 2 does not belong to RM. 11

He seems optimistic about the eventual success of Hilbert’s program at thispoint, as his concluding remark notes with satisfaction that Weyl had al-ready provided an important if partial first step toward confirmation.

The important point is that the manipulation of symbols has to be rule-like to be valid, even though the symbols themselves need have no necessarymeaning; but, and this is important, the choice of symbols and the rules toguide their valid manipulation has a goal. It is widely accepted that mathe-matics deals with impersonal truths. ‘‘Meaning,’’ however, is that whichattaches to our personal views. Since mathematical truth has to be commu-nicated to other people, who may each have different ‘‘meanings,’’ it is thena necessary consequence that mathematical communication itself must bedevoid of ‘‘personality.’’ In other words, the program of Hilbert was, forvon Neumann, just a consequence of the need to find those rules that willdetermine what we can agree on independently of our personal ‘‘mean-ings.’’ It is a minimalist program of communication.

This openness to alternatives is suggested by von Neumann’s com-ments in the ensuing discussion. The intervention comes at Carnap’s at-tempted reconciliation of the several views of mathematics. Von Neumannimmediately spoke up to express his doubts whether ‘‘all’’ of mathematicscan be enclosed within the framework suggested by Carnap.12

With regard to your interpretation of the concept of consistency, Iwould like to remark that I doubt whether that really works. Thesituation is this: In Hilbert’s work symbols without actual meaningare introduced. But the introduction of these ‘‘meaningless’’ sym-bols is for Hilbert not an end in itself. The agreeable experienceswith positive whole numbers do not entitle one to an optimisticview toward subsequent results. [Even] if Hilbert’s proof of consis-tency is successful, it is questionable whether that yields a possibil-ity for interpretation. For an axiom system to be free fromcontradiction, it is enough that each finite subset of it be free fromcontradiction. Therefore one attempts to indicate a possible inter-

11 Ibid., 65.12 Ibid.

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pretation for finite subsets of the system. [But] the everlasting fluc-tuation of these provisional interpretations shows that one cannotarrive at a definitive interpretation from them without further con-siderations. Thus one can indeed arrive at a consistency proofwithout finding an interpretation for the mathematics. So I do notbelieve that the proof of consistency suffices.

After Godel’s initial long statement, which hints around his famous incom-pleteness result, von Neumann gets a chance to reiterate his convictions.He bluntly states his doubts about the necessary coherence of our intuitionwith formalism. ‘‘It is not settled whether all rules of inference that areintuitionistically permissible may be formally reproduced.’’ This remarkwas made well before Godel’s famous announcement of his undecidabilityresult and shows that von Neumann’s mind had been prepared for receivingsuch news. And it is at this point, after von Neumann’s statement, thatGodel makes his claim in clear language. ‘‘One can (assuming the consis-tency of classical mathematics) even give examples of propositions . . .which are really contentually true but are unprovable in the formal systemof classical mathematics.’’ The readiness with which von Neumann ac-cepted the full import of Godel’s results, and indeed his own independentdiscovery of the second incompleteness result, clearly shows an open mind,and one ready for the failure of Hilbert’s program.13 For John von Neu-mann, the axiomatics of set theory was only a procedural posture.14

It may help to combine the implications of sections II and III as follows.From von Neumann’s views on scientific method, as indicated in his lectureon ‘‘The Mathematician,’’ he was a follower of the Mach-Poincare-Duhemview that Science is successful when it summarizes many facts into a fewprinciples, i.e. that

13 Von Neumann interpreted the negative import of Godel’s second Incompleteness resultmore strongly than Godel himself. See Kurt Godel, Collected Works, vol. 5, ed. by Solo-mon Feferman et al. (Oxford: Clarendon Press, 1986–2003). This point is raised by Wil-fred Seig in his Introduction to the correspondence, which is valuable because it translatesparts of von Neumann’s writings in German, which are not otherwise readily available.14 Von Neumann himself admits as much in a letter to Carnap of June 7, 1931, in Johnvon Neumann: Selected Letters, 85: ‘‘There are some programmatic publications by Hil-bert in which he claims that certain things have been proven or almost proven while thisis in fact not even approximately the case (continuum problem and so on). Therefore Iwould like neither to quote these nor to correct or ignore them, and I therefore believethat the best course of action is to include a general reference to the handbook article.’’

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A. Science is an economical summary of the world;15

From his attitude to Hilbert’s Program and to the foundational aspects oflogic and set theory, we find von Neumann arguing thatB. Mathematics is interpersonal communication of those parts of sciencethat can be made unambiguous.Since most scientific thought is expressed in mathematical form, the twostatements are commonly conjoined, but it is helpful to separate them forclarity. If we now interpolate between these statements, and note that nei-ther A nor B contains a claim of uniqueness, we find that von Neumannwould be open to the assertion that ‘‘More than one mathematical sum-mary of the world, i.e. more than one science, may exist.’’ This is no puzzlefor von Neumann, because science is what science does—it is only a toolfor manipulating the world. von Neumann’s lowbrow espousal of the viewthat science is known only by its applications is thus consistent with hisapproach to mathematics.

