Table Of ContentVita Mathematica
18
Hugo Steinhaus
Mathematician
for All Seasons
Recollections and Notes,
Vol. 1 (1887–1945)
Translated by Abe Shenitzer
Edited by Robert G. Burns,
Irena Szymaniec and
Aleksander Weron
Vita Mathematica
Volume 18
Editedby
MartinMattmuRller
Moreinformationaboutthisseriesathttp://www.springer.com/series/4834
Hugo Steinhaus
Mathematician
for All Seasons
Recollections and Notes, Vol. 1 (1887–1945)
Translated by Abe Shenitzer
Edited by Robert G. Burns, Irena Szymaniec
and Aleksander Weron
Author
HugoSteinhaus(1887–1972)
Translator
AbeShenitzer
Brookline,MA,USA
Editors
RobertG.Burns
YorkUniversity
Dept.Mathematics&Statistics
Toronto,ON,Canada
IrenaSzymaniec
Wrocław,Poland
AleksanderWeron
TheHugoSteinhausCenter
WrocławUniversityofTechnology
Wrocław,Poland
VitaMathematica
ISBN978-3-319-21983-7 ISBN978-3-319-21984-4 (eBook)
DOI10.1007/978-3-319-21984-4
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Foreword to the First Polish Edition (1992)
You hold in your hands a record of the memories of Hugo Steinhaus, eminent
mathematician,afounderofthePolishSchoolofMathematics,first-ratelecturerand
writer,andoneofthemostformidablemindsIhaveencountered.Hissteadfastgaze,
wry sense of humor (winning him enemies as well as admirers), and penetrating
critical and skeptical take on the world and the people in it, combined in an
impressionofbrilliancewhenI,forthefirsttime,conversedwithhim.Iknowthat
many others, including some of the most eminent of our day, also experienced a
feelingofbedazzlementinhispresence.
InmyfirstconversationwithprofessorSteinhaus,heattemptedtoexplaintome,
someone who never went beyond high school mathematics, what that discipline
is and what his own contribution to it was. He told me then—I took notes for
later perusal—the following, more or less. It is often thought that mathematics
is the science of numbers; this is in fact what Courant and Robbins claim in
their celebrated book What is Mathematics?. However, this is not correct: higher
mathematicsdoesindeedincludethestudyofnumberrelationsbutawelterofnon-
numericalconceptsbesides.Theessenceofmathematicsisthedeepestabstraction,
thepurestlogicalthought,withthemind’sactivitymediatedbypenandpaper.And
there is no resorting to the senses of hearing, sight, or touch beyond this in the
exerciseofpureratiocination.
Moreover, of any given piece of mathematics it can never be assumed that
it will turn out to be “useful”. Yet many mathematical discoveries have turned
out to have amazingly effective applications—indeed, the modern world would
be nothing like what it is without mathematics. For instance, there would be no
rocketsflyingtootherplanets,noapplicationsofatomicenergy,nosteelbridges,no
Bureaux of Statistics, international communications, number-based games, radio,
radar, precision bombardment, public opinion surveys, or regulation of processes
ofproduction.However,despite allthis, mathematicsis notatits heartan applied
science: whole branches of mathematics continue to develop without there being
anythoughtgiventotheirapplicability,orthelikelihoodofapplications.Consider,
forinstance,“primes”,thewholenumbersnotfactorableasproductsoftwosmaller
wholenumbers.Ithaslongbeenknownthatthereareinfinitelymanysuchnumbers,
v
vi ForewordtotheFirstPolishEdition(1992)
andamongthemthereare“twins”,suchas3and5,5and7,11and13,17and19,
andsoon.Itisprobablethatthereareinfinitelymanysuchtwins,butnoonehasas
yetmanagedto provethis, despitea greatmanyattempts, allwithoutthe slightest
potentialpracticalapplicationinview.
The late Zygmunt Janiszewski, a brilliant mathematician, wrote: “I do mathe-
maticsinordertoseehowfaronecangetbymeansofpurereason.”
The number of problems thought up by mathematicians but still waiting to be
solvedisunlimited.Andamongthoseforwhichsolutionsarefound,onlyafewwill
find practical application. But it is mathematical abstraction that attracts the best
minds—thosecapableofthepurestkindofhumanmentalactivity,namelyabstract
thought. Hugo Steinhaus was of the opinion that the progress of mathematics is
like a greatmarchforwardof humanity.Butwhile the greatmass of mankindhas
reached no further than the level of the cave-dweller, and a few have attained the
level of the best of the middle ages, and even fewer the level of the eighteenth
century,thequestionarisesastohowmanyhavereachedthepresentlevel.Healso
said: “There is a continuing need to lead new generations along the thorny path
which has no shortcuts. The Ancients said there is no royal road in mathematics.
But the vanguardis leaving the greatmass of pilgrimsfurtherand furtherbehind,
theprocessionisevermorestrungout,andtheleadersarefindingthemselvesalone
faroutahead.”
* * *
However,HugoSteinhaus’srecollectionsaretobereadnotsomuchinorderto
learnanymathematics—althoughone cangleanfromtheminterestingfactsabout
what mathematicians have achieved. The main reasons for reading them are as
follows:First,heledaninterestinglife,activeandvaried—althoughthisisnottosay
thatitwasaneasyonesincetheepithet“interesting”asusedoflifeinourpartofthe
worldhasoftenenoughbeenaeuphemismforexperiencesonewouldnotwishon
anyone.Second,hisgreatsenseofhumorallowshimtodescribehisexperiencesin
unexpectedways. Third, his vast acquaintance—peoplefascinated him—included
many interesting, important, and highly idiosyncratic individuals. And fourth and
last, he always said what he thought,even thoughthis sometimesbroughttrouble
onhim.Sincehehadnodefiniteintentionofpublishinghiswritings,itfollowsthat
he was even franker in them. This truth-telling in response to difficult questions,
thisreluctancetosmoothedges,notshrinkingfromassertionsthatmayhurtsome
andinduceinothersuneasyfeelingsofmoraldiscomfiture:thisisperhapsthemain
virtueofthesenotes.
The following were the chief character traits of the author of these notes: a
sharpmind,arobustsenseofhumor,agoodlyportionofshrewdness,andunusual
acutenessof vision. For him, there was no spoutingof slogans, popularmyths,or
propaganda, or resorting to comfortable beliefs. He frequently expressed himself
bluntly,evenviolently,onmanyofthequestionsofhistime—forinstance,questions
concerninginterwarpoliticsasitrelatedