John Von Neumann by S. Ulam
Transcript of John Von Neumann by S. Ulam
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J O H N V O N N E U M A N N
1903-1957
S. ULAM
In Jo hn vo n Ne um an n ' s dea th on Fe br ua ry 8 , 1957, the wor ld of
mathemat ics los t a most or ig ina l , penet ra t ing , and versa t i le mind.
Science suffered the loss of a universal intel lect and a unique inter
pre ter of mathemat ics , who could br ing the la tes t (and develop la tent )
appl ica t ions of i t s methods to bear on problems of phys ics , as t ron
omy, b io logy, and the new technology. Many eminent voices have
al ready descr ibed and pra ised h is cont r ibut ions . I t i s my a im to add
here a brief account of his life and of his work from a background of
personal acquaintance and f r iendship extending over a per iod of 25
years .
* * *
Jo hn von N eu m an n ( Joh nn y , a s he w as un ive r sa l ly know n in th i s
count ry) , the e ldes t of three boys , was born on December 28 , 1903,
in B udapes t , H ungary , a t t ha t t im e pa r t o f the A us t ro -H ungar i an
em pire . H is fami ly wa s we l l - to-do; h is fa ther , M ax von N eu m an n,
was a banker . As a smal l chi ld , he was educated pr iva te ly . In 1914,
a t the outbreak of the Fi rs t World War , he was ten years o ld and
en te red the gym nas ium .
Budapes t , in the per iod of the two decades around the Fi rs t World
War , proved to be an except ional ly fer t i le breeding ground for sc ien
tific talent. It will be left to historians of science to discover and ex
pla in the condi t ions which ca ta lyzed the emergence of so many br i l
l i an t i nd iv idua l s (the i r nam es abound in the anna l s o f m a the
m at i c s and phys ic s of the p resen t t im e) . Jo hn ny w as p rob ab ly the
most br i l l iant s ta r in th is cons te l la t ion of sc ient i s t s . When asked
abou t h i s ow n op in ion on w ha t con t r ibu ted to th i s s t a t i s t i ca l ly un
l ike ly phenomenon, he would say tha t i t was a coinc idence of some
cul t ura l fac tors which he could not ma ke prec ise : an externa l pres
sure on the whole soc ie ty of th is par t of Cent ra l Europe , a subcon
scious feel ing of extreme insecuri ty in individuals , and the necessi ty
of producing the unusual or fac ing ext inc t ion . The Fi rs t World War
had shat te red the exis t ing economic and socia l pa t te rns . Budapes t ,
former ly the second capi ta l of the Aust ro-Hungar ian empire , was
now the pr inc ipa l town of a smal l count ry . I t became obvious to
Received by the editors February 8, 1958.
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many scientists that they would have to emigrate and find a living
elsewhere in less restricted and provincial surroundings.
According to Fellner,
1
who was a classmate of his, Jo hn ny 's u nusu al
abilities came to the attention of an early teacher (Laslo Ratz). He
expressed to Johnny's father the opinion that it would be nonsensical
to teach Johnny school mathematics in the conventional way, and
they agreed that he should be privately coached in mathematics.
Thus, under the guidance of Professor Krschak and the tutoring
of Fekete, then an assistant at the University of Budapest, he learned
abo ut the problems of ma them atics. When he passed his "m atur a" in
1921, he was already recognized a professional mathematician. His
first paper, a note with Fekete, was composed while he was not yet
18.
During the next four years, Johnny was registered at the Uni
versity of Budapest as a student of mathematics, but he spent most
of his time in Zurich at the Eidgenssische Technische Hochschule,
where he obtained an undergraduate degree of "Diplomingenieur in
Chemie," and in Berlin. He would appear at the end of each semester
at the University of Budapest to pass his course examinations (with
out having attended the courses, which was somewhat irregular).
He received his doctorate in mathematics in Budapest at about the
same time as his chemistry degree in Zurich. While in Zurich, he
spent much of his spare time working on mathematical problems,
writing for publication, and corresponding with mathematicians. He
had contacts with Weyl and Polya, both of whom were in Zurich.
At one time, Weyl left Zurich for a short period, and Johnny took
over his course for that period.
It should be noted that, on the whole, precocity in original mathe
matical work was not uncommon in Europe. Compared to the United
States, there seems to be a difference of at least two or three years in
specialized education, due perhaps to a more intensive schooling sys
tem during the gymnasium and college years. However, Johnny was
exceptional even among the youthful prodigies. His original work
began even in his student days, and in 1927, he became a Privt
Dozent at the University of Berlin. He held this position for three
years until 1929, and during that time, became well-known to the
mathematicians of the world through his publications in set theory,
algebra, and quantum theory. I remember that in 1927, when he
came to Lww (in Poland) to attend a congress of mathematicians,
his work in foundations of mathematics and set theory was already
famous. This was already mentioned to us, a group of students, as
an example of the work of a youthful genius.
1
This information was communicated by Fellner in a letter recalling Johnny's
early studies.
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In 1929, he transferred to the University of Hamburg, also as a
Privat-Dozent, and in 1930, he came to this country for the first
time as a visiting lecturer at Princeton University. I remember
Johnny telling me that even though the number of existing and pro
spective vacancies in German universities was extremely small, most
of the two or three score Dozents counted on a professorship in the
near future. With his typically rational approach, Johnny computed
that the expected number of professorial appointments "within three
years was three, the num ber of D ozents was40 He also felt that the
coming political events would make intellectual work very difficult.
He accepted a visiting professorship at Princeton in 1930, lectur
ing for part of the academic year and returning to Europe in the
summers. He became a permanent professor at the University in
1931 and held this position until 1933 when he was invited to join
the Institute for Advanced Study as a professor, the youngest mem
ber of its permanent faculty.
Johnny married Marietta Kovesi in 1930. His daughter, Marina,
was born in Princeton in 1935. In the early years of the Institute, a
visitor from Eu rope found a wonderfully informal and ye t intense
scientific atmosphere. The Institute professors had their offices at
Fine Hall (part of Princeton University), and in the Institute and
the University departments a galaxy of celebrities was included in
w ha t q uite possibly co nstituted one of th e grea test conc entrations of
brains in mathematics and physics at any time and place.
It was upon Johnny's invitation that I visited this country for the
first time at the end of 1935. Professor Veblen and his wife were
responsible for the pleasant social atmosphere, and I found that the
von Neumann's (and Alexander's) houses were the scenes of almost
constant gatherings. These were the years of the depression, but the
Institute managed to give to a considerable number of both native
and visiting mathematicians a relatively carefree existence.
Johnny's first marriage terminated in divorce. In 1938, he re
married during a summer visit to Budapest and brought back to
Princeton his second wife, Klara Dan. His home continued to be a
gathering place for scientists. His friends will remember the in
exhau stible hospitality an d the atm osphe re of intelligence and w it one
found there. Klari von Neumann later became one of the first coders
of mathematical problems for electronic computing machines, an art
to which she brought some of its early skills.
With the beginning of the war in Europe, Johnny's activities out
side the Institute started to multiply. A list of his positions, organ
izational memberships, etc., will be found at the end of this article.
Th is mere enum eration gives an idea of the enorm ous am ou nt of work
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Johnny was performing for various scientific projects in and out of
the government.
In October, 1954, he was named by presidential appointment as a
member of the United States Atomic Energy Commission. He left
Princeton on a leave of absence and discontinued all commitments
with the exception of the chairm ansh ip of the IC BM Co m m ittee.
Adm iral Strau ss, chairm an of the Com mission and a friend of Jo hn ny 's
for many years, suggested this nomination as soon as a vacancy oc
curred. Of Johnny's brief period of active service on the Commission,
he writes:
"During the period between the date of his confirmation and the late autumn , 1955,
Johnny functioned magnificently. He had the invaluable faculty of being able to take the
most difficult problem, separate it into its compo nents, whereupo n everything looked
brilliantly simple, and all of us w ondered why w e had not been a ble to see through to the
answer as clearly as it was possible for him to do. In this way, he enormously facilitated
the work of the Atomic E nergy Com mission"
Johnny, whose health had always been excellent, began to look
very fatigued in 1954. In the summer of 1955, the first symptoms of a
fatal disease were discovered by x-ray examination. A prolonged and
cruel illness gradually put an end to all his activities. He died at
Walter Reed Hospital in Washington at the age of 53.
Johnny's friends remember him in his characteristic poses: stand
ing before a blackboard or discussing problems at home. Somehow,
his gesture, smile, and the expression of the eyes always reflected
the kind of thought or the nature of the problem under discussion.
He was of middle size, quite slim as a very young man, then increas
ingly corpulent; moving about in small steps with considerable ran
dom acceleration, but never with great speed. A smile flashed on his
face whenever a problem exhibited features of a logical or mathe
matical paradox. Quite independently of his liking for abstract wit,
he had a strong appreciation (one might say almost a hunger) for
the more earthy type of comedy and humor.
