John Von Neumann by S. Ulam

7/26/2019 John Von Neumann by S. Ulam

1/49

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.

1


2/49

2

S. ULAM

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.


3/49

JOHN VON NEUMANN, 1903-1957

3

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


4/49

4

S. ULAM

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


5/49


5

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 .


6/49

6

S. ULAM

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 -


7/49


7

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].


8/49

8

S. ULAM

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


9/49


9

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


10/49

10

S. ULAM

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.


11/49


11

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-


12/49

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.


13/49


13

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."


14/49

14

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


15/49


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.


16/49

16

S. ULAM

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.


17/49


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


18/49

18

S. ULAM

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


19/49


19

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.


20/49

20

S. ULAM

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


21/49

JOHN

VON NEUMANN 1903-1957

21

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.


22/49


23/49


23

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


24/49

24

S. ULAM

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,


25/49


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.


26/49

26

S. ULAM

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


27/49


27

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.


28/49

28

S. ULAM

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

Documents

John Von Neumann by S. Ulam