NOTES ON MATHEiviATICIANS

5. Norbert Wiener (1894 - 1964)

Y. K. Leong

University of Singapore

To the general public, Norbert Wiener is known as the

founder of the science of cybernetics (the theory of control

and communication in the anima1 and the machine). Perhaps

less well-known to the laymen is the fact that Wiener is a

mathematician of the first rank who con:tributed towards the

advancement of electrical engineering, physics and biophysics.

In doing so, he was upholding the tradition of the great

mathematical universalists of the past three centuries. His been

mathematics had often1,inspired by physical problems and his

mathematical. thinking was t:;uided by a deep physical intuition.

Yet his mathematical works are among the purest of pure mathe­

matics. So pure that G. H. Hardy!_- 1J , the British mathematician

and the c..trchetypal protagonist of "art for art's sake'', thought

that Wie'Ier 1 s :·engineering-inspired 11 c lair.ls were, if not sheer

humbug, i~re professional politicking; Wiener was also a

phi~osop·1er interested in the implications of the new automated

tc:c .mole» y, . .:;.nd an unusual autobiographical writer who has

;iv =.:n u:: a fl" ·"ink and det0iled account of the emotional and

int~llec :ual vicissitudes of his life and career in his two books 7 a·' .- .. , l h .. r~] '-·c· •1'0 i y l_u_\ c..na I am a mat,emat?.-o?.-an L"-·

In the ::.ladoh' of brilliance. Norbert vJiener was born the

eldest snn of Leo Wiener and Bertha Kahn Wiener on 26 November

18S 1f in ':olumbia, Missouri. His father• wa.s a Russian Jew who

migc:1ted to the United States a penniless young man of

eig~teen after an unprom1s1ng start in Europe. His intelligence

and his talent for languages soon turned him from a peddler

int~ a f~reign language teacher at the University of Missouri.

- 8 2 -

A self-made man w;ith no formal university education, he

11evertheless became.a distinguished philologist and a

!Ia:cvard. University professor of Slavic languages who trans­

lated into English twenty-four volumes by the Russian writer

Lr:.;o Tolstoy C 4]. The mother Bertha Kahn Wiener was the

,_bughter of a Jewish immigrant from Germany and had a non­

Jewish grandparent.

Born into an intellectual tradition that could be

traced back to a Grand Rabbi and even supposedly to the

philosopher Moses Maimonides [ 5], Norbert Wiener soon felt

the psychological pressures of the intellectu~l training and

disciplines imposed by an exacting father who would accept

nothing short of perfection. These demands instillad in

young Norbert feelings of awe and humiliation and an emotional

dependence on the family. He was an extremely sensitive person ,

and his adolescence was marked by a social ineptitude and

j nsensitivity that w~s compounded by an agonlSlng awareness

- f his Jewish background, the overt and subtle racial prejudice s

'f that time and the need to achieve (and invariably intertwined ,

''1 0 fear of failure) that is often compulsively felt by a prodigy.

~~7iener' s p:recoci ty was manifested \'7hen he began reading

~ t an oarly age. By the age of six, he was an avid reader

0 f the ~ooks in his father's collection and was fascinated

by zcclogy, botany, physics and chemistry in addition to the

11Sual t~les of travel and adventure. His omnivorous appetite

for books was so great that by the age of eight his eyesight

had deteriorated badly. His childhood ambition was to b e come

a naturalist. Surprisingly, his precocity in mathematics was

far beyond his poor manipulative skills in arithern tic. His

father soon discovered that the ordinary school could n o t

resolve this unusual situation of his precocious son. Sc

up to the age of ten, young Wiener's education was directed

b y the father who worked out a pr'os;ramme on Ivlathematics

(algebra and geometry) and languages (Latin and German).

-· 8 3 -

'' - ·- l . ~ , ,E- father was supplemented by that of

two tutors, one lh chemistry and the other in Latin and

German . Still, he had his full share of play with the

neighbournood children during this period of his childhood.

His three years, (1903 - 1906 ) at Ayer High School

in Harvard,Massachusetts, brought him fond memories in his

later years. The next three years (1906 -1909) saw him

working for his bachelor's degree at Tufts College. His

successful graduation left him in a state of physical

exhaustion together with a feeling of bitterness at not

being elected to the honour society Phi. Beta Kappa. This

bitterness affected him so deeply that he was to reject

many hono~rs b:.:latedly bestowed upon him.

In 19 0 9 , barely fifteen, h e enrolled ln Har'V:ird

University to do graduate work in zoology. However, his

physical slowne ss set against his intellectual impatience

made him a disa ster in the laboratory. Hith his dream to

become a biolo gist shattered, he went to Cornell University

the f o llowing year after having won a scholarship. His

father haJ ur ge d him to switch to philosophy, but the

~ca J emic year 1t Cornell saw him emotionally disoriented

3 n d intel ~ectually lost . His scholarship was not renewed

0 nd h e r eturne l to Harvard in 1911 to work for a doctorate

i n philosophy. He vJrot':.: his ?h .D. thesis on mathematical

l cc; ic HhC' l he nas h a rdly nineteen.

Up to ttis time, his mathematical education had been

meagre. At T~fts, the mathematics course s ~ere designed

f o r engin e erin ~ students, except for a special reading

c o urse, which ·.Jas beyond him, in the theory o f equations.