He seemed to combine in his mind several abilities which, if not
contradictory, at least seem separately to require such powers of con
centration and memory that one very rarely finds them together in
one intellect. These are: a feeling for the set-theoretical, formally
algebraic basis of mathematical thought, the knowledge and under
standing of the substance of classical mathematics in analysis and
geometry, and a very acute perception of the potentialities of modern
mathematical methods for the formulation of existing and new prob
lems of theoretical physics. All this is specifically demonstrated by
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his br i l l iant and or iginal work which covers a very wide spectrum of
con tempora ry s c i en t i f i c t hough t .
His conversat ions with f r iends on scient i f ic subjects could las t for
hours . There never was a l ack of sub jec t s , even when one depar ted
f rom ma thema t i ca l t op i c s .
Johnny had a v iv id in te res t in people and de l igh ted in goss ip . One
often had the feel ing that in his memory he was making a col lect ion of
human pecu l ia r i t i es as i f p repar ing a s ta t i s t i ca l s tudy . He fo l lowed
a lso the changes brought by the passage of t ime . When a young man,
he ment ioned to me severa l t i rnes h i s be l ie f tha t the p r imary mathe
m at ica l powers dec l ine a f te r th e age of ab ou t 26 , b u t th a t a ce r ta in
more prosa ic shrewdness deve loped by exper ience manages to com
pensate for th is gradual loss , a t leas t for a t ime. Later , th is l imit ing
age was s lowly ra ised.
He engaged occas iona l ly in conversa t iona l eva lua t ions of o ther
sc ien t i s t s ; he was , on the whole , qu i te generous in h i s op in ions , bu t
of ten ab le to damn by fa in t p ra i se . The expressed judgment was , in
genera l , ve ry cau t ious , and he was ce r ta in ly unwi l l ing to s ta te any
final op in ions ab ou t o the r s : " L e t R ha da m an ty s and Minos . . .
judge . . . . " Once when asked , he sa id tha t he would cons ider Erhard
S c h m i d t a n d H e r m a n n W e y l a m o n g t h e m a t h e m a t i c i a n s w h o e s
pecia l ly inf luenced him technical ly in his ear ly l i fe .
Johnny was r ega rded by many a s an exce l l en t cha i rman o f com
mi t t e e s ( t h i s pecu l i a r con t empora ry ac t i v i t y ) . He wou ld p re s s
s t rongly h i s t echnica l v iews , bu t defe r ra ther eas i ly on persona l o r
o rgan i za t i ona l ma t t e r s .
In spi te of his great powers and his ful l consciousness of them, he
lacked a ce r ta in se l f -conf idence , admir ing grea t ly a few mathemat i
c ians and physic is ts who possessed qual i t ies which he did not bel ieve
he himself had in the highest possible degree. The qual i t ies which
evoked this feel ing on his par t were , I fe l t , re la t ively s imple-minded
powers of in tui t ion of new truths , or the gif t for a seemingly i r ra t ional
percept ion of proofs or formulat ion of new theorems.
Qu i t e aware t ha t t he c r i t e r i a o f va lue i n ma thema t i ca l work a r e ,
to some ex ten t , pure ly aes the t ic , he once expressed an apprehens ion
tha t the va lues pu t on abs t rac t sc ien t i f ic ach ievement in our p resen t
c iv i l i z a t i on migh t d imin i sh : " The i n t e r e s t s o f human i ty may change ,
the present cur iosi t ies in sc ience may cease , and ent i re ly dif ferent
th ings may occupy the human mind in t he fu tu re . " One conve r sa t i on
cente red on the ever acce le ra t ing progress o f t echnology and changes
in the mode of human l i fe , which g ives the appearance of approaching
some essen t ia l s ingula r i ty in the h i s to ry of the race beyond which
human a f fa i r s , a s we know them, could no t con t inue .
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His fr iends enjoyed his great sense of humor. Among fel low scien
t i s t s ,
he could make i l luminat ing , of ten i ronica l , comments on h is
tor ica l or soc ia l phenomena wi th a mathemat ic ian ' s formula t ion ,
exh ib i t ing the hum or inhe ren t in som e s t a t em ent t rue on ly in the
vacuous se t . These o f t en cou ld be apprec ia t ed on ly by m a them at i
c ians .
He cer ta in ly d id not cons ider mathemat ics sacrosanct . I re
m em be r a d iscuss ion in Los Alam os, in conne ct ion w i th some phys ica l
problems where a mathemat ica l a rgument used the exis tence of e rgod-
ic t rans form at ion s and f ixed po ints . H e rem arke d wi th a su dden
smile , "M od ern m ath em at ic s can be appl ied af ter a l l I t i sn ' t c lear ,
a priori , is i t , th a t i t could be so . . . . "
I would say that his main interest af ter science was in the s tudy of
his tory . His knowledge of ancient h is tory was unbel ievably de ta i led .
He remembered, for ins tance , a l l the anecdot ica l mater ia l in Gibbon 's
Decline and Fal l and l iked to engage after dinner in historical discus
s ions .
O n a t r ip sou th , t o a m ee t ing of the A m er ican M athe m a t i ca l
Socie ty a t Duke Univers i ty , pass ing near the ba t t le f ie lds of the Civi l
War he amazed us by h is fami l ia r i ty wi th the minutes t fea tures of
the ba t t les . This encyclopedic knowledge molded his v iews on the
course of fu ture events by inducing a sor t of analy t ic cont inuat ion .
I can test i fy that in his forecasts of pol i t ical events leading to the
Second World War and of mi l i ta ry events dur ing the war , most of h is
guesses were amazingly correc t . Af ter the end of the Second World
War , how ever , h i s apprehens ions o f an a lm os t im m edia te subsequen t
ca lami ty , which he cons idered as ext remely l ike ly , proved for tunate ly
wrong. There was perhaps an inc l ina t ion to take a too exclus ive ly
ra t ional point of v iew about the cases of h is tor ica l events . This
tendency was poss ib ly due to an over- formal ized game theory ap
p roach .
A m ong o the r accom pl i shm ent s , Johnny w as an exce l l en t l i ngu i s t .
He remembered h is school Lat in and Greek remarkably wel l . In
addi t ion to Engl ish , he spoke German and French f luent ly . His lec
tures in th is count ry were wel l known for the i r l i te rary qual i ty (wi th
very few charac ter i s t ic mispronuncia t ions which h is f r iends ant ic i
pa ted joyful ly , e .g . , " in teghers" ) . Dur ing h is f requent v is i t s to Los
Alamos and Santa Fe (New Mexico) , he d isplayed a less per fec t
knowledge of Spanish , and on a t r ip to Mexico, he t r ied to make
himsel f unders tood by us ing "neo-Cast i l ian , " a c rea t ion of h is own
Engl ish words wi th an "e l" pref ix and appropr ia te Spanish endings .
B efore the w ar , Johnny spen t the sum m ers in Europe on vaca t ions
an d lec tur ing ( in 1935 a t Cam brid ge Un ivers i ty , in 1936 a t th e
In s t i tu t H en r i Po inca r in Pa r i s ) . O f ten he m en t ioned th a t pe r -
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sonally he found doing scientific work there almost impossible be
cause of the atmosphere of political tension. After the war he under
took trips abroad Only unwillingly.
Ev er since he came to th e U nited S tates , he expressed his apprecia
tion of the opportunities here and very high hopes for the future of
scientific work in this country.
To follow chronologically von Neumann's interests and accomplish
ments is to review a large part of the whole scientific development of
the last three decades. In his youthful work, he was concerned not
only with mathematical logic and the axiomatics of set theory, but,
simultaneously, with the substance of set theory itself, obtaining
interesting results in measure theory and the theory of real variables.
It was in this period also th a t he began his classical work on q ua ntum
theory, the mathematical foundation of the theory of measurement
in quantum theory and the new statistical mechanics. His profound
studies of operators in Hubert spaces also date from this period.
He pushed far beyond the immediate needs of physical theories, and
initiated a detailed stud y of rings of opera tors, which has independ ent
mathematical interest. The beginning of the work on continuous
geometries belongs to this period as well.
Von Neumann's awareness of results obtained by other mathe
maticians and the inherent possibilities which they offer is astonish
ing. Early in his work, a paper by Borel on the minimax property led
him to develop in the paper, Z ur Theorie der Gesellschaft-Spiele,
2
ideas which culminated later in one of his most original creations,
the theory of games. An idea of Koopman on the possibilities of
treating problems of classical mechanics by means of operators on a
function space stimulated him to give the first mathematically
rigorous proof of an ergodic theorem. Haar's construction of measure
in groups provided the inspiration for his wonderful partial solution
of Hubert's fifth problem, in which he proved the possibility of in
troducing analytical parameters in compact groups.
In the middle 30's, Johnny was fascinated by the problem of hy-
drodynamical turbulence. It was then that he became aware of the
mysteries underlying the subject of non-linear partial differential
equations. His work, from the beginning of the Second World War,
concerns a study of the equations of hydrodynamics and the theory
of shocks. The phenomena described by these non-linear equations
are baffling analytically and defy even qualitative insight by present
2
Paper [17].