IV. ECONOMICS16

If we take the period between 1940 and 1955 as marking the time whenvon Neumann was involved with economics, three issues stand out: optimi-zation as the basis of individual behavior; equilibrium concepts for socialinteraction, and the use of mathematics in economics. At a time when verylimited use of mathematics was being attempted, von Neumann was adefinite advocate for more extensive mathematical modeling, but the situa-tion is not so clear on the other issues. Paul Samuelson is closely linkedwith the emphasis upon optimizing economic agents, while John Nash isassociated with the notion of social equilibrium based upon the noncooper-ative behavior of individuals—both are Nobel laureates in Economics.Folklore has it that von Neumann had disagreements with both Nash andSamuelson, thus suggesting that he preferred cooperative social action andthat he had doubts about optimization. I am unable to find any grounds forthe assertion that there was direct opposition with John Nash and/or withPaul Samuelson, but the issue merits further discussion.

One has to begin by looking at von Neumann’s growth model. This

15 Among well known recent mathematicians both Gregory Chaitin and Stephen Wolframappear to support such a view of science.16 This entire section is so greatly indebted to the comments made by Kenneth Arrow thatI will not clutter it with repeated references—but my debt remains.

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paper preceded his work in hydrodynamics and it avoids methodologicalpronouncements.17 However, the preamble reveals von Neumann’s aware-ness of the need for topology and fixed points,18 of the emergence of max-min problems, of maximization, of duality, and of efficiency—all very sig-nificant for showing that von Neumann himself had pioneered the use ofsome of the tools critical to modern mathematical economics.

At the very beginning of the Theory of Games, ‘‘von Neumann’’ pro-vides a careful and spirited defense of the use of mathematics in economics.

Mathematics has actually been used in economic theory, perhapseven in an exaggerated manner. In any case its use has not beenhighly successful. This is contrary to what one observes in othersciences: There mathematics has been applied with great success,and most sciences could hardly get along without it. Yet the expla-nation for this phenomenon is fairly simple19

‘‘Von Neumann’’ goes on to point out that every objection now being madeagainst the use of mathematics would have been made with equal cogencyagainst those branches of physical science which today successfully rely onmathematics. For example, heat was felt to be an ordinary quantity capableonly of being known as warmer or colder and yet eventually came to beprecisely measurable. Whether or not economics was significantly differentfrom physics was a point to be revealed only in the course of the actualdevelopment of economic theory. ‘‘Applicability’’ was not an inappropriatecriterion, but it was to be answered only after the trial had been made. Theproper scope and limits of mathematics was thus an ‘‘empirical’’ question.The proper procedure was therefore to learn what had led to success in thephysical sciences and to imitate this method as far as possible.

The great progress in every science came when, in the study of prob-lems which were modest as compared with ultimate claims, methods weredeveloped which could be extended further and further.

17 In his article on von Neumann in the New Palgrave (MacMillan 1987) Gerald Thomp-son notes that the paper was originally presented at Princeton in 1932 but was so farahead of its time that it made no impression. Von Neumann eventually published it somesix years later. For an alternative appreciation of von Neumann see Mohammed Dore, MH I, Sukhamoy Chakravarty, and Richard Godwin, eds., .John von Neumann and mod-ern economics (Oxford: Clarendon Press, 1989).18 The growth model was subsequently solved by more elementary mathematics, but thisdoes not affect the point.19 John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behav-iour (New York: John Wiley, 1944), 3.

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Very frequently the proofs are lacking because a mathematicaltreatment has been attempted of fields which are so vast and socomplicated that for a long time to come—until much more empir-ical knowledge is acquired—there is hardly any reason at all toexpect progress more mathematico.20

In subsequent years this strand of thought, reinforced by von Neumann’swork in turbulence, grew to be increasingly important. The last sentencequoted, where ‘‘von Neumann’’ admits that the mathematical methodsheretofore used were quite inadequate for the task at hand, is perhaps oneof the most significant differences between his approach and the mathemat-ical economics of the next half-century. What von Neumann meant by thischallenge is not easy to divine, but if we take it to be the use of the computerto assist hypothesis formation and verification (a position defended atlength in the next section), the last decade has seen considerable supportfor von Neumann’s vision.21 Whether it be at the Santa Fe Institute, wherecomplex system theory is being applied to economics, in the new field ofexperimental economics, or in the increasing popularity of computationaleconomics and agent based modeling, we see a variety of approaches thatare in the spirit of von Neumann’s methodological openness and his reli-ance upon the computer.