At Cornel : , he took but did not understand a cours e in the

theory of func::i ons o f a complex variable. At Harvard, h e

studied a x ioma -:::ic systems unclcr the American mathematician,

E. V. Hunt ing t on [s J , and his thesis was written under the

s uDer.visi r) !1 of Ka rl Schmidt of Tufts Co llege (he had be e n

~r eve nted fr orr working with the philosopher Josiah Royc e [7]

the l a:: t e: r· '.J :i.ll ness). Nevertheless, in the summer of 1913

- 84 -

prior to his departure for Cambridge University on a

travelling fellowship , Wiener read,at Huntington's

Moder•n algebra by Maxime B'Ocher [8 J and Projective

by Oswald Veblen [gj and J . W. Young [10]

suggestion,

geometry

From philosophy to mathematics . ~he Wiener family (complete

with two sons and two daughters) planned to spend the winter

of 1913 in Europe. Ecwever, \tliener vrould g:o to Cambridge

University to study with Bertrand Russell [I IJ , Jart of

whose work on logic had been dealt with in Wiener's Ph.D.

thesis. His short stay at Cambridge enlarged his perspectives

in mathematics and physics, and he settled comfortably into

the new environment of the individualistic and eccentric

Cambridge dons.

Though essentially a logician, Russell had deep insight

into the relevance of mathematics and physics to philo~cphy,

At his suggestion, Wiener took courses by H. F. Baker [12] , Hardy, J. E . Little~ll/"ood [13] and lJ. Mercer [i4J. Hardy

impressed the young logician by his clarity', interest and

intellectual power, and his course on analysis (particularly

the Lebesgue integral) had a strong influence on Wiener's

mathematical development . Wi8ncr was also introduced to the

three revolutionary papers of Albert Einstein0Sl, on relativity,

the photoelec~ric effect and Brownian m~tion, and to the work

of Niels Bohr ClG] on the atomic nucleus. (~ar~ly did the young

Wiener suspect that he would make a profound contribution to

the theory of Brownian motion ten years later). All these

were in aJditinn to the courses given by Russell. During this

+ inle, Viiener ha.d sensed the impossibility of building a

cc:nplcte c:nd consistent logical system. This '..Jas borne out

Kurt G~del [17] who showed years later that the consistency

~ .J :f' ti· c ari tliernetical syste1n implies its own incompleteness

(in the senEe -hat there are arithemetical statements which

, ... ,...,n_,t be ~)roveJ or disproved within thE' system).

- 8 s -

The May term (19 1 4) saw some changes in t•Jiener's plan .

Russell would be away in Ha~ard; so Wiener decided to spend

the rest of the academic year at G~ttingen University which

had become the beehive of mathematical and scientific aci:ivi ties.

At Gottingen, he studied mathematics with Edmund Landau [18] and David Hilbert C1sJ, and philosophy with Edmund Husserl[20] .

He benefitted little from i:he formal courses but the many

contacts vlith various kinds of people helped him in his

social adjustment towards other people . Above all, the meetings

of the l1athematical Society taught him that "mathematics was

not only a subject to be done in the study but one to be dis­

cussed v.1ith and lived with." C2].

On his return to the United States, he was again awarded

a travelling fellowship by Harvard. He again chose Cambridge

University and arrived amidst the gathering storms of the

First 1/Jorld · \-Jar. The once congenial atmosphere was now

highly charged with a heavy gloom and a demoralizing bleakness.

Shortly after, he returned to the United States .

Until the end of the war, he did not obtain any proper

job in mathematics . A possible open1ng 1n the philosophy

department at Harvard was closed to him after it was discovered

that·a janitor had leaked out to him the actual grades of his

examination results when he was a student. He was at C~lumbia

Tiniversity for a short period and was an assistant in philosophy

1t Harvar•d 'dhere he also gave a fr·ee series of Docent LectureE

l : 1] on logic. His impressions and experiences at Harvard did

1ut endear that academic. institution or its academicians to

him. On his father's advice, he switched to a career in

mathematics and obtained an instructorship at the University

o f Maine but this experience turned out to be a nightmare

f or him. He left teaching and made som8 unsuccessful attempt2

at enlisting for military seryice . Subsequently, he worked

for a brief period as an apprentic~ engineer and then as a

hack writer for ~cyclopedia Americana .finally, he helped

in the computation of ballistic tables at the Aberdeen Proving

Ground in Maryland. Though the work was not mathematically

significant, he found stimulating the intellectual life

with the other scientists and mathematicians working on

the same project. With the signing of the armistice, he

1,.;as discharged from the army in vJhich he managed to get

enlisted as a private in the last days of the war . Next

came a short and abortive stint at reporting for the Boston

Herald.

vfuen the war was raglng in Europe and on the high seas,

Wiener continued his research work in logic although he did

rry his hand at some of the most difficult problems in

mathematics such as the four-colour problem, Fermat's last

theorem and the Riemann hypothesis. There was an attempt

in 1915 to set up the axiomatic foundations of what is now

known as topology. He found his results unsatisfactory and

abandoned this field of research. He was comparatively a

la.tecomer to modern mathematics. His real understanding of

modern mathematics was first acquired by a chance reading of

the m~thematics books given to his sister Constance by the

parents of her fiance G. M. Green when Green, a budding Harvard

mathematician, succumbed tc the post-war influenza epidemic. were

Among those booksAThSorie des equations integrates by Vito

Vol terra [ 2 2] , Funktionen theoric by VJ. F. Osgood [2 3 J, Lebesgues

1 L24] book on the theory of integration and Fr~chet's

r2 sl book on the theory of functional s. The final break in '- .. 1

iener's hitherto unce~tain career came in the spring of 1919

through the food offices of Osgood who secured for him a

~osition in the mathematics department of the Massachusetts

I~stitutc of Technology (M.I.T.).