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m ethods . N um er ica l w ork seem ed to h im the m os t p rom is ing w ay to
obtain a feel ing for the behavior of such systems. This impelled him
to s tudy new poss ib i l i t ies of computa t ion on e lec t ronic machines ,
ab in i t io . He began to work on the theory of comput ing and planned
the work, to remain unf in ished, on the theory of automata . I t was
a t the outse t of such s tudies tha t h is in teres t in the working of the
nervous sys tem and the schemat ized proper t ies of organisms c la imed
so much of h is a t tent ion .
This journey through many f ie lds of mathemat ica l sc iences was not
a resul t of rest lessness. Neither was i t a search for novelty, nor a
des i re for applying a smal l se t of genera l methods to many diverse
specia l cases . Mathemat ics , in cont ras t to theore t ica l phys ics , i s not
confined to a few central problems. The search for uni ty, i f pursued
on a pure ly formal bas is , von Neumann cons idered doomed to fa i lure .
This wide range of cur ios i ty had i t s bas is in some metamathemat ica l
mot iva t ions and was inf luenced s t rongly by the wor ld of phys ica l phe
no m ena the se wil l pro bab ly defy formal iza t ion for a long t ime to come.
Mathemat ic ians , a t the outse t of the i r c rea t ive work, a re of ten
confronted by tw o conf lic ting mo t iv a t i on s : the f irs t is to c on t r ib ute
to the edif ice of exist ing workit is there that one can be sure of
ga in ing r ecogn i t ion qu ick ly by so lv ing ou t s t and ing p rob lem sthe
second is the desire to blaze new trai ls and to create new syntheses.
This la t te r course i s a more r i sky under taking, the f ina l judgment of
value or success appear ing only in the fu ture . In h is ear ly work,
Johnny chose the f i r s t of these a l te rna t ives . I t was toward the end of
his life that he felt sure enough of himself to engage freely and yet
pa ins takingly in the crea t ion of a poss ib le new mathemat ica l d is
c ip l ine . This was to be a combinator ia l theory of automata and or
ganisms. His i l lness and premature dea th permi t ted h im to make only
a beginning.
In h is cons tant search for appl icabi l i ty and in h is genera l mathe
mat ica l ins t inc t for a l l exact sc iences , he brought to mind Euler ,
Po inca r , o r in m ore r ecen t t im es , pe rhaps H erm ann Weyl . O ne
shou ld r em em ber tha t t he d ive r s i ty and com plex i ty o f con tem pora ry
problems surpass enormously the s i tua t ion confront ing the f i r s t two
named. In one of h is las t a r t ic les , Johnny deplored the fac t tha t i t
does not seem poss ib le nowadays for any one bra in to have more than
a passing knowledge of more than one-third of the f ield of pure mathe
m at i c s .
Early work, set-theory, algebra.
The f irs t paper , a joint work with
Fekete , dea ls wi th zeros of cer ta in minimal polynomials . I t concerns
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a general izat ion of Fejr 's theorem on locat ion of the roots of Tcheby-
scheff poly nom ials . I t s da t e is 1922. Von N eu m an n wa s not qui te
e ighteen when the ar t ic le appeared.
A no the r youth ful w ork i s con ta ined in a pap er ( in H un ga r ian wi th
a G erm an sum m ary) on un i fo rm ly dense sequences o f num bers . I t
conta ins a theorem on the poss ib i l i ty of re-order ing dense sequences
of points so they wi l l become uni formly dense ; the work does not ye t
in dic ate the future de pt h of form ulat io ns no r is i t techn ical ly diff icult ,
but the choice of subject and the conciseness of technique in proofs
begins to indica te the combinat ion of se t - theore t ica l in tu i t ion and
the a lgebra ic technique of h is fu ture inves t iga t ions .
The se t - theore t ica l or ienta t ion in the th inking of a grea t number f
young m a them at i c i ans i s qu i t e cha rac te r i s t i c o f th i s e ra . The g rea t
ideas of George Cantor , which found their frui t ion f inal ly in the
theory of rea l var iables , topology and la ter in analys is , through the
work of the grea t Frenchmen, Bai re , Bore l , Lebesgue , and o thers ,
were not ye t commonly par t of the fundamenta l in tu i t ions of young
mathemat ic ians a t the turn of the century . Af ter the end of the
Fi rs t World War , however , one not ices tha t these ideas became more ,
as i t were , na tura l ly ins t inc t ive for the new genera t ion .
P a p e r [2 ] in the Acta Szeged on t ransf in i te ordina ls a l ready shows
von N eu m an n in h is cha rac te r i s t ic form an d s ty le in dea l ing wi th the
a lgebra ic t rea tment of se t theory . The f i r s t sentence s ta tes f rankly:
"The a im of th is work i s to formula te concre te ly and prec ise ly the
idea of Cantor ' s ordina l numbers . " As the preface s ta tes , the here to
fore som ew hat va gu e formula t ion of Can tor h imself i s replaced by
defini t ions which can be given in the system of axioms of Zermelo.
Moreover, a r igorous foundation for the defini t ion by transfini te in
du ct io n is out l ined . T h e in t rodu ct ion s t resses the s t r ic t ly formal ist ic
a p p r o a c h , a n d v o n N e u m a n n s t a t e s s o m e w h a t p r o u d l y t h a t t h e
sym bo ls . . . ( for "e t ce ter a" ) an d sim ilar expression s are nev er em
ployed. This t rea tment of ordina l numbers , la ter a l so cons idered by
Kuratowski , i s to th is day the bes t in t roduct ion of th is idea , so im
por t an t fo r " cons t ruc t ions" in abs t r ac t s e t t heory . Each o rd ina l
number by von Neumann 's def in i t ion i s the se t of a l l smal ler ordina l
numbers . This leads to a most e legant theory and moreover a l lows
one to avoid the concept of order type , which i s vague insofar as the
se t of al l ord ered sets s imilar to a given on e does no t exist in ax iom atic
se t theory .
Paper [S] on Pri i fer 's theory of ideal algebraic numbers begins to
indica te h is fu ture breadth of in teres ts . The paper dea ls wi th se t
theore t i ca l ques t ions and enum era t ion p rob lem s abou t r e l a t ive ly
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prime ideal components. Prfer had introduced ideal numbers as
ideal solutions of infinite systems of congruences. Von Neumann
starts with methods analogous to Krschak and Bauer's work on
Hensel's -adic numbers. Here again, von Neumann exhibits the
techniques which were to become so prevalent in the following
decades in mathematical researchof continuing algebraical con
structions, originally considered on finite sets, to the domain of the
infinitely enumerable and the continuum. Another indication of his
algebraic interests is a short note [39] on Minkowski's theory of
linear forms.
A desire to axiomatizeand this in a sense more formal and precise
than that originally considered by logicians at the beginning of the
20th centuryshows through much of the early work. From around
1925 to 1929, most of von Neumann's papers deal with attempts to
spread the spirit of axiomatization even through physical theory.
Not satisfied with the existing formulations, even in set theory itself,
he states again quite frankly in the first sentence of his paper [3]
on the axiomatization of set theory:
3
The aim of the present work
is to give a logically unobjectionable axiomatic treatment of set
theory ; the next sentence reads, I would like to say something a t
first about difficulties which make such a construction of set theory
desirable.
The last sentence of this 192 S paper is most interesting. Von
Neumann points out the limits of any axiomatic formulation. There
is here perhaps a vague forecast of Gdel's results on the existence of
undecidable propositions in any formal system. The concluding
sentence is: We cannot, for the present, do more than to state tha t
8
About this paper, Professor Fraenkel of the Hebrew University in Jerusalem
wroteme the following:
"Around
1922-23,
being then professor at Marburg University, I received from
Professor
Erhard Schmidt, Berlin (on behalf of the Redaktion of the Mathematische
Zeitschrift)
a long manuscript of an author unknown to me, Johann von Neumann,
with
the title
Die Axiomatisierung derM engenlehre,
this being his eventual doctor
dissertation
which appeared in the Zeitschrift only in 1928, (Vol. 27). I was asked to
express
my view
since
it seemedincomprehensible.I don't maintain that I understood
everything,but enough to see that this was an
outstanding
work and to
recognize
ex
ungue leonem.
While answering in this sense, I invited the young scholar to visit me
(in
Marburg) and discussed things with him, strongly advising him to prepare the
ground
for the understanding of so technical an essay by a more informal essay
which
should stress the new
access
to the problem and its fundamentalconsequences.
He
wrote such an essay under the title,
Eine
Axioma tisierung der
M engenlehre,
and I
published
it in 1925 in the Journal fr Mathematik (vol. 154) of which I was then
Associate
Editor.
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JOHN VON NEUMANN, 1903-1957
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there are here objections against set theory itself, and there is no
way known a t presen t to avo id the se difficulties. " (One is reminded
here, perhaps, of an analogous statement in an entirely different
domain of science: PaulFs evaluation of the state of relativistic
quantum theory written in his Handbuch der Physik article and the
still mysterious role of infinities and divergences in field theory.)
His second pape r [18] on this subject ha s the title,The axiomatiza
tion of set theory {An axiomatization of set theorywas th e 1925 title) .