While von Neumann’s basic insight—that mathematics can be ex-pected to be useful, and that its applicability was a question only to beanswered through testing—has been vindicated, his contribution is limitedto general guidelines because there is some uncharacteristic vagueness in thedetails of von Neumann’s economics. He used a very pessimistic solutionconcept, the minmax, in formulating the theory of games. If this was morethan a mathematical convenience, one would expect him to attempt toapply the minmax in social situations. But we have a remarkable quotefrom Kenneth Arrow (yet another Nobel laureate) that ‘‘von Neumann washere recently. He stated that he did not agree with Wald in applying min-max ideas to the treatment of uncertainty.’’22 So perhaps the minmax justhappened to be mathematically convenient? This thought leads naturally tothe purported differences with Paul Samuelson and John Nash.

I quote at some length the only known contact between von Neumann

20 Ibid., 5.21 I am indebted to Ed McPhail for this point.22 Quoted in Philip Mirowski, Machine Dreams: Economics becomes a Cyborg Science(Cambridge and New York: Cambridge University Press, 2002), 128n.

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and Samuelson, if only to emphasize how little it tells us about any potentialantagonism.

I had only one encounter with the formidable John von Neu-mann, who of course was a giant of modern mathematics and whoin addition proved himself to be a genius in his work on the hydro-gen bomb, game theory, and the foundations of quantum mechan-ics. . . . Sometime around 1945 von Neumann gave a lecture atHarvard on his model of general equilibrium. He asserted that itinvolved new kinds of mathematics which had no relation to theconventional mathematics of physics and maximization. I pipedup from the back of the room that I thought it was not all thatdifferent from the concept we have in economics of the opportu-nity-cost-frontier, in which for specified amounts of all inputs andall but one output society seeks the maximum of the remainingoutput. Von Neumann replied at that lightning speed which wascharacteristic of him: ‘‘Would you bet a cigar on that?’’ I amashamed to report that for once little David retired from the fieldwith his tail between his legs. And yet some day when I passthrough Saint Peter’s Gates I do think I have half a cigar still com-ing to me—only half because von Neumann also had a validpoint.’’23

Samuelson’s own approach involved traditional calculus, which re-quired that demand equal supply for every available good, while von Neu-mann had insisted earlier that goods in excess supply, ‘‘free goods,’’ shouldbe incorporated into the model and this required dealing with inequalitiesas well as equalities. Did the equality/inequality issue separate von Neu-mann and Samuelson? The role of inequalities in motivating some of vonNeumann’s reservations about Samuelson is certainly possible. But theremust be more. In the 1930s Tjalling Koopmans and Leonid Kantorovichhad each independently modeled activity analysis, which included inequali-ties, and so did the mathematical work of Theodore Motzkin. So the useof inequalities cannot have been quite novel. Given von Neumann’s ownrecognition of free goods in the growth model, inequalities alone cannotaccount for the deprecatory remarks about Mathematical economics madeto Morgenstern (in private) and should not have kept von Neumann from

23 The Collected Scientific Papers of Paul Samuelson, 3 vols., ed. by Joseph E. Stiglitz andRobert C. Merton (Cambridge, Mass.: MIT press, 1966–1972), 3:14–15.

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writing a book review of Samuelson. Optimization and equilibrium are theonly other viable candidates to account for the difference with Samuelson.As one cannot really engage in analysis without some sort of equilibrium inmind, optimization seems to be the more probable candidate. Since vonNeumann had already used both efficiency and duality in his growthmodel,24 one cannot properly argue that von Neumann was opposed tomaximization. The seeming reluctance to put optimization front and centerdoes suggest some difference in emphasis.

An alternative explanation would look at the difference more as a con-sequence of von Neumann’s mathematical habits rather than of any princi-pled objection to optimization. In connection with his own work onhydrodynamics, von Neumann eloquently explained to Oswald Veblen whyit was desirable to have a ‘‘global’’ or teleological view of the subject.

More specifically: since the equations of classical point mechanicscan be put into a variational form, it was to be expected that any-thing that stems from classical point mechanics will also possesssuch a formulation. This is indeed so for all the three fields men-tioned above—quantum mechanics, hydrodynamics and electro-dynamics. The great virtue of the variational treatment, ‘‘Ritz’smethod,’’ is that it permits efficient use, in the process of calcula-tion, of any experimental or intuitive calculation. It is importantto realize that this is not possible, or possible to a much smallerextent, if one performs the calculation by using the original formof the equations of motion—the partial differential equations. In-deed, some general insight into the nature of the problem can beincorporated into even such a calculation, like symmetry, station-arity, similitude properties, although even in these cases such ‘‘sim-plifying assumptions’’ frequently lead to the oddest and entirelyextraneous complications. But any simple experimental knowl-edge about the approximate shape of the solution, or the qualita-tive position of certain salient features is much harder to make useof.25

Von Neumann recognized how ‘‘global’’ approaches, of which optimiza-tion is one, can provide an entry for ‘‘experimental or intuitive calculation’’;however, he himself was unable to provide such insights. So his attempts to

24 And since the Theory of Games contains the axiomatics of expected utility theory.25 von Neumann, Collected Works, 6:358.