~;I'Oh7th and fruition. During the early twenties, the mathematics

lepartmePt of M.I. T. was relatively unknown. It was mainly

scrvic~ department devoted to the teaching of calculus to

engineering students. The younger members of the department

were cnt~usiastically engaged in research, but it had no high

Jtanding in research . This environment proved to be a godsend

to the young hliener who often felt ·insecure about hif.3 own

abilities and suspicious of the higher echelon of the academic

hierarchy. He managed with a heavy teaching load of twenty

o~ -· (J/ -·

hours a week and got along well·with his students, his

eccentricities notwithstanding. He discussed freely with

his colleagues and his self doubt wat allaye~ by their

encouragement.

Frechet's book had inspired Wiener to ask the young

m::l.'t:hema tic ian I .A. Barnett [ 2 6] for a good research problem

u1 functjonal analysis. The latter suggested "the problem

uf integr::J.tion in function space il. This "completely

influenced the whole course of Wiener's work and his greatest

c.chievements all stemmed from this problem. 11 [ 2 7] Wiener

also consuJ.ted O.D.Kellog [2i] of Harvard on potential theory

and solved significant problems in that field. Unfortunately,

two doctorate students of Kellog were attacking the same

problems but making less progress than Wiener. And when

Wiener was asked to hold back the publication of his results

so as not to affect the theses of those students he was very

upset and unhappy. Anyway, N. Levinson~9] tells us that

this matter was satisfactorily settled.L 28].

At the request of his colleagues in the electirical

engineering department, Wiener undertook to lay the mathematical

foundatic)ns of tl1c formal calcul·us of cc.,mmunication engineering

developed by Oliver Heaviside [30] some twenty years before.

His efforL:s cu·l.minated in his formulation of the theory of

generallzt;d ha:L'monic anlaysis. A problem in this field led

him ~o a ~ti ll greater discovery of general results 1n the

;o-called Tnub~rian theorems [31].

All thess achievements were made within the first decade

_f ~icner's career at M.I.T., 2nd he had already established

~orld-widG reputation ~~en he was in hi~ early thirties.

:',n t 1-.e fe1 t and for a lon; time remembered that it was not

his fellow Americans who gave him recognition for his early

successes. His rise in the academic ranks of MIT was slow.

He became an assistant professor in 1924, an associate professor

in 1929 and finally a professor in 1932.

- 88 -·

The city of Strasbourg had just been re-Gallicized aftdr

the trauma of the First \vorld 'vJar, and was selected as the venue

of the International Mathematical Congress [32]of 1920. However,

~s a punishment, the Germans were to be exclud~d from the corigress.

Even the scien·ti.fic community could not then forget the war-time

hatred and bitterness. Wiener did not care about the politics

of that time. He was only too eager to resume scictitific contact

with Europe after such a long lull. He still had an emotional

attachment for Europe and he wanted to be independent. Before

the start of the congress ln SeptE:mber , he spent some time workiJl S

with the Fr'ench analyst, Maurice rr-echet. It was during this

period that he discove.red, independently cf ·the Polish mathema"'dci.J.n

Stefan Banach [33] , axioms for vector spaces. However, Wiener

did not continue to do much r'esearch in this field of Banach spac , .. ::>.

In 1926, Wiener married Margaret Engemann who, at the a.;:rc

of fourteen, migrated to America from Germany with her widowed

mother and sisters. She was to give him the emotional support

and understanding that was so essential to him in his climb t o

conquer challenging mathematical heights. He was a keen travellc·' .

Before marriage, he had been to England and Europe several tim0 s~

at times by himself and at other times w~th his sisters. Shortly

after his marriage, he went to G~ttingen as a Guggenheim Fellow.

Because 6f her teaching job, the newly-wedded wife could only J Ol n

him in Europe later. With his wife and two daughters, Wiener su~nt

1931-1932 at Cambridge University and 1935-1936 at Tsing Hua

University in Peking. His stay at Tsing Hua was partly arran ~~~

by his former Ph.D . student, Y.W.Lee [34] ~ an electrical e n. ci.nc' r

by training with whom he had patented ~n invention.

In 19 3 3, for· his work on generalized harmonic analysis

and Tauberian theorems, Wiener shared with Marston Morse [3 s] the American Hathematical Societj' s Bbcher prize, which i::-~ av.J crd f:·. ~

every five years t c:: American mathematicians for outstanding resecJ.~:..:h -

- 3 9 -

He had strong Vlews against· the construction and use of

nuclear bombs, and resented the intrusion of the military

establishment into the laboratory in an atte~pt to control the

aissemination of ~nowledge. His work on prediction and computing

·~~chines gave him a vision of a future society in which automation

'-' 'J:'_ d replace a considerable amount of human labour. The moral

i 3 3'-:'2s arising out of automation loomsd c·-:i::-~ously · over his men.t=:l

)<; r·izons, and he felt it his duty to speak about the social

p roblems of automation to union officials and administrators.

In 1946, Wiener went to France to attend a conference on

harmonic analysis. When he reached Paris, Freymannn of the French

publishing company, Hermann et Cie, met him and solicited for

a Hriting up of his ideas on control and communication, Here

was a good opportunity for him to present a unified view-point on

the scattered but related ideas that have been nutured over a

period of more tha~ ten years. Two years later, his well-known

book Cybernetics [46 J ~Jas published.