The conciseness of the system of axioms is surprising, the introduc
tion of objects of the first and second type corresponding, respec
tively, to sets and properties of sets in the nave set theory; the
axioms tak e only a little more th an one page of pri nt. T his is sufficient
to build up practically all of the naive set theory and therewith all of
modern mathematics and constitutes, to this day, one of the best
foundations for set-theoretical mathematics. Gdel, in his great work
on the independence of the axiom of choice, and on the continuum
hypothesis, uses a system inspired by this treatment. It is noteworthy
that in his first paper on the axiomatization of set theory, von Neu
mann recognizes explicitly the two fundamentally different directions
taken by mathematicians in order to avoid the antinomies of Burali-
Forti, Richard and Russell. One group, containing Russell, J. Knig,
Brouwer, and Weyl, takes the more radical point of view that the
entire logical foundations of exact sciences have to be restricted in
order to prevent the appearance of paradoxes of the above type.
Von Ne um ann says, "th e general impression of their activity is almost
crushing." He objects to Russell's building the system of mathe
matics on the highly problematic axiom of reducibility, and objects
to W eyl's and Brou wer's rejection of wh at he considers as the greater
part of mathematics and set theory.
He has more sympathy with the second less radical group, naming
in it Zermelo, Fraenkel, and Schoenflies. He considers their work,
including his own, as far from complete, stating explicitly that the
axioms appear somewhat arbitrary. He states that one cannot show
in this fashion that antinomies are really excluded but while nave
set theory cannot be considered too seriously, at least much of what
it contains can be rehabilitated as a formal system, and the sense of
"formalistic" can be defined in a clear fashion.
Von Neumann's system gives the first foundation of set theory on
the basis of a finite number of axioms of the same simple logical
structure as have, e.g., the axioms of elementary geometry. The
conciseness of the system of axioms and the formal character of the
reasoning employed seem to realize Hubert's goal of treating mathe-
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12 S. ULAM
matics as a f ini te game. Here one can divine the germ of von Neu
mann ' s f u tu r e i n t e r e s t i n compu t ing mach ines and the " mechan iza
tion" of proofs.
Star t ing with the axioms, the eff ic iency of the a lgebraic manipula
t ion in the der iva t ion of mos t o f the impor tan t no t ions of se t theory
is a s to un d ing ; t he economy of t he t r e a tm en t s eems to ind i ca t e a
more fundamenta l in te res t in b rev i ty than in v i r tuos i ty fo r i t s own
sake . I t the reby he lped prepare the grounds for an inves t iga t ion of
the l imits of f ini te formalism by means of the concept of "machine"
o r " a u t o m a t o n . "
4
I t s eems cu r ious t o me tha t i n t he many ma thema t i ca l conve r sa
t ions on topics belonging to se t theory and al l ied f ie lds , von Neumann
even s eemed to t h ink fo rma l ly . Mos t ma thema t i c i ans , when d i s
cussing problems in these f ie lds , seemingly have an intui t ive f rame
work based on geomet r ica l o r a lmos t t ac t i l e p ic tures o f abs t rac t se t s ,
t r ans fo rm a t ions , e t c . Von N eu m an n gave t he impre s sion of ope ra t i ng
sequ ent ia l ly by pure ly formal ded uc t io ns . W h a t I me an to say is
tha t the bas i s o f h i s in tu i t ion , which could produce new theorems
and proofs jus t as wel l as the "naive" intui t ion, seemed to be of a
type t ha t i s much r a r e r . I f one ha s t o d iv ide ma thema t i c i ans , a s
Poincar p roposed , in to two typesthose wi th v i sua l and those wi th
aud i to ry i n tu i t i onJohnny pe rhaps be longed to t he l a t t e r . I n h im ,
the " a ud i to ry s ense , " howeve r , p robab ly was ve ry abs t r a c t . I t i n
vo lved , r a the r , a complemen ta r i t y be tween the fo rma l appea rance o f
a co l lec t ion of symbols and the game p layed wi th them on the one
hand , and an in te rpre ta t ion of the i r meanings on the o ther . The fore
go ing d i s t inc t ion i s som ew hat l ike th a t b e twee n a men ta l p ic tu re of
the phys ica l chess board and a menta l p ic tu re o f a sequence of moves
on i t , wr i t t en down in a lgebra ic no ta t ion .
In conversa t ions , some qu i te recen t , on the presen t s ta tus o f
founda t ions o f ma thema t i c s , von Neumann seemed to imp ly t ha t i n
his view, the s tory is far f rom having been told . Gdel 's d iscovery
should lead to a new appro ach to the un de rs ta nd ing of the ro le of
formal ism in m ath em at ic s , ra t he r tha n be cons idered as c losing the
subjec t .
Pape r [16 ] t r ans l a t e s i n to s t r i c t l y ax ioma t i c t r e a tmen t wha t was
do ne inform ally in pa pe r [2] . T h e f irst pa r t of th e pa pe r deals with
the in t roduc t ion of the fundamenta l opera t ions in se t theory , the
founda t ion of the theor ies o f equ iva lence , s imi la r i ty , we l l -o rder ing ,
and finally, a proof of the possibil i ty of definit ion by finite or trans-
4
This was, of course, very much in Leibniz's mind.
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JOHN VON NEUMANN, 1903-1957
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f in i te induct ion , inc luding a t rea tment of ordina l numbers . Von
Neumann r ight ly ins is t s a t the end of h is in t roduct ion to the paper
tha t t ransf in i te induct ion was not r igorous ly in t roduced before in any
axiomat ic or non-axiomat ic sys tem of se t theory .
Pe rhaps the m os t in t e res t ing o f von N eum ann ' s pape r s on ax io -
mat ics of se t theory i s [23] . I t has to do wi th a cer ta in necessary and
sufficient condit ion which a property of sets must sat isfy in order to
define a set of sets . The condit ion is that there must not exist a one-
to-one correspondence be tween a l l se ts and the se ts which have the
proper ty in ques t ion . This exis tent ia l pr inc ip le for se ts had been as
sumed as an axiom
6
by von N eum ann and som e o f the ax iom s as
sumed in o ther sys tems, in par t icular the axiom of choice , had been
derived from i t . Now i t is shown that , vice versa, these other axioms
im ply von N eum ann ' s ax iom , w hich the reby i s p roved cons i s t en t ,
provided the usual axioms are .
N o .
[ l2] h is grea t paper in the Mathemat ische Zei t schr i f t , Zur
Hilbertschen Beweistheorie, is de vo ted to th e prob lem of th e freedom
from cont radic t ion of mathemat ics . This c lass ica l s tudy conta ins an
exposi t ion of the pr imi t ive ideas under ly ing mathemat ica l formal isms
in general . I t is s t ressed that the whole complex of problems, origi
na ted and deve loped by H uber t and a l so t r ea t ed by B ernays and
Ackermann, have not been sa t i s fac tor i ly so lved. In par t icular , i t i s
pointed out tha t Ackermann 's proof of f reedom from cont radic t ion
ca nn ot be carr ied th ro ug h for classical ana lysis . I t is replac ed by a
rigorous f ini tary proof for a certain subsystem. In fact von Neumann's
proof show s (al th ou gh th is is no t s ta te d explici t ly) th a t f initely
i te ra ted appl ica t ion of quant i f ie rs and propos i t ional connect ives to
an y f initary ( i .e . , dec idable ) rela t ion s is con sisten t . T hi s is no t far
f rom the l imi t of what can be obta ined on the bas is of Huber t ' s or ig i
na l pro gra m , i .e . , w i th s t r ic t ly f in ita ry m etho ds . B ut von N eu m an n
at tha t t ime conjec tured tha t a l l of analys is can be proved cons is tent
w i th the sam e m ethod . A t the p resen t t im e , one canno t e scape the
impress ion tha t the ideas in i t ia ted by the work of Huber t and h is
school , developed wi th such prec is ion , and then revolut ionized by
6
Gdel says about this axiom:"The great interest which this axiom has lies in the
fact that it is a maximum principle, somewhat similar to Hubert's axiom of complete
ness in geometry. For, roughly speaking, it says that any set which does not, in a cer
tain well defined way, imply an inconsistency exists. Its being a maximum principle
also explains the fact that this axiom implies the axiom of choice. I believe that the
basic problems of abstract set theory, such as Cantor's continuum problem, will be
solved satisfactorily only with the help of stronger axioms of this kind, which in a
sense are opposite or complementary to the constructivistic interpretation of mathe
matics."
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S. ULAM
Gdel , a re not ye t exhaus ted . I t might be tha t we are in the mids t
o f ano the r g rea t evo lu t ion : the "na ive" t r ea tm en t o f se t t heory and
the fo rm a l m e tam athem at i ca l a t t em pts to con ta in the se t o f our
in tu i t ions about inf in i ty are , I th ink, turning toward a fu ture "super
se t theory . " Severa l t imes in the h is tory of mathemat ics , the in
tu i t ions or , one might be t te r say , common vague fee l ings of leading
mathemat ic ians about problems of exis t ing sc ience , la ter became in
corpora ted in a formal "super sys tem" deal ing wi th the essence of
problems in the or ig ina l sys tem.
Von Neumann pursued his in teres t in problems of foundat ions of
m ath em at ic s unt i l th e end of h is l ife. A qua r ter of a ce ntu ry af ter the
appearance of the above series of papers , one can see the imprint of
this work in his discoveries in the plans for the logic of computing
m ach ines .