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minimize the role of philosophy was partly a defense of his own ‘‘symbol-processing’’ style of thought, even as he recognized that others with intu-ition were able to attain those breakthroughs to which his ‘‘symbol process-ing’’ syntactic approach was not privy. The polite refusal to reviewSamuelson’s Foundations of Economic Analysis is consistent with thisreading.

Samuelson’s book is very interesting and very detailed, and clearlyrepresents the result of a great deal of work. It would be improperto accept the reviewing in any different spirit from this. Besides,the whole subject of the use of mathematical methods in econom-ics is one which I wouldn’t like to deal with in print except after avery careful study of the corpora delicti and of my correspondingformulations. The methodological questions which are involvedare very delicate and it is very easy with respect to them to sin byoverstatement as well as understatement.26

The ‘‘arguments’’ with John Nash are reported thus by Sylvia Nasar.

Nash started to describe the proof he had in mind for an equilib-rium in games of more than two players. But before he had gottenout more than a few disjointed sentences, von Neumann inter-rupted, jumped ahead to the yet unstated conclusion of Nash’sargument, and said abruptly, ‘‘That’s trivial, you know. That’s justa fixed point theorem.’’27

Keeping in mind the vagaries of reports from earlier decades and the possi-bility that von Neumann only meant that he could see the proof to be sim-ple (for him!), we have to ask how far this could also be a seriousobjection.28 To object to the Nash solution by pointing out that it was‘‘just’’ a fixed-point goes against von Neumann’s own professed methods.As noted earlier, in the Theory of Games, the example of heat measurementas a paradigm was urged upon us—would von Neumann have dismissed

26 von Neumann to Morgenstern, Oct 31,1947, in von Neumann: Selected Letters, 128.27 Sylvia Nasar, A Beauftiful Mind (New York, N.Y.: Simon & Schuster, 1998), 94. Thenote tells us that this is from the memory of Harold Kuhn in 1997.28 We do not know how Nash explained his own approach and what von Neumann wasdismissive about. Only with the passage of time did the Nash solution emerge victorious.Robert Leonard, ‘‘Reading Cournot, Reading Nash: the Creation and Stabilisation of theNash Equilibrium,’’ in Economic Journal 104 (1994): 492–511.

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the absolute zero as ‘‘just’’ a calibration problem? What von Neumannshould have done, following the arguments he so repeatedly made, wasto see how far this non-cooperative Nash approach could illuminate realeconomic problems and then make a final judgment.29 That he failed to doso is partly explained by a later letter to Harold Kuhn, in which he elabo-rates upon his incredibly ambitious view of what mathematical economicsis supposed to achieve.

I am not aware of any experimental effort towards the determina-tion of actual human behavior in actual n-person games, otherthan the ones you are familiar with. The problem is certainly adifficult one. I have, of course, thought about it in the past, andthe picture which I used was this: I think that nothing smaller thana complete social system will give a reasonable ‘‘empirical’’ pic-ture. Here, over relatively long periods of time, one can meaning-fully assert that the ‘‘system’’ has not changed, while the positionsof various participants within it may have changed many times.This would seem to me to be the analogue of a single solutionand an ‘‘exploration’’ of the imputations that belong to it. Afterrelatively long times, there occur discontinuous changes, ‘‘revolu-tions,’’ which produce a different ‘‘system.’’ It would seem to meto be true, that the imputations belonging to a single ‘‘system,’’i.e., solution, have a certain stability relatively to each other (i.e.,they do not dominate each other), while no such condition is satis-fied in the relationship between imputations belonging to consecu-tive, different ‘‘systems.’’30

This is a breathtaking prospect. Von Neumann seems to want to incor-porate all coalitions and their machinations within his mathematics of soci-ety—even before the theory of selfish individuals acting alone had beenproperly addressed. Is it not curious how someone who constantly urgedthe use of mathematics for its iterative clarity, which called for step by stepprogress, somehow set his eyes on a distant, and perhaps unreachable, goal

29 There is also a brief reference by von Neumann to a game invented by Nash (probably‘Hex’) in a letter to Tuckerman, but this has no bearing on the concerns of this paper.von Neuman to Tuckerman, September 25,1950, in von Neumann: Selected Letters, 248.30 von Neumann to Harold Kuhn, April 14,1953, in ibid., 170. As von Neumann alsorepeats several times that the methods employed in the Theory of Games are tentative,limited, beginnings, we have curious mixture of pronouncements. See also von Neumannto Wold, October 28, 1946, in ibid., 283.

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even as the process of mathematization in economics was just beginning?This would partially explain his constant repetition of the idea that entirelynew mathematics was called for to make real progress in mathematical eco-nomics.