The word 1 'cybern ~.:::tic s 11 ha.d been coined by Hiener himself

from the Greek word for steersman ana. ne t-Jcts then unaware thr.tt

the E'rench physicist Andrb Ampe:r:'e [ 4 7] had used the same ~TOrd before, but in a sociological sense. The seeds of this new

discipline \vere sown as far back as 19 3 3 vJhen he came to know

the l'1s;-dcan physiologist Arturo Rosenblueth [t~8] who was at

. - _:,v c~_.r d Hedical Schoc'l before his re·turn to Hexico in 1944 .

• · ~0 Llg those; prc~~rJcU' year's, Rosenblueth conducted a monthly

·rJ . . s a f discussion meetings in which a scientific paper would

~cad ~nd a frc;e dnd unrestrained discussion, with emphasis

.. n :-:2 thodology ~ t-~ould c::nsue. t>iiener Has introduced to those

~~et ~ ng s by the Mexican physicist, Manuel Sandova Vallarta [4~ -· ~ ;-· .I. T. , and. subsequently became an active participant. His

f ruitful collaboration with Rosenblueth continued through the

v~r. After the war, they took turns to visit each other.

·- C) 2 ·-·

There was an exchange of ideas on co1nputing machines

b:::tween Wiener and John von Neumann [50 J ~..;rho v-1as then heading

the Computer Project at the Institute for Advanced Study.

In the late vJinter of 1943 - 1944, they convened at Princeton

what might be called the inaugural meeting of cybernetics.

-rt prepared a common ground for a multi tude of disciplines.

S~bsequently, in 1946, the Macey Foundation series of conferences

on cyhernetics was launched in a meeting in New York and lasted

ror several years. Cybernetics was thus firmly planted. It

1as grown and developed since then.

In 1944, Wiener was in Mexico to attend a meeting of the

I-;exican t-'iathema·tical Society. This was the first of many visits

to Mexico in· the course of his joint research with Rosenblueth.

In 1953, Wiener was invited by the Indian government to undertake

a seVen··\.·Jeek lecture trip in India. He published in 1959 a novel_

1' he ter:mter in which the main charac-ter vJas inspired by Heaviside.

M.I.T. appointed Wiener as Institute Professor in 1959.

This was a distinguished position ivhich allmved him access to

any department in the institute. He retired in 1960. He was

to receive from President Johnson the National Medal of Science

shortly before his death on 18 March 1964.

Running through Wiener's multifarious

:_:-:)Ei:r·ibutlons is a corrnnon thread of motivation and intuition that

""':·iirJ:ls from his interest in ·the applications of mathematics

~J physics and engineering. In electrical engineering, Heaviside

~,, ;•is idol. At heart, tviener was a pure mathematician vperha.>~>

':~ ~- ini -!:L-<.1 tralrnng as a logician made exacting demands on his

/:,.s Levinson puts it:, 11 • • • once I;Jiener ta.ckled a

~- ··,J.~.le!n) 1·1is trleat:::i.entVJc?..S rigorous, general e.nd aesthetic.

• -T'-J~Ds the framing of his tl·1eories in the full generality and

-bstradtion of the Lebesgue integral delayed their accessibilit?

·;.::he: engin<.::er. Yet he could do it nc either way 11• [51 J.

Out of a problem of integration 1n function spaces arose

his greatest ;:,-Jork on Brownian motion and random processes. h11en

he contemplated this problem as an instructor at M.I.T. the

physical theory of Brownian motion was thought to be finished.

It was his physica::i_ insig!-"t that made him revive this subject

and bu~ld a whole edifice whose ramifications continue to spread

and multiply more than fifty years later. Mark Kac Cs2] points

out that even Wiener's brief but significant work in potential

theory done arounc the same time as that on 3rownian motion has

turned out to be closely related to the latter work. It is

\<7Drthwhile noting that his probabilistic analysis was done more

than ten years before the axiomatic formulation of probability

by the RussL:m mathematician A.1LKolmogorov [53] in 1933.

The breakdown of classical harmonic analysis in the study

of white light led \viener to develop a 17 generalized 11 harmonic

analysis. Some of his ideas have since been incorporated into

v;c•l'ks c:r1 ·:ii.(t~ics. VJ11ile he \'-las in GOttingE;n as a Gttggertheim ,..., - - · . - c ,. - -, 9 2.., h B . . h th t . . ' E I .h !.- .- 4-1 .reLL01JJ J.n J.J.c:o- . .L. , , t e rJ.-tJ.s. rna "ema 1Clan .~-~. ·. ng. am_ 0 J pointed ~ut to him the similarity between the so-called Tauberia~

t:heor.e,ms and certain resul ·ts in l·Jiener' s ner,y analysis,

Ta·..;bc:rian theoreos Pcre hitherto rather scattered results to \.Jh:i -=~

Hardy and Littl ewocd made numerous contributions, but there did

not seem to be a general theory. Wiener then saw that his

generalized analysis ~rcvided the tool for a general theory of

"J'.::. Iberian theor(::ms. At the sam.:?- tim(::, his methods could also

i:);:; used to give proof of the prime-number theorem

More than ten years later, his generalized harmonic analysis

agaJ.n found i 7~3 'i.-:ay into his ~·Jar--time work on prediction theory.