Para l le l to the work on foundat ions of mathemat ics , there come
specif ic resul ts in set theory i tself and set- theoret ical ly motivated
theorems in rea l var iables and in a lgebra . For example , von Neumann
shows the existence of a set M of real numbers, of the power of the
cont inuum, such tha t any f in i te number of the e lements of
M
are
a lgebra ica l ly indep end ent . T he proof is g iven ef fec tive ly w i tho ut th e
axiom of choice . In a pap er in F un da m en ta M ath em at ic ae , [14] the
same year , a decomposi t ion of the in terval i s g iven in to countably
many dis jo in t and congruent subse ts . This so lved a problem of Ste in-
hausa specia l ingenui ty i s requi red to have such a decomposi t ion
on an in tervalthe corresponding cons t ruc t ion of Hausdorf f for the
circumference of a circle is much easier . (This is due to the fact that
the circumference of a circle may be regarded as a group manifold. )
In pape r [28] on the gene ra l m easure theory , i n Fundam enta
(1928) ,
the prob lem of a f initely ad di t iv e m ea sur e is t re ate d for su b
se ts of groups . The paradoxica l decomposi t ions of the sphere by
Hausdorf f and the wonderful ly s t range decomposi t ions of Banach
and Tarski a re genera l ized f rom the Eucl idean space to genera l non-
Abel ian groups . The af f i rmat ive resul t s of Banach on the poss ib i l i ty
of a measure for all subsets of the plane are general ized to the case
of subsets of a commutat ive group. The f inal conclusion is that al l
so lvable groups are "measurable" ( i . e . such measure can be in t ro
duced in them ) .
The problems and methods of this ar t icle form one of the f i rs t in
s tances of a t rend which developed s t rongly s ince tha t t ime, tha t of
genera l iz ing the se t - theore t ica l resul t s f rom Eucl idean space to more
genera l topologica l and a lgebra ic s t ruc tures . The "congruence" of
tw o se ts i s un ders too d to me an equivalence und er a t ransfo rm at ion
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JOHN VON NEUMANN, 1903-1957
IS
belonging to a g iven group of t ransformat ions . The measure i s a
genera l addi t ive se t funct ion . Again , the formula t ion of the problem
presages th e wo rk of H aa r and th e s tu dy of Hau sdorf f -Ban ach-
Tarsk i pa radox ica l decom pos i t ions .
6
In the sam e "a nn us mirab i l i s , " 1928, the re appe ars the ar t ic le on
the theory of games. This is his f i rs t work on what was to become
la te r an im p or tan t com bina to r i a l t heory w i th so m an y a pp l i ca t ions
and developments v igorous ly cont inuing a t the present t ime. I t i s
ha rd to be lieve th a t beginning wi th 1927, s imu l taneou s ly wi th th e
work d iscussed above , he could have publ ished numerous papers on
the m a them at i ca l founda t ions o f quan tum theory , p robab i l i t y in
s t a t i s t i ca l quan tum theory , and som e im por tan t r e su l t s on r ep re
sen ta t ion o f con t inuous g roups
Theory of functions of real variables, measure theory, topology,
continuous groups. Professor Halmos ' a r t ic le descr ibes von Neu
m ann ' s im por tan t con t r ibu t ions to m easure theory . We sha l l b r i e f ly
mention some of his resul ts in this f ield viewed against the back
ground of h is o ther work.
Paper [35] solves a problem of Haar. I t concerns the select ion of
representa t ives f rom c lasses of funct ions which are equivalent up to a
set of measure zero from l inear manifolds over products of powers of
fin ite sys tem s. T h e problem is genera l ized to me asures o the r th an
Leb esgu e ' s and an analog ous problem is so lved af f irmat ive ly .
[45] conta ins a proof of an impor tant fac t in measure theory: Any
Boolean mapping be tween two c lasses of measurable se ts (on two
m easu re spaces) which preserves the i r measu res is gen era ted b y a
po in t t r ans fo rm at ion p rese rv ing m ea sure . Th i s r e su l t is im po r tan t in
showing the equivalence of ra ther genera l measure spaces , when they
are separable and comple te , to Eucl idean spaces wi th Lebesgue meas
ure ,
and permi ts one to reduce the s tudy of Boolean a lgebras of meas
u rab le se t s to o rd ina ry m easures .
In [5 l ] von N eum ann p roves the un iqueness o f the H aar m easure
as cons t ruc ted by A. Haar ( in Ann. of Math . vol . 34 , pp . 147-169) , i f
one requi res e i ther le f t or r ight invar iance of the (Lebesgue- type)
m easure under g roup m ul t ip l i ca t ion . The theorem on un iqueness i s
proved for compact groups . A cons t ruc t ion d i f ferent f rom tha t of
Haar i s employed to in t roduce h is measure . This paper precedes the
construct ion of a general theory of almost periodic funct ions on
separable topologica l groups and a l lows a theory of the i r or thogonal
represen ta t ions .
6
Recently pushed to the most extreme minimal form by R. M. Robinson.
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In paper [54] the ordinary not ion of comple teness , usual ly def ined
only for metr ic spaces, is general ized for l inear topological spaces.
In teres t ing examples of spaces which are not metr ic but comple te are
produced. Such cases involve , of course , non-separable spaces . The
pap er a lso con ta ins a novel con s t ruc t io n of pseudo-m etr ics and convex
spaces.
In a joi nt pa pe r w ith P . Jo rd an [59] , a solut ion is given to a ques
t ion ra ised by Frchet of the charac ter iza t ion of genera l ized Huber t
space among l inear metr ic spaces . The condi t ion which i s necessary
an d suff icient , s t r en gth en ing a resul t of Frc he t , is : A l inear m etr i c
space L is isometric with a Hubert space if and only i f every 2-dimen-
s ional l inear subspace i s i sometr ic wi th a Eucl idean space .
The resul ts of [35] are general ized in a joint paper with M. H.
Sto ne [60 ] an d dea l w ith select ion of rep res en tat ive e lem ents from
residual classes in an abstract r ing modulo a given lef t- and r ight-
idea l . The ar t ic le conta ins a number of theorems on representa t ions
of Boolean r ings modulo an ideal .
In the Russ ian "Sb ornik , " [64] , von Ne um an n deals again wi th th e
problem of the uniqueness of Haar ' s measure . The previous proof of
uniqueness was accompl ished through a cons t ruc t ive process d i f ferent
f rom tha t of Haar , which conta ined no arbi t rary e lements and led
automat ica l ly to the uniqueness of the measure . In th is paper an in
dependen t t r ea tm en t o f un iqueness o f the l e f t - and r igh t - inva r i an t
exter ior measure i s g iven for loca l ly compact separable groups . (A
dif ferent proof was obta ined s imul taneous ly by Andr Wei l . )
In a jo in t pa per wi th K ura tow ski , [69] , prec ise an d s t rong re sul t s
are obta ined on the projec t iv i ty of cer ta in se ts of rea l numbers de
f ined by t ransf in i te induct ion . The ce lebra ted se t of Lebesgue ,
7
shown
previou s ly by K ura tow sk i to be of projec t ive c lass 3 , i s shown t o b e
a difference of two analyt ic sets and therefore of the second project ive
class . A genera l theorem is proved on the analyt ic charac ter of se ts
( in the sense of Hausdorff ) ob ta ine d b y cer ta in gen era l con s t ruc t io ns .
This resul t i s l ike ly to p lay an impor tant ro le in the s t i l l incomple te
theory of projec t ive se ts .
T h e Mm oire in Com posi t io M ath em at ic a , [75] , on infin ite d i rec t
products conta ins an a lgebra ic theory of opera tors and a measure
theory fo r such sys t em s , so im por tan t in m odern abs t r ac t ana lys i s .
I t sum ma rizes some of the previou s wo rk on th e a lgeb ra of funct ional
opera tors and topology of r ings of opera tors , inc luding the non-
sepa rab le hyper -H i lbe r t spaces . Methodo log ica l ly and in the ac tua l
con s t ruc t ion s , th is paper i s bo th a forerunner of an d a good in t ro duc -
7
Journal de Mathmatiques, 1905, Chapter VIII.
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JOHN VON NEUMANN, 1903-1957
17
t ion to much of the recent work in mathemat ics dea l ing , so to say ,
wi th the pyramiding of a lgebra ica l not ions . Star t ing wi th a vec tor
space , one deals f i r s t wi th the i r products , then wi th l inear opera tors
on these s tructures; and f inal ly with classes of such operators whose
algebraical propert ies are invest igated again "on the f i rs t level ." Von
N eum ann in t ended to d i scuss the ana logy o f th i s e l abora te sys t em
w i th the theory of hy perq uan t i za t io n in qu an tum theory , and con
s idered the pa per in pa r t icu lar as a m ath em at ic a l pre par a t ion for
dea l ing w i th non-enum erab le p roduc t s .
The paper [24] is, I believe, the first one in which a very significant
cont r ibut ion i s made to the complex of ques t ions or ig ina t ing in
H u be r t ' s fifth pro blem : the poss ib i l i ty of a chan ge of para m eter s in a
cont inuous group so tha t the group opera t ion wi l l become analyt ic .