The enormous hopes for mathematical economics expressed by vonNeumann are followed by lapses from clarity. Scholars have noted how thegame theory espoused in 1944 (both zero-sum two-person and cooperative)makes heavy assumptions on the rationality of the players and are at vari-ance with their occasional statement of the need to look at behavior empiri-cally. Or consider again von Neumann’s repeated claim that economicconcepts are not clear and that this was what prevented mathematicalbreakthroughs in economics.

What seems to be exceedingly difficult in economics is the defini-tion of categories. If you want to know the effects of the produc-tion of coal on the general price level, the difficulty is not so muchto determine the price level or to determine how much coal hasbeen produced, but to tell how you want to define the level andwhether you mean coal, all fuels, or something in between. Inother words, it is always in the conceptual area that the lack ofexactness lies. Now all science started like this, and economics, asa science, is only a few hundred years old. The natural scienceswere more than a millennium old when the first really importantprogress was made.31

Perhaps von Neumann was being too hasty here. He failed to ask why theconcepts are not clear. Is it for some remediable faults, such as a lack oftrying, or is it due to a lack of intelligence, or is it inherent in the subject?If the subject is composed of many causal factors, too numerous to be use-fully listed, then a degree of indefinite aggregation32is normally employed,and so we will always face von Neumann’s question—‘‘whether you meancoal, all fuels, or something in between.’’ Further, if the relative strength ofthese causal factors is constantly subject to unpredictable shifts as coal, oiland natural gas shift in relative importance, then clarity of the sort von

31 von Neumann, ‘‘The Impact of Recent Developments in Science on the Economy andon Economics,’’ Speech to the National Planning Association (1956), in Collected Works,6:101. The claims of doing something entirely new are frequently made by von Neumannand seem to be a characteristic fault—see S. M. Ulam, ‘‘John von Neumann: 1903–1957,’’ in Bulletin of the American Mathematical Society 64 (1958): 38.32 Meaning an aggregation that is instinctively done without any guidance from rules.

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Neumann was seeking will frequently fail. The assertion that the physicalsciences have been gathering data for centuries seems to indicate a misun-derstanding of the root issue, since the physical data refer to an unchangingstructure while economic data come from structures that are continuallyevolving. We know that that von Neumann emphasized applications andempirical facts as the guide for theory. Yet his own understanding of eco-nomics began with the Austrian School, whose methodological basis (called‘‘praxeology’’) minimizes the importance of both facts as well as of ma-thematization (largely because the economic structure keeps changing). Per-haps von Neumann never quite reconciled the conflicting claims of hiseconomic mentors with his own belief in the potentially universal usefulnessof a utilitarian mathematics.33

V. HYDRODYNAMICS34

One thinks first of Quantum Physics whenever von Neumann’s relationshipto physical theory is concerned because of the famous Mathematical foun-dations of Quantum Mechanics, which he wrote around 1930. Yet thisbook is not concerned with empirical anomalies and eschews any novelpredictions. It is concerned solely with providing a coherent mathematicalapparatus for the known results of quantum physics.35 More than onescholar has remarked that the Mathematical Foundations was more a con-tribution to philosophy than to physics.36 Ulam reports that von Neumannwas proudest or happiest with his work on quantum mechanics, a volumethat has many philosophical asides.37 The goal of science as explicit and

33 It is curious that no one appears to have picked up on the potential of von Neumann’scontribution to Social Choice. In 1947, when asked how votes should be apportioned byCongress so as to be fair, von Neumann is said to have suggested five possibilities; small-est divisors, equal proportions, harmonic means, major fractions, and greatest divisors(McRae, 337). The entire report and the reaction of Congress bears further examination.34 This section borrows heavily from my earlier paper, ‘‘John von Neumann, ScientificMethod and Empirical Economics,’’ The Journal of Economic Methodology 1 (1994):279–94.35 See the comments of Freidrichs as quoted by McRae, 141.36 Stanislaw Ulam is said to have considered it as having largely philosophical significance,and Miklos Redei seems to reinforce in his essay, ‘‘Mathematical Physics and Philosophyof Physics (with special consideration of J. von Neumann’s work),’’ in History and Philos-ophy of Science: New Trends and Perspectives, ed. Michael Heidelberger and FriedrichStadler (Dordrecht and Boston: Kluwer Academic, 2002), 239.37 One of the more suggestive such issues is the reality of ‘‘consciousness’’ within vonNeumann’s formulation. One wonders if this could have a relationship with the philoso-phy of George Berkeley? In an appreciation written on the 100th birthday of von Neu-mann, Stephen Wolfram suggests that the von Neumann formulation gave a greater feel

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communicable knowledge is clear from Ulam’s characterization of thequantum mechanics as38 ‘‘an attempt to make a rational presentation of aphysical theory which, as originally conceived by the physicists, was basedon non-universally communicable intuitions.’’39

However, it would be a mistake to consider this mathematical contri-bution to be representative of von Neumann’s scientific approach, espe-cially since the details of the change can be demonstrated. The preface tothe Theory of Games is dated January 1943; the evolution towards empiri-cally based theory, only hinted at in the introduction to the Theory ofGames, develops into intense appreciation of the role of experiment inguiding theory in a paper on ‘‘Oblique Reflection of Shocks’’ for the Bureauof Ordinance in 1943. Many have observed that von Neumann was deci-sively influenced towards applications and the role of computations by hiswork for the military. The studies on turbulence show a repeated series ofreferences to our inability to solve the important questions ‘‘mathemati-cally’’; this is linked with the desire to learn more by experiment—considerhis perception of wind tunnels as analog computers—and to try out compu-tational ways of narrowing down the mathematical possibilities so as toarrive at a practical solution.