In ~is 1942 classic The extrapolation~ interpolation and amootA)~g

of stationary time series _. h •2 introduced statistical cc..r!ce r;~s

into the study of the transmission of messages and noise! thereby

layL:;:; the foundatio'l. -For communication and control theory. !__t

that time, h E· ~·?<'18 unavtarc that Kolmogol'OV had earlier published

SliTllL? ·:· ~- · tl''.Tnat ic-3.1 idea.:.:; on prediction theory. But only \vie ner

ificant use of the theory in engineering. His stu~'

of linear and non-linear filters in electrical engineering

stirred up deeper questions on the learning, self-o~ganizing

and reproducing abilities of natural systems.

A product of war--time work \vas ·the information theory

developed by Wiener and Claude E. Shannon [5~ The former

approached it via the electric circuit, carrying a continuous

current, while the latter considered a message as a discrete

sequence of yeses and noes. The feE:dback: mechanism that was

discovered in conjunc:·tion with his project on fire -control

reminded ~7iener of a clos12ly related phenomenon in biology -

that of homeost-:ls{s, the process by Hhich the internal environ-­

Dent of a living organism is :naintained at a 1'1eal thy level.

This conglomeration of ideas found synthesis in his book on

cybernetics. The underlying philosophy of this synthesis is

founci_ lr' the follmving passage fro111 his autobiography. [57],

'' W,:::. a.Pe Sl'-iimming upstream against a great torrent of

. cl:"sor'.c:a:n:i.zation ~ \vhich tends to re.duce ever'ything to the heat-·

c1ei-1.t:h of ecpilibl"ium and sameness described in the second laV-7

cf thermodyn.::unics. tvhat HaxvJell [5s]., Boltzmann [59] and _ Gibbs C6o] rnea.nt by this heat deatfl in physics has a counterpart in the

ethics of Kirkeu1ard [ 61} t•J11o poil).ted out that we live in a

chaotic moril universe. In this, our main obligation is to

-s tablish arbitrary enclaves of order and system. These enclaves

·,-Jill not remain th·o;re indefinitely by a.ny momentum of their· o~·;n

after we hav~ once established them. Like the Red Queen, we

cannot s·tay V-?hc:re we are Hi thout running as fast as we can.

n\·Je .:H'<:o ne-t: :·i._3l: ting for a definitive victory

indefinite future. It lS ·the greatest possible victory to be~

to continue to be, and to hav2 been. No d12feat can deprive us

of ·the s~ccess of having existed for some moment of tlme 1n a

~nlv~~s~ that seems indifferent to us.

- 0 5 ...

"This is no defeatism, it iE? rather a sense of tragedy

1n a vmrld in which necessity is, represented by an inevitable

disappearance of differentiation. The declaration of our own

nature and the attempt to build up an enclave of organization

in the face of nature's overwhelming tendency to disorder is

an insolence against the gods and the iron necessity that they

lmpose. Here lies tragedy, but ·he~e lies glory too' 1•

A survey and evaluation of his many contributions to our

knowledge is given in a special memorial issue of the Bulletin

of -the American Mathematical Society [ 6 2} and in a collection

.f sel<::cted pco.pers [6 3]. The latter contains his most important

~atha~3~i.2al works and gives a cross-section of his achievements .

. \.rnong the biological research which he carried out (in

c ollaboration with others and, in particular, with Rosenblueth)

J r stimulated is that on biological regulation, characterization

of the electroencephalogram as a time series and prosthetic

devic 2s. However, after the publication of his book on cybernetics

h s did not keep a bPeast of the advances in molecular biology while

ethers took up the challenges posed by the flow of int~a-and

·intGr-cellular information.

Wiener's success is that he has not only pl~nted the sGeds

cf knowledge buT also fertilized the ground in which seeds sown

oy ~thers ma y germinate and grow. If, eventually, his fundamental

achievements bc~ome particular undergrowths in a luxuriant for est,

let us bear in mind these words of Kac, 11 The fate of all great

wm~ r is 'to be subsumed j the more attention it attracts the greater

t ha chances of becoming engulfed in a cascade of generalizations - "1

a.nC. extensions . 11 L 6 4 J .

.. .. s 5 -·

Fi~ally , Wiener has t his to say about the creative urrre

ln Ela therna tics .

" Hathematics lS too arduous and univi tin.;: a field to

3ppeal to those to whom it does not give great rewards . These

rewards are of exactly the same character as those of the artist .

To see a difficult~ uncompromising material take livinc shape

and meaning is to be Pygmalion, whether the material is stone

or hand , stonelike losic. To see meanin[ and understandinc

come where there h<J.s been no meaning and no understa.ndin:.::; is

to share the ~.;rork of a demiurge . !Tc· amount of technical correctnc~c-:s

and no amount of labour can re;)lace this creative moment , wh,.:::ther

in the life of a mathematician or in that of a painter or musician.

Bound up with it lS a judgement of values, quite parallel to the

jud;;emi:;nt cf values that belon;:~s ·to the paintE~r or muslcle.n.

Neither the artist nor the mathematician may be able to tell you

vih:::~ t constitutes the difference betvJeen a significant piece of

work and an inflated trifle; but if he lS never able to recognjze

this in his own heart , he is no artist and no mathematician." 65

Note~ and references

.[1] Godfrey Harold Hardy ( 18 77-194 7 ), British mathematician;

studied at Cambridge; '\..Jorked at Cambrid8e a.nd Oxford;

contributed to analysis and analytic theory of numbers.