The work deals wi th subgroups of the group of l inear t ransformat ions
of w-dim ensional space an d th e res ul t is aff i rm ative: E ve ry such con
t inuous g roup has a norm al subgroup , loca l ly r ep resen tab le ana ly t i
cal ly and in a one-to-one way by a f ini te number of parameters .
This i s the f i r s t of the theorems showing tha t the group proper ty
prevents the "pathologica l" poss ib i l i t ies common in the theory of
functions of a real variable. The resul ts of the paper, la ter general ized
and simplif ied by E. Cartan for subgroups of general Lie groups,
g ive de ta i led ins ight in to the s t ruc ture of such groups by the repre
senta t ion of e lements as products of exponent ia l opera tors . They
show tha t every l inear manifold which conta ins wi th every two
matrices / , V a l s o t h e i r c o m m u t a t o r UVVU, is an infinitesima l
group of an ent i re group G. This paper i s h is tor ica l ly impor tant ,
as preceding the work of Cartan, a later paper of Ado, and of course,
von Ne um an n ' s own pape r [48] wh ere H ub er t ' s fifth problem is
solved for compact groups .
This ce lebra ted resul t i s based on and s t imula ted by a paper of
Haar ( in the same volume of Annals of Math . ) where an invar iant
m e asure func tion i s i n t roduced in con t inuo us g roups . V on N eum an n
shows, (us ing an analogue of the Peter -Weyl in tegra t ion on groups
and employing the theorem on approximabi l i ty of funct ions by l inear
combinat ions of a f ini te number of eigenfunctions of an integral
ope ra to rthe m e thod o f E . Schm id t ' s d i s se r t a t ionand w i th an in
genious use of Brou w er ' s theo rem on invar ian ce of region in E ucl idean
^ -d im ens iona l spa ce ) tha t eve ry com pac t an d w -d im ens iona l topo
logica l group i s cont inuously i somorphic to a c losed group of uni tary
matrices of a f ini te dimensional Euclidean space.
The method of th is a r t ic le a l lows one to represent more genera l
(not necessar i ly ^-dimensional ) groups as subgroups of inf in i te
products of such w-dimensional groups . In the second par t of the
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pa pe r, an exa mp le is given of a f inite dim ension al non -co m pac t g ro up
of t ransformat ions ac t ing on Eucl idean space in such a way tha t no
change of parameters in the space wi l l make the g iven t ransforma
t ions analyt ic . I t was a lmost twenty years before the solu t ion of
Hilbert 's f i f th problem was completed, to include the "open" ( i .e . ,
non-com pac t ) ^ -d im ens iona l g roups , by the w ork o f Montgom ery
and G leason . V on N eum ann ' s ach ievem ent r equ i red an in t im a te
knowledge of both the se t - theore t ica l , rea l var iable techniques , a fee l
ing for the spi r i t of Brouwer ian topology, and a rea l unders tanding
of the technique of in tegra l equat ions and the ca lculus of matr ices .
A combinat ion of v i r tuos i ty in the mode of abs t rac t a lgebra ic
th inking and the employment of analyt ica l techniques can be seen
in the jo in t pa pe r [SO] wi th Jo rd an a nd W igner on an a lgebra ic
genera l iza t ion of the q ua nt um mec hanica l formal ism. This is con
ceived as a possible s tar t ing point for future general izat ions of the
qu an tum m echan ica l t heor i e s and dea l s w i th com m uta t iv e bu t no t
associa t ive hypercomplex a lgebras . The essent ia l resul t i s tha t a l l
such formal ly rea l f in i te and commutat ive r -number sys tems
are mere ly matr ix a lgebras , wi th one except ion . This except ion , how
ever , seems too narrow for the genera l iza t ions needed in quantum the
ories.
An unpubl ished resul t , announced in the Bul le t in of the American
M a t h e m a t i c a l S o c i e t y
[14,
Ap pend ix 2] con ta ins the theo rem on t he
simplici ty (of the component of uni ty) of the group of al l homeo
morphisms of the sur face of the 3-dimensional sphere . The ac tua l
theo rem i s th a t , g iven tw o a rb i t r a r y hom e om o rph i sm s A, B (ne i ther
equ al to id en ti t y) th er e ex ists a f ixed num be r (23 is suff icient ) of
conjugates of the f i rs t one whose product is equal to
B.
Hilbert space, operator, theory, rings of operators. A deta i led ac
c o u n t of v o n N e u m a n n ' s fu n d a m e n t a l a n d c o m p r e h e n s iv e t r e a t m e n t
of these topics is presented in the ar t icles in this volume by Professor
Murray and Professor Kadison. His f i r s t in teres t in th is subjec t
a lso s temmed f rom work on r igorous formula t ions of quantum theory .
In 1954, in a que s t ion nai re which von N eu m an n answ ered for th e
Nat ional Academy of Sciences , he named th is work as one of h is
th ree con t r ib u t ions to m a the m a t i c s th a t he cons ide red m o s t im
por tant . In sheer bulk a lone , papers on these subjec ts comprise
roughly one- th i rd of h is pr in ted work. These conta in a very de ta i led
analys is of proper t ies of l inear opera tors and an a lgebra ica l s tudy
of classes (r ings) of operators in infini te-dimensional spaces. The
resul t fulf i l ls his avowed purpose stated in the book, Mathematische
Grundlagen der Quantenmechanik [47a] , of dem ons t ra t ing th a t t he
ideas or ig ina l ly in t roduced by Hi lber t a re capable of cons t i tu t ing
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an adequate bas is for the phys ica l cons idera t ions of quantum theory ,
and tha t no need exis t s for the in t roduct ion of new mathemat ica l
schemes for these phys ica l theor ies . Von Neumann 's unbel ievably
detai led and meticulous work of classif icat ion of the propert ies of
l inear i ty for uni tary spaces resolves many problems for unbounded
ope ra to r s . I t g ives a com ple te theo ry of hyp erm axim al t r ans fo rm a
t ions and b r ings H uber t space a lm os t a s com ple te ly w i th in the
grasp of the mathemat ic ian as i s the case wi th the f in i te d imensional
Euc l idean space .
His in teres t in th is subjec t was cont inuous throughout h is sc ien
t i f ic l i fe . Even up to the end, in the midst of work on other subjects ,
he obta ined and publ ished resul t s on the proper t ies of opera tors and
spec t ra l theo ry . Pa pe r [ lO] wa s publ i shed in 1950, and w r i t ten in
honor o f the 75 th b i r thday o f Erha rd Schm id t . ( I t w as Schm id t w ho
first introduced him to the fascinat ions of this subject . ) No one has
done m ore than von N eum ann , a t l eas t i n the un i t a ry case and fo r
l inear t ransformat ions , towards the resolut ion of the myster ies of
non-compactness . Future work in th is d i rec t ion wi l l be based on h is
resul ts for a long t ime to come. This work is now being vigorously
con t inued by , am ong o the r s , h i s co l l abora to r s and fo rm er s tuden t s
M ur ra y in par t icu lar and one i s ent i t led to expect f rom the m fur ther
valuable ins ight in to the proper t ies of l inear opera tors .
Theory of latt ices , continuous geometry.
Birkhoff 's ar t icle ,
von
Neum ann and lattice theory,
presents the work on these subjec ts . Here
again , von Neumann 's in teres t was s t imula ted by the poss ib i l i ty of
app ly ing these new com bina to r i a l and a lgebra ic schem es to quan tum
the ory . Lat t ic e theory , a ro un d 1935, wa s be ing developed and gen
eral ized by Garret t Birkhof from the original formulat ions of
D edek ind . A t abou t the sam e t im e , an a lgebra ic and se t - theore t i ca l
s tudy o f B oo lean a lgebras w as sys t em at i ca l ly under t aken by M. H .
Stone . I remember tha t in the summer of 1935,Birkhoff, Stone , and
von N eum ann , on the i r w ay f rom a m a them at i ca l m ee t ing in Mos
cow, s topped in Warsaw and presented shor t ta lks a t a meet ing of the
Warsaw Mathem at i ca l Soc ie ty on the new deve lopm ent s in these
fields with novel formulat ions of the logic of quantum theory. The
ensuing discussions led one to expect far-reaching applicat ions of the
genera l Boolean Algebra and la t t ice theory formula t ions of the
language o f quan tum theory . V on N eum ann re tu rned to these a t
tempts severa l t imes la ter in h is work, but most of h is thoughts in
th is d i rec t ion are in unpubl ished notes .
8
8
Professor Givens is preparing an edition of the lecture notes to be published
shortly by the Princeton Press. Another paper on continuous geometry written in 1935
is being published in the Annals of Mathematics.
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His work on cont inuous geometr ies and geometr ies wi thout points
was mot iva ted by the be l ie f tha t the pr imi t ive not ions of quantum
theory deal wi th such ent i t ies ; obvious ly , the "universe of d iscourse"
consists of certain classes of identified points or linear manifolds in
H ub er t space . (Th is i s no ted ex pl ic it ly by D irac in h is book.)