As our first results are already rather paradoxical, early experi-mental tests become highly desirable. In the subsequent develop-ments the incomplete-ness of the theory and the extrememathematical difficulties and ambiguities which beset it gavethe experimental approach an even more far-reaching and quitepeculiar significance: Experiments became necessary in orderto provide guidance for the mathematical development of thetheory—which, had it been carried on ‘‘more mathematico,’’would have presented overwhelming difficulties.40

This paper is also notable because it contains a long section (section 12)dealing carefully with the experimental evidence.41

of ‘‘completeness’’ than later appeared [citation???]. the URL below http://forum.wolframscience.com/showthread.php?s�&threadid�139.38 Ulam, ‘‘John von Neumann: 1903–1957,’’ 2239 But it was not entirely a ‘‘game.’’ So von Neumann refused to follow up on a model ofDirac he considered ‘‘artificial and arbitrary,’’ von Neumann to Ortvay, in Redei, Letters,195 [Is this from a letter in Redei ed., 2005, or in the article by Redei, 2002???]40 Collected Works,6: 238–941 I have been unable to get expert guidance on this point but I assume von Neumann isreferring to the Navier-Stokes equations, which are still unsolved in general, and whichare one of the problems noted among the Clay Prize collection.

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Two years later, in a paper delivered at the Institute for AdvancedStudy, von Neumann considered the interaction between theory and experi-ment more carefully from a methodological point of view. He begins witha frank admission of ignorance and then argues that: ‘‘The theory is in amost unsatisfactory state, and it is very questionable whether our presentmathematical methods are at all adequate to handle it. Actually, at some ofthe most ‘strategic’ points more light has been thrown on the situation byexperimental work than by theoretical work.’’ At about the same time, vonNeumann states explicitly the considerable inadequacies of the availablemathematics in the face of considerable experimental complexity.

The question as to whether a solution which one has found bymathematical reason really occurs in nature and whether the exis-tence of several solutions with certain good or bad features can beexcluded beforehand, is a quite difficult and ambiguous one. . . .Mathematically, one is in a continuous state of uncertainty, be-cause the usual theorems of existence and uniqueness of a solutionthat one would like to have, have never been demonstrated andare probably not true in their obvious forms.42

Later he emphasized the same issue in different ways,

Thus there exists a wide variety of mathematical possibilities influid mechanics, with respect to permitting discontinuities, de-manding a reasonable thermodynamic behavior, etc., etc. Thereprobably exists a set of conditions under which one and only onesolution exists in every reasonably stated problem. However, wehave only surmises as to what it is and we have to be guided almostentirely by physical intuition in searching for it. It is therefore im-possible to be very specific about any point. And it is difficult tosay about any solution which has been derived, with any degree ofassurance, that it is the one which must exist in nature.43

The belief that we can discover those regularities ‘‘which must exist in na-ture’’ indicates a metaphysical attitude of confidence in human abilitieswhich will be discussed in the next section. In urging us to look at facts andlearn von Neumann is also suggesting that we are so constituted as to be

42 Collected Works,6: 29, emphasis added.43 Ibid.

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attracted to significant regularities among the mass of unorganized facts.So von Neumann was asking us not to rely on either deduction or inductionalone, but rather to accept something like the viewpoint of C. S. Pierce,who suggested the name ‘‘abduction’’ for the inferential process involved.Abduction involves the belief that we are made so that we can recognizeregularities if we approach facts with a minimum of presuppositions and insuggesting that facts are useful not only in testing theories but also in dis-covering theories. That there was something in the nature of reality to guideus along our investigations appears to be clearly expressed by Goldsteinand von Neumann’s recognition of the fact that the physical insights ofRiemann and Plateau were important for advances in solving elliptic differ-ential equations.

Such advances have been made in the theory of nonlinear partialdifferential equations, are also covered by this principle, just inwhat seems to us to be the most decisive instances. Thus, althoughshock waves were discovered mathematically, their precise form-ulation and place in theory and their true significance has beenappreciated primarily by the modern fluid dynamicists. The phe-nomenon of turbulence was discovered physically and is stilllargely unexplored by mathematical techniques.44

In his last review of turbulence, what may be considered his most matureview of the subject, von Neumann provided a historical and evaluativeoverview of the subject and returned to the incompleteness of the theoreti-cal approaches.45 ‘‘The entire experience with the subject indicates that thepurely analytical approach is beset with difficulties, which at this momentare still prohibitive.’’ Von Neumann considered this to be due to an intu-itive understanding that was ‘‘still too loose.’’ He suggested a way out byusing computers extensively.