[ 2 J norbert '07iener, Ex-p1oodigy, Simcn c:md Schuster. 19 S ~ ~

M.I.T . Press, 1964 .

[3] Norbert Wiener, I am a mathematician~ Victor Gollancz,

London, 1956 .

[ 4 J Lc:o Tolstoy (1828-1910), Russian novelist, moral

philosopher and social reformer; studied at Kazan

·university, fou:ht in Crimean War; well-known for his

neve 1 s (especially War and lea co .~ A nnn. Krrre1L t.oa > .=1nd

for his philosophy of hum~nism.

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[ 5) Moses Maimonides (1135-1204), Jewish philosopher;

lived in Cordoba (Spain); migrated to Cairo (Egypt)

after Hohammedan invasion; became royal physician;

did definitive work on the Judaic laws of the Talmud.

[ 6} Edmund Vermilye Huntington (borm 18 71+), American

lo~ician; studied at Harvard and Strassburg; worked at Harvard; conributed to postulational theory of mathematics.

[7] Josiah Royce (1855-1916), American philosopher; studied

at California, Leipzig, G6ttingen, Johns Hopkins~ worked

at California, Harvard; contributed to philosophy and

psycholop:y.

r P J L ..J (1867-1918), American mathematician;

studied at Harvard, Gottingen; worked at Harvard, Paris;

contributed to theory of differential equations.

[ 9] Os\vald Veblef1 (born 18 8 0), American mathematic ian; studie(

at Im.,ra, Harvard, Chi cage; v10rked at Iowa, Chicago,

Princeton, Institute of Advanced Study (Princeton);

contributed to geometry and topology.

[101 John ~·Jesley Young (1879-1932), American mathematician;

studied in Germany and at Ohio State University, Cornell

Un~versity; worked at Northwestern, Princeton, Illinois

Kansas, Chicago·, Dartmouth Collese; contributed to

postulational theory of ~eometry.

[11] Dertrand Russell (1872-1970), Dritish logician and

philosopher; Nobel laureate in literature (1950);studied

and worked at Cambridse; contributed to logic and foundation

~~ ~~thematics, philosophy, pedarogy and literature.

[12] :~~nry fi'ederick Baker (1866-1956), British mathematician;

s~u~ied and worked at Cambridge; contributed to geometry 1

the·:)ry of alcebraic and multiply periodic functions~ i..i:·:~cr·y

of differential equations, dynamical astronomy.

- 9 8 -

Ll3] ,_,o.~,_, LJe"'c,o:.. · Littlewood (born 1885), British mathematician;

[:J c.• 1.di:::"j at Cambridge; worked at Ivlanchester, Cambridge;

contributed to theory of functions.

James 1','-".1.'-'<::~ ,:~.33--1:332), British mathematician; studied

~nd worked at Liverpool and Cambridge; contributed to

~heory of integral equations.

Ll5] Albert Einstein (1879-1955), German-Darn American physici 3c;

Nobel ~aureate (1922); worked in Berne, Z~rich, Prague,

Berlin, Princeton; contributed to theories of Brownian mot ior,

photo- ~ _ec~ricit~ special relativity, general relativity.

[16~ Niels Bohr (1885-1962), Danish physicist; Nobel laureate

(1922); studied at University of Copenhagen; worked at

Ca~bridge; Manchester, Copenhagen; was in Washington and

LOP Alamos dur·ing Second vJorld War; contributed to quant ! __ n

·-t:t· , .; _.) and dl.:r:lec:.r physics.

[ J 7 ~ K1: : • t Godcl C:xn"n 19 0 6) , Czech-born American logician;

studi~d at Vienna; worked at Vienna and Institute of

i\dv·anced Study( ,...~.,inceton.); c_ontributed to mathematical

logic ci. :r!r1 f < ' J:•c' :ctions of mathematics.

i)8J TdmunC::. Lanuau (1877-193:3) ,German mathematician; studied

in Berlin and Munich; worked at Berlin, Gottingen;

contributed to analytic number theory.

r, J l L .. ~. .~ J David Hilbert (1861~1943) German m~thematician and . . } .

theoret~ca ~ physicist; c ontributed to number theory, set

theory, logic, [eometry, integral equations and mathematic~ ]

physics.

(201 Edmund Husserl (1859-1938), German philosopher; stud i e ~

at Leipzi~, Berlin, Wien, Viennd; worked at Halle,

+~~ en, Frcinurg im Breisgan;coniributed to philoso9hy

- 9 9 ·-

[2 1] io r s ome years before the Second World War, Harvard

Uni versity i mp lemented the system of Docent Lectures

alon2 tha t o f the German system. Every Harvard Ph.D.

was r rante d the iight to g1ve a series of lectures

i n hi s own f ield without pay. Harvard students would

be fr e e t o attend them but they would not be taken into

a cco unt f en· a degree.

[22] Vito Vo lterra (1860-1940), ItaliAn nathematician1 child

pro digy; studie d at Flo rence and Pisa; worked at Pisa,

Turin, Rome, Paris; f o u ght in First World War with the

Air force; livec mainly o utside Italy after 1932; contributed

t o analysis, the6ry o f integral equations , mathematical

bio logy~ mathematical physics.

[23] Willia m Forg Os good (1864-1943), American mathematician;

studied at Harvard , · G~ttingen, Erlangen (Germany); worked

a t Harvarc and Pekin g ; c ontributed to theory of functions.