Some of this work was considered for presentat ion in col loquium
lectu res ; an acco un t of i t is con ta ined in the P r ince to n I ns t i tu te Lec
tures ; some remains in manuscr ip t form. In conversa t ions wi th h im
touching upon these problems, my impress ion was tha t , beginning
ab ou t 1938, von N eu m an n fe lt th a t th e new fac ts and prob lems of
nuclear physics gave r ise to problems of an ent irely different type
an d m ad e i t less im po r ta nt to ins is t on a math em at ic a l ly f lawless
formula t ion of a quantum theory of a tomic phenomena a lone . Af ter
the end of the war , he would express sent iments , somewhat s imi lar
to r em arks r epor t ed ly m ade by E ins t e in , t ha t t he bew i lde r ing w ea l th
of nuc lea r and e l em enta ry pa r t i c l e phys ic s m ake p rem ature any a t
tempt to formulate a general quantum theory of f ields, at least for
the t ime being.
Theoretical physics . Pro fesso r V an H ove desc r ibes von N eum ann ' s
work in Von Neum ann's contributions to quantum theory.
In the ques t ionnai re for the Nat ional Academy of Science men
t ioned ear l ie r , von Neumann se lec ted as h is most impor tant sc ien
t i f i c con t r ibu t ions w ork on m a them at i ca l founda t ions o f Q uan tum
Theory and the Ergod ic Theorem ( in add i t ion to the Theory o f
Opera tors d iscussed above) . This choice , or ra ther res t r ic t ion , might
appear cur ious to most mathemat ic ians , but i s psychologica l ly in
teres t ing . I t seems to indica te tha t perhaps h is main des i re and one
of h is s t ronges t mot iva t ions was to he lp re-es tabl i sh the ro le of
m a t h e m a t i c s o n a
conceptual level
in the ore t ical ph ysics. T h e drif t ing
apar t of abs t rac t mathemat ica l research and of the main s t ream of
ideas in theoret ical physics s ince the end of the First World War is
unden iab le . V on N eum ann o f t en expressed conce rn tha t m a them at i c s
might not keep abreas t of the exponent ia l increase of problems and
ideas in physical sciences. I remember a conversat ion in which I ad
vanced the fear tha t a sor t of Mal thus ian d ivergence may take p lace :
the physical sciences and technology increase in a geometrical rat io
and mathemat ics in an ar i thmet ica l progress ion. He sa id tha t th is
indeed
might
be the case . Later in the d iscuss ion, we both managed
to c l ing , however , to the hope tha t the mathemat ica l method would
remain for a long t ime in conceptual control of the exact sciences
Art ic le [7] i s pub l i shed jo in t ly wi th H ub er t and N ord heim . Ac
cording to the preface, i t is based on a lecture given by Hubert in the
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winter of 1926 on the new developments in quantum theory, and pre
pared with the help of Nordheim. According to the introduction,
important parts of the mathematical formulation and discussion are
due to von Neumann.
The stated aim of the paper is to introduce, instead of strictly
functional relationships of classical mechanics, probability relation
ships.
It also formulates the ideas of Jordan and Dirac in a con
siderably simpler and more comprehensible manner. Even now, 30
years later, it is difficult to overestimate the historical importance and
influence of this paper and the subsequent work of von Neumann in
this direction. The great program of Hubert in axiomatization gains
here another vital domain of application, an isomorphism between a
physical theory and the corresponding mathematical system. An ex
plicit statement in the introduction to the paper is that it is difficult
to understand the theory if its formalism and its physical interpreta
tion are not separated concisely and completely. Such separation is
the aim of the paper, even though it is admitted that a complete
axiomatization was at the time impossible. May we add here paren
thetically that such complete axiomatization of a relativistically in
variant quantum theory, embracing its application to nuclear phe
nomena is still to be achieved.
9
The paper contains an outline of the
calculus of operators which correspond to physical observables, dis
cusses the properties of Hermitean operators, and altogether forms a
prelude to the Mathematische Begrundung der Quantenrnechanik.
Von Neumann's precise and definitive ideas on the role of statisti
cal mechanics in quantum theory and the problem of measurement
are introduced in [lO].
His well-known book,
[4 7 a ] ,
gives both the axiomatic treatment,
the theory of measurement, and statistics in detailed discussions.
At least two mathematical contributions are of importance in the
history of quantum mechanics: The mathematica l treatment by
Dirac did not always satisfy the requirements of mathematical rigor.
For example, it operated with the assumption tha t every self-ad joint
operator can be brought into diagonal form, which forced one to
introduce for those operators where this cannot be done, the famous
improper functions of Dirac. A priori it might seem, as von
Neumann states, that just as Newtonian mechanics required (at that
9
For an excellent succinct summary of the present state of axiomatizations of
non-relativistic
quantum theory in the domain of atomic phenomena, see the article
by George Mackey,
Quantum mechanics and Hilbert
space,Amer. M ath. Mon thly,
October,
1957,
still based essentially on von Neumann'sbook,
M athematischeGrund-
lagen der Quantenrnechanik.
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JOHN VON NEUMANN, 1903-1957
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l eas t
one
logica l ly and m ath em at ic a l ly c lear bas is for a r igorous t re a t
m en t .
Analysis , numerical work, work in hydrodynamics. An ear ly paper
is [33] . In i t , a fundamenta l lemma in the ca lculus of var ia t ions due
to Rad i s proved by means of a s imple geometr ica l cons t ruc t ion
( the lemma asser t s tha t a funct ion
z=f(x, y)
satisfies a Lipschitz
condi t ion wi th a cons tant A i f no p lane whose maximal inc l ina t ion i s
grea ter than A meets the boundary of the sur face def ined by the
given funct ion in three or more points ) . The paper i s a l so in teres t ing
in tha t the method of proof involves d i rec t geometr ic v isual iza t ions
som ew ha t r a re in von N eum ann ' s pub l i shed w ork .
T h e p a p e r
[41 ]
conta ins one of the impress ive achievements of
mathemat ica l analys is in the las t quar ter century . I t i s the f i r s t pre
cise mathematical resul t in a whole f ield of invest igat ion :a r igorou s
t rea tment of the ergodic hypothes is in s ta t i s t ica l mechanics . I t was
s t imula ted by the d iscovery by Koopman of the poss ib i l i ty of reduc
ing the s tudy o f H am i l ton ian dynam ica l sys t em s to tha t o f ope ra to r s
i n H u b e r t s p a c e . U s i n g K o o p m a n ' s r e p r e s e n t a t i o n , v o n N e u m a n n
proved what i s now known as the weak ergodic theorem, or the con
vergence in measure of the means of funct ions of the i te ra ted , meas
ure-prese rving t ran sfor ma t ion on a me asu re space . I t is th is theo rem ,
s t r eng thened shor t ly a f t e rw ards by G . D . Birkhoff, in the form of
convergence a lmost everywhere , which provided the f i r s t r igorous
mathemat ica l bas is for the foundat ions of c lass ica l s ta t i s t ica l me
chan ics . T he sub sequ en t dev e lopm ent s in th i s fie ld and the num erous
genera l iza t ions of these resul t s a re wel l -known and wi l l not be men
t ioned here in de ta i l . Again , th is success was due to the combinat ion
of von Neumann 's mastery of the techniques of the se t - theore t ica l ly
inspi red methods of analys is and those or ig ina t ing in h is own work on
opera to r s on H uber t space . S t i l l ano the r dom ain o f m a them at i ca l
physics became accessible to precise and general considerat ions of
modern analys is . In th is ins tance again , a grea t in i t ia l advance was
scored, but , of course, here the s tory is real ly quite unfinished; a
m athem at i ca l t r ea tm en t o f the founda t ions o f s t a t i s t i ca l m echan ics ,
in th e case of classical dy na m ics , is far from c om ple te I t is ve ry well
to have the ergodic theorems and the knowledge of the exis tence of
metr ica l ly t rans i t ive t ransformat ions ; these fac ts , however , form
only a bas is of the subjec t . Von N eu m an n of ten expressed in conversa
t ions a fee l ing tha t fu ture progress wi l l depend on theorems which
w ould a l low a m a them at i ca l ly sa t i s f ac to ry t r ea tm en t o f the sub
sequen t pa r t s o f the sub jec t . A com ple te m a them at i ca l t heory o f the
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B ol tzm ann equa t ion and p rec i se theorem s on the r a t e s a t w h ich sys
t em s t end tow ards equ i l ib r ium a re needed .
Im p o r t an t work i s con ta ined in the ar t ic le [56] , a jo in t wo rk wi th
S. Bochner . The use of opera tor - theore t ica l methods a l lows a ra ther
profound discussion of the propert ies of part ial different ial equations
of the type A^>=d/dt,
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JOHN VON NEUMANN, 1903-1957
25
this paper has a pra isewor thy e lementary charac ter of expos i t ion not
a lways found in h is work on Hi lber t space .
Work belonging to th is same order of ideas i s cont inued in h is
j o i n t p a p e r [91 ] w i th B arg m a nn and M ontg om ery . I t con ta ins an
account of var ious methods of so lv ing a sys tem of l inear equat ions
an d i s or iented tow ard s the poss ib il i ties , a l rea dy beginning to ap pe ar
a t tha t t ime, of computa t ions involving the use of e lec t ronic ma
chines.