Under these conditions there might be some hope to ‘‘break thedeadlock’’ by extensive, but well-planned, computational efforts.. . . This should, in the end, make an attack with analytical meth-ods, that is more truly mathematical, possible.

These few quotes should serve to show that von Neumann took no sidesin the theory-experiment dispute. As far as research on Turbulence was

44 Ibid., 5:4.45 Ibid., 6:469.

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concerned, he was ready to try whatever worked. The difference of empha-sis with Clifford Truesdell and the school of Rational Mechanics—trueaxiomatisers—is worth noting. Garrett Birkhoff, another eminent mathe-matician concerned with ‘‘practical’’ issues wrote about the proper ap-proach to turbulence by arguing for a mixture of mathematical logic,empirical intuition, and metaphysics. After describing the problems of theo-retical hydrodynamics as boundary value problems for sets of appropriatepartial differential equations, Birkhoff goes on to say,

However, the boundary-value problems of rational hydrodynam-ics are exceedingly difficult, and progress would have been muchslower if rigorous mathematics had not been supplemented by var-ious plausible intuitive hypotheses. Of these, the following havebeen especially suggestive:(A) Intuition suffices for determining which physical variables re-quire consideration.(B) Small causes produce small effects, and infinitesimal causesproduce infinitesimal effects.(C) Symmetric causes produce effects with the same symmetry.(D) The flow topology can be guessed by intuition.(E) The processes of analysis can be used freely: the functions ofrational hydrodynamics can be freely integrated, differentiated,and expanded in series (Taylor, Fourier) or integrals (Laplace, Fou-rier).(F) Mathematical problems suggested by intuitive physical ideasare ‘‘well set [i.e., possess unique solutions].’’46

This is a considerable list of requirements for real-world problems to fulfill,and one can well understand why von Neumann, who was trying to under-stand experimental phenomenon, would have argued so firmly for the lim-ited nature of mathematical solutions.

VI. CONCLUSION

John von Neumann had a very lowbrow approach to method. If it workedin practice, there must be a real reason for practical success and no method-

46 Garrett Birkhoff, Hydrodynamics: A Study in Logic, Fact and Similitude, 2nd rev. ed.(Princeton, NJ: Princeton University Press, 1960), 2. There is an interesting exchangebetween Theodore von Karman and von Neumann on the last point in 1951, CollectedWorks, 6:353.

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ological stricture about rigor was going to stop von Neumann from usingthe successful result.

In my own experience, which extends over only some thirty years,it [ideas of rigor] has fluctuated so considerably, that my personaland sincere conviction as to what mathematical rigor is, haschanged at least twice. And this in a short time of the life of oneindividual!Those portions of mathematics which had been questioned andwhich had been clearly useful, specifically for the internal use ofthe fraternity—in other words, when very beautiful theories couldbe obtained in those areas—that those were after all at least assound as, and probably somewhat sounder than, the constructionsof theoretical physics. And after all, theoretical physics was allright; so why shouldn’t such an area, which had possibly evenserved theoretical physics, even though it did not live up to 100percent of the mathematical idea of rigor, why shouldn’t it be pur-sued? This may sound odd, as well as a bad debasement of stan-dards, but it was believed in by a large group of people for whomI have some sympathy, for I’m one of them.47

The real world dictated, I use this word advisedly, what was to be‘‘explained.’’ The task of the intellectual, whether mathematician, physicist,or economist, was to find those ideas which would enable us to remain ‘‘inconceptual control of the world.’’ Why was mathematics so important tohim? Von Neumann himself provides the answer.

The attitude that theoretical physics does not explain phenomena,but only classifies and correlates, is today accepted by most theo-retical physicists. This means that the criterion of success for sucha theory is simply whether it can, by a simple and elegant classify-ing and correlating scheme, cover very many phenomena, whichwithout this scheme would seem complicated and heterogeneous,and whether the scheme even covers phenomena which were notconsidered or even not known at the time when the scheme wasevolved. (These two latter statements express, of course, the unify-ing and the predicting power of a theory.) Now this criterion, as

47 ‘‘The Role of Mathematics’’ (1954), in Collected Works, 6:480–81. In 1947 he hadnoted three changes.

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set forth here, is clearly to a great extent of an aesthetical nature.For this reason it is very closely akin to the mathematical criteriaof success, which, as you shall see, are almost entirely aesthetical.48

Nothing is ever really explained—all we succeed in doing is economiz-ing on the information we need to keep in conceptual control of the physi-cal world. Mathematics is the most compact way of maintaining thisconceptual control—that is all. However, there is no reason to expect thatsuch an economical answer will always exist. It is not enough to have amathematical formulation of the problem; one must actually be able toproduce workable solutions. And here von Neumann was remarkablyforceful in insisting that mathematics alone was frequently unable to pro-ceed because solutions could not be guaranteed to exist, or they may not beunique, or they may involve too many parameters for anything useful to besaid. He noted how the theoretical possibility of nuclear fission was dis-counted until the experimental results forced everyone to look again.