[ 24] He nri Le bes gue (1 87 5-1941), French mathematician; studied

at t h e Ec o l e N.ormale Sup~rieure; worked at Rennes, Poitiers,

Par i s ; c ontributed t o analysis and theory o f Fourier series.

[25] Maur i ce R en~ Fr~chet (born 1878), Fr~nch mathematician;

studied . / . . . . 1n Lyc e e Salnt Louls; worked at Nantes, Rennes,

Po itiers, Strasbc urg; contributed to set theory, analysis, . -· '• .

t opo l ogy, probability, matr:~.matical statistics.

[ 2E] ~saac Al bert Barnett (be rn 1894), British-born American

ntathe matician, stud ied a t Chicago; worked at hlashins t o n,

Harva r d , Saskatchewan, Cino:.llnati; contributed to calculus

of v a ria tions, integral equaiions, classical theory of

n umbers.

I2 7J Norman Le vinson, 11 \.Yiener 1 s life 11, BuZZ. American Mathe matic'" l

Soci ety ., No. 1, Part T-,-. .J.. ...1... ' 1966.

[28] Olive r D. Ke llogg (1878 --1932), American mathematician;

wo rked at Ha rvard; c ontributed t o theory 0 f intez ral and

d i f f e rential equations , po tential theory.

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[29] Norman Levinson (born 1912), American mathematician;

studied and worked at M.I.T., contributed to theory of

complex variables, transform theory, differential equations.

[30} Oliver Heaviside (1850-1925), British mathematical

physicist; self-taught; received Ph.D (honJ from G~t±ingen

University; worked as telegrapher; contributed to electro­

magnetic theory, telegraphy.

[31] In 1897, A.Tauber proved that if a0

+ a1

x + a2

x 2 + .•.

t'<:;nds to A as x__,l-0, and a :: o(l/n) (i.e. na _,.0), n n

then a + a, + ()

a 2 + .•. = A. Hardy and Littlewood rs~laced

hy-:-0t:hesis a :: 0(1/n) by thehypothesis an= 0(1/n) n

(i. 3. there is a constant K such that n1aJ ~ K for n ~n0 ).

The International Mathematical Congress is held every

four years by the International Mathematical Union.

At each congress, one or more mathematicians (below the

age of 40) would be awarded the prestigious Fields Medal

for significant contributions to mathematics. The Congress

of 1974 has held in Vancouver (Ca.nada), and ~th<:; recipients

of the Fields Medal at that congress were the American

Mumford and the Italian Bombieri.

[33] Stefan Banach (1892-1945), Polish m~thematicain; studied and

worked at Lwow (Poland); contributed to functional analysis,

topology.

[34] Yuk-Wing Lee (bo rn 1904); Chinese-born American electric2l

en'-,ineer; studied at r·1. I. T. ; worked in Nmv York, Shanr:hai,

Tsin,cr, Hua; now at H.I.T.: contributed to communication

p S J f'i,·:r'ston l'·1orsc (born 18 9 2), /\merican mathematician; studied

at Colby Collese and Harvard; worked at Harvard, Cornell,

Brown, Institute of Advanced Study (Princeton); contribute~

to theory of dynamical systems, analysis in the larfe,

function theory.

- 101 ·-

the

[36] Raymond E.A.C.Paley (1907-1933), Britisl1 mathematician;

worked at Cambridge and M.I.T~: was killed in a skiing

accident in the Canadian Rockies; contributed to theory

of Fourier transforms.

[27}

/ Szasz (bnrn 1884), Hungarian-born American mathematician;

studied at Budapest; worked at Frankfurt, M.I.T.J Cincinnati;

contributed to analysis, theory of approximation, number

theory.

[39] Hans Rademacher ·(born 18 9 2), German-born American mathematician;

studied at Gottinzen ( ·:wo:rked at Berlin, ' Hamburg, Breslau,

Pennsylvania; coniributed to analysis and analytic number

theory.

[ ] J •

40 George Polya (born 1887), Hungarian-born American mathematicla~

worked ln G~ttingeni - Z~rich, Stanford; contributed to

analysis, probability theory; number theory, applied

mathematics.

[41] Gabor Szes~ (born 1895), Hungarian-born American mathematician;

studied at Vienna; worked at Berlin, K6nigsberg, Washington,

~tanford; contributed to pure and applied mathematics.

f4 2 l K ~rl Me nger (born 1902), Austrian-born American mathematician;

studied at Vienna; worked at Amsterdam, Vienna, University

o f No tre Dame and Illinois Institute of Technology;

contributed t o geometry, algeb~a, analysis.

[43] Eberhard Hopf (born 1902), Austrian mathematician; studied

in Germany and at Harvard; worked at M.I.T. and in Germany.

[44] Julian Himely Bi gelow, American electrical engineer and

computer scientist; worked at International Business

Machines; first chief engineer of Institute for Advanced

Study Electronic Computer Project.

- 102 -

(4~ Harold Urey (born 1893), American chemist; Nobel laureate

(1934); studied at Mon~ana and California~worked at

Montana, Copenhagen, Columbia, Chicago, Califo~nia;

contributed to huclear chemistry, chemistry of the orlglns

of ~arth, meteorites, moon and solar system.

u b . t \'' ,.., b . . ~or er Nlener, Gy ernet~cs~ Mii.T. Press, Cambridge,

Hassachusetts, 1961.

[47) An-'~r.e Marie A.mpere (1775-1835), flrench mathematical

physicist; worked in Lyons and Paris; contributed to

electomagnetic theory.