In problems of appl ied analys is , the war years brought a need for
quick es t imates and approximate resul t s in problems which of ten do
no t p resen t a ve ry "c l ean" appea rance , t ha t i s t o say , a re m a the
m at i ca l ly ve ry inhom ogeneous , t he phys ica l phenom ena to be ca l
cula ted involving, in addi t ion to the main process , a number of ex
terna l per turbat ions whose ef fec t cannot be neglec ted or even sepa
ra ted in addi t ional var iables . This s i tua t ion comes up of ten in ques
t ions of present day technology and forces one, at least ini t ial ly, to
resor t to numer ica l methods , not because one requi res the resul t s
w i th h igh accuracy bu t s im ply to ach ieve qua l i t a t ive o r i en ta t ion
Thi s f ac t , pe rhaps som ew ha t dep lo rab le fo r a m a them at i ca l pur i s t ,
was rea l ized by von Neumann whose in teres t in numer ica l analys is
increased grea t ly a t tha t t ime.
A jo in t wo rk wi th H . H . Go lds t ine , [94] , prese nts a s tu dy of the
problem of the numer ica l invers ion of matr ices of h igh order . Among
other th ings , i t a t tempts to g ive r igorous er ror es t imates . In teres t ing
resul t s a re obta ined on the prec is ion achievable in inver t ing matr ices
of o rde r ^150 . Es t im a tes a re ob ta ined " in the gene ra l case . " ( "G en
e ra l " m eans tha t under p l aus ib le a s sum ed s t a t i s t i c s , t hese e s t im a tes
hold with the exception of a set of low probabil i ty. )
In a subsequent paper on th is subjec t , [109] , the problem is re
considered in an effort to obtain optimum num er ica l es t im ates . Given
a m a t r i x A = (an) (i, j = 1, 2, n) w hose e l em ent s a re independen t
random var i ab les , each norm al ly d i s t r ibu ted , t he p robab i l i t y tha t
the upper bound of this matr ix exceeds 2.72o-w
1/2
w h e r e a is the dis
persion of each variable, is less than .027 X2~~
n
n"
1/2
.
The deve lopm ent o f the f a s t e l ec t ron ic com put ing m ach ines w as
prompted pr imar i ly by the need of a quick or ienta t ion and answer to
problems in mathemat ica l phys ics and engineer ing . There i s , as a
byproduc t , an oppor tun i ty fo r som e l igh te r w ork Thus , fo r exam ple ,
one can now t ry to sa t i s fy , to a modes t extent , some of the cur ios i ty
which is fel t about certain interest ing sequences of integers , e .g. , to
mention the s implest ones, the frequency of the sequence of digi ts in
the deve lopm ent o f e an d 7r, carr ied to m an y th ou sa nd s of places.
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One such computa t ion , per formed on the machine a t the Ins t i tu te for
Advanced Study, g ives the f i r s t 2 ,000 par t ia l quot ients of the cube
roo t of 2 in i t s dev elop me nt as a cont inue d f rac t ion . Jo hn ny w as in
teres ted in such exper imenta l work no mat ter how s imple-minded
the problem; in one discussion in Los Alamos on such quest ions, he
asked to be g iven " in teres t ing" numbers for computa t ion of the i r
con t inued f r ac t ion deve lopm ent . I nam ed the qua r t i c i r r a t iona l i ty
y
g iven by the equa t ions
y = l/(x+y)
w here
x = l/(l+x)
as one in
w hose deve lopm ent the re m igh t appea r som e cur ious r egu la r i t i e s .
C om puta t ions o f m any o the r num bers w ere p lanned , bu t i t i s no t
known to me whether th is l i t t le projec t was ever pursued.
Game theory.
This subjec t forms a new, rapidly developing
ch ap ter in pre sen t -da y m ath em at ic a l researc h; i t i s essent ia l ly a
crea t ion of von Neumann 's . His fundamenta l work in th is f ie ld wi l l
be descr ibed e lsewhere in th is volume by A. W. Tucker and H. W.
K uhn and I sha l l con ten t m yse l f w i th r em ark ing tha t i t p resen t s
some of his most fecund and influential work. I t was Borel , in
a no te in the C om pte s-R en du s in 1921, wh o firs t formu la ted a ma th e
mat ica l scheme descr ib ing s t ra tegies in a game between two players .
The subjec t can , however , be da ted as rea l ly or ig ina t ing in the paper
of von N e um an n , [17] . I t is t he re th a t t he fundam enta l "m in im ax "
theorem is proved and the genera l scheme of a game between
n
p laye r s
(n^
2) i s form ula ted . Such sch em ata , qu i te ap ar t f rom the i r
in teres t and appl ica t ions to ac tua l games in economics , e tc . in
t roduced a weal th of novel combinator ia l problems of pure ly mathe
m a t i ca l i n t e res t . T he theo rem t h a t M in M ax = M ax M in and the
corol lar ies on the existence of saddle points of funct ions of many
var iab les is con ta ined in h is 1937 pap er [72] . T he y are shown to be a
consequence of a general izat ion of Brouwer 's f ixed-point theorem and
of the fol lowing geometrical fact . Let 5, T be tw o non -em pty , convex ,
c losed, and bounded se ts conta ined in the Eucl idean spaces R
n
and
R
m
r e spec t ive ly . Le t S XT be the d i rec t prod uct of these se ts and V,
W two c losed subse ts of i t . Assume tha t for every e lement x of S
the se t Q(x) of all y s u c h t h a t (x
y
y) belongs to F is a closed convex
and non-empty se t . Analogously , for every y in T the se t P(y) of all
x
s u c h t h a t
(x, y)
be longs to
W
a lso has th is proper ty . Then the se ts
V a n d W have a t leas t one point in common. This theorem, fur ther
d i scussed by K aku tan i , N ash , B row n and o the r s , p l ays a cen t ra l
role in the proofs of existence of "good strategies."
Game theory, including now a study of infini te games (f i rs t formu
la ted by M az ur in Po land aro un d 1930) i s in v igorous m ath em at ic a l
development. I t suff ices to refer to the work contained in the three
v o l u m e s , C o n t r i b u t i o n s t o G a m e T h e o r y
[102; 113; 114] ,
to poin t
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out the wealth of ideas, the variety of ingenious formulations in
purely mathematical context, and the increasing number of im
portant applications; it abounds in simply stated problems still un
solved.
Economics. The now classical treatise by Oskar Morgenstern and
John von Neumann, Theory of games and economic behavior [90]
contains an exposition of Game Theory in its purely mathematical
form with a very detailed account of applications to actual games;
and together with a discussion of some fundamental questions of
economic theory introduces a different treatment of problems of
economic behavior and certain aspects of sociology. The economist
Oskar Morgenstern, a friend of von Neumann's in Princeton for
many years, interested him in aspects of economic situations,
specifically in problems of exchange of goods between two or more
persons, in problems of monopoly, oligopoly and free competition.
It was in a discussion of attempts to schematize mathematically
such processes that the present shape of this theory began to take
form.
Th e present num erous applications to "operational research," prob
lems of communications and the statistical estimation theory of A.
Wald either stem from or are drawing upon the ideas proposed and
worked out in this monograph. We cannot outline in this article
even the scope of these investigations. The interested reader may
find an account of it in, e.g., L. Hurwicz's The theory of economic
behavior
11
and J . Marshak's Neumann's and M orgenstern's new ap
proach to static
economics.
12
Dynamics, mechanics of continua, meteorological calculations.
In two papers written jointly with S. Chandrasekhar [84and 88] the
following problem is considered. A random distribution of mass
centers is assum ed; these m ight be , for example, stars in a cluster or a
cluster of nebulae. These masses are mutually attracting and in mo
tion. Th e problem is to develop the statistics of the fluctuating
gravitational field and the study of the motions of individual masses
subject to the changing influence of the varying local distributions.
In the first paper, the problem of the rate of the fluctuations in the
distribution function for the force is solved through ingenious calcula
tions,
and a general formula is obtained for the probability distribu
tions W(F, ) of a gravitational field strength F and an associated
rate of change which is the derivative of F with respect to time.
Among the results obtained is the theorem that for weak fields the
11
American Economic R eview vol. 35 (1945) pp. 909-925.
12
Journal of Political Economy vol. 54 (1946) pp. 97-115.
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prob ability of a change occurring in the field acting a t a given ins tan t
of time is independent of the direction and magnitude of the initial
field, while for strong fields, the probability of a change occurring in
the direction of the initial field is twice as great as in a direction at
right angles to it.
The second paper is devoted to a statistical analysis of the speed of
fluctuations in the force per unit mass acting on a star which moves
with a velocity V with respect to the centroid of the nearby stars.
This problem is solved on the assumption of a uniform Poisson dis
tribution of the stars and a spherical distribution of the local veloc
ities.
It is solved for a general distribution of different masses. An
expression is derived for the correlations in the force acting at two
very close points. The method gives the asymptotic behavior of the
space correlations. Von Neumann was long interested in the phe
nomenon of turb ulence . Th e w riter remem bers discussions in 1937 on
the possibility of a statistical trea