Mathematics economizes knowledge and keeps Man in control of hisenvironment. But why was it so important for Man to be in control? Howdid he see Man’s place in the Universe? Von Neumann never made a theol-ogy of science or believed in mankind as some abstract god. He did not likedeterminism in Science and wrote to George Gamow that, ‘‘I shudder atthe thought that highly efficient purposive organizational elements, like theprotein, should originate in a random process.’’49 He never commented onthe need for a new ethics or new values as would someone who doubtedthe fundamental moral philosophy of his age. His method was clearly op-portunistic—as in ‘‘The Mathematician’’ and his work on turbulence.

The family tried to have von Neumann go to school like everyoneelse—he never skipped a grade and tried to be as ‘‘normal’’ as possible.(Hedid receive special private coaching to bring out his undoubted genius). Helived through some very turbulent times and took various lessons to heart.But he never lost the desire for fun. One of his more endearing phrases toStan Ulam was ‘‘die gojim haben den folgenden Satz beweisen’’ (the gentileshave proved the following theorem). He made it a point to insist that hehad the best parties in Princeton. He loved racy jokes and driving recklessly.When guests came he remarked with sorrow how the European emigreslike himself brought so much high culture and wished to impart it to their

48 ‘‘The Mathematician,’’ op. cit., 1:7.49 Heims, 154.

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children but were stymied by the addiction to chewing gum. Pudgy, brainy,and utterly bourgeois.

Man is to be master of the Universe—this was an implicit article offaith for von Neumann. At a first approximation, he could see nothingwrong with Technology. He looked forward to the control of nuclear en-ergy, to its use in transmuting matter, and even in global climate control!There is no dismay at the loss of the humane life, nor any glory at exceedingthe bounds of Prometheus. Overall, the minimal references to the Hellenicheritage in his writings is surprising—and when he does refer to Euclid, hedoubts if the axiomatization was really as significant as subsequent genera-tions have made it out to be. McRae notes the Latin verse von Neumannquoted near the end to be a Latin declamation and perhaps indicative of aClassical bent, but this is a mistake, as the quoted verse is actually part ofa powerful hymn, ‘‘Dies Irae.’’50

He also believed in Human Nature. This is not just a contrast with theSocialist view of the indefinite malleability of human nature, which he mayhave seen closely in some of its unpleasant manifestations, but as a firmfaith in the human capacity for evil. Furthermore, he was skeptical of ourability to produce mechanistic or deterministic explanations of human be-ings or their decisions.51 ‘‘The human intellect has many qualities for whichno automatic approximation exists. The kind of logic involved, usually de-scribed by the word ‘intuitive,’ is such that we do not even have a decentdescription of it.’’ His ‘‘right wing’’ politics are well known; but whetherthis extended beyond his anti-communism is somewhat doubtful. After em-phasizing how the scale of all effective military operations had becomeglobal, how any attempt to halt the destructive ‘‘progress’’ of Science wasfutile, von Neumann does not attempt any grand solutions. With humility,he goes back to the everyday capacities of human beings and suggests ‘‘op-portunistic measures’’—that word again!—as the only solution to the di-lemma of the nuclear age.52

What safeguard remains? Apparently only day-to-day—or per-haps year-to-year—opportunistic measures, a long sequence ofsmall, correct, decisions. . . . To ask in advance for a complete

50 McRae, 379. I am grateful to Bruce Swann of the Classics Library in tracing the refer-ence to Dies Irae.51 ‘‘The Impact of Recent Developments in Science on the Economy and on Economics.’’Speech to the National Planning Association (1956), Collected Works, 6:100.52 Ibid., 6:518–19.

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recipe would be unreasonable. We can specify only the humanqualities required: patience, flexibility, intelligence.

Von Neumann believed that Man is to dominate, that the fundamental val-ues of the Western world were good, that Man has an inherent and ineradi-cable capacity for evil and that Life is a blessing. Are these beliefsindependently selected or were they part of larger philosophical complex?It is well known that von Neumann, who had nominally converted to Ca-tholicism as a young man, asked for a confessor when he found out abouthis fatal illness. One hesitates to distinguish between cause and effect in‘‘explaining’’ individual beliefs—did doubts about the fate of man lead vonNeumann to religion or was it the other way around?—but, regardless ofthe causation, the possibility that von Neumann’s picture of the world co-hered because of a religious worldview requires further study.53

University of Illinois.

53 Von Neumann may have been interested in the fact that the ‘‘Invisible Hand’’ wasoriginally a reference to God.

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John von Neumann and Scientific Method

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