[4~ Arturo Rosenblueth (born 1900), Mexican physiologist;

studied at Mexico, Paris; worked at Harvard, Mexico; ' '

contributed to neurophysiology and electrophysiology.

[49] 1·1anuel Sandova Vallarta (born 1899), Hexican theoretical

physicist; studied at H.I.T.; worked at H.I.T. and

Mexico.; contributed to , theory Qf cosmic radiation and

relativity.

[50] John von Neumann (1903-1957), Hungarian-born mathematician

and theoretical physicist; studied i.rt Budapest, ~erlin,

Z~rich, G6ttingen ; worked at Berlin, Hamburg, Institute

f~r Advanced Study (Princeton); contributed to set theory,

algebra,analysis, game theory, computer science, mathematical

physics, See Y.K.Leong, nNotes on , mathematicians: 3. John

von N 2 u: : 1m · " · 1 a them a tic a l .:-1 e d Z c y 3 ( 1 9 7 5 ) , 9 0 -1 0 6 •

[51) Norman Levinson, 11 Introduction and some comments on ~viener 's

sel•2cted papers", in Selected papers of I! orbert Wiener_,

M.I.T. Press, Cambridge, Massachusetts~ 1964, pp . 1,2.

[52] Hark Kac (born 191Lf); Polish-born American mathematici::.n;

studied in Lwow (Poland); worked in Lwow, Cornell University

Rockefeller University; contributed to theory of probabilit~

mathematical statistics, analysis, numter theory, statistic~l

mechanics.

- 103 -

[53] Andrei Nikolaevich Kolmogorov(born 1903), Russian

mathem~tician; studied at Moscow; worked at Paris,

Moscow; contributed to analysis, probability,mathematical

statistics.

[54] Al bert ldmund Ingham (1900-1967), British mathematician;

s tu:.i ied at Cambridge;worked at Gotfingen, Leeds, Cambridp:e;

co~ tr ibuted to analysis and analytic number theory.

["55] The Prime Number Theorem states that if Jt (x) lS the number

of ]rimes not exceeding the real number x, then

(.n(x)log x) /y -~ 1 a.s x~o<> . This Has con~.ectured by Legendre

(1J52-1833) and Gauss (1777-1855), and was proved by

Hadamard (1865-1963) and, simultaneously, de la Vall~e

Poussin (1866-1962) using the theory of entire functions.

An elementary number-theoretic proof was given by Selberg

and Erdos in 1947.

[56] Claude Elwood Shannon (born 1916), American applied

mathematician; studied at Hichi ;:can and M.I.T.; worked

1n Be ll Te l ephone Laboratories; now ~t MIT; contributed

to communicatio n theory, mathematical cryptosr'aphy,

theory of computinr machines.

[57] I am a mathematician~ pp. 324,325. See

! ::. ;JJ lc .::r~,.... s C:l.,crk I·1a.xwell (1831-1879), British mathematical

p;ysicis: ; studied at Edinburgh, Cambridge; worked at

Ab e r deen, Lo ndon and Cambr~dge; contributed to electro­

ma znetic theory; statistical mechanics; thermodynamics.

[59] Lu~wi& E~uar: Bo ltzmann (1844-1906), Austrian theoretical

physicist; studied at Vienna, worked at ~ienna, Graz

f,Auc ~~l·a), 1~.:unl'c11. Telrzl~· • • u - • • w . J.'' i;, l contributed to electromagnetic

theory, thermodynamics, statistical mechanics.

(GO] Josiah Willard Gibbs (1838-1903), American mathematician

physicist; studied at Yale, Paris, Berlin, Heidelber[;

worked at Yale; contributed to thermodynamics, statistic~!

mechanics, optics.

- 104 -

[61] S¢ren Kirkegaard (1813-1855), Danish theologian and

ph5 .. lospher'; ·studi~o: at Copenhagen and entered the past:::-ral

seminary; rebelled against the philosophical tradition

of his time; made li'ttle impact during his life-time;

originator and fore-runne~ of modern existentialist

philosophy.

[62] Bull. American Mathematical Society, Vol. 72, No. 1,

?ar·t II, 1966.

[6 3] Selected papers of. V orbert ~liener, with contributions

by Y.W.Lee, N.Levinson, W.T.Martin, M.I.T. Press,

Cambridge, Massachusetts, 1964.

[64] ~ C:P r62] , p, 68.

[ss] Se e ['2] , p. 212.

~ * . * * * * * * *

'We do not claim for mathematics the prerogative

of a Queen' Of Science~ there are other fields which

are of the same or even higher importance in education.

But ma~hematics sets the standard of objective truth

for all ini~liect~al endeavours; science and technology_

bear ~~~~ess to its practical usefulness. Besides

Zcr ;J 'Jag'4 and music 'it 1.:s one of the primary manifestation ;;

of the free creative power of the human mind, and it 1-s

the universal organ for world-understanding through

t hr oretical construction. Mathematics must therefore

1' cw·a ·in em essential elf3ment of the knowledge and abilities

.: . · · we hav .;c to , teach., of the cu Z ture we have to

.:::- 1 · 'i t , to the next (jenerat1.:on. Only he who knows

· · ·:1t ma·Uwma ·t-~e-s is, and what. its function in our pr e s e i.i::

c ivil~ z~tio~ is, can give sound advice for the

imrroverzent of our mathematical teaching. '

Hermann Weyt (1885 ~ 1955)

- 105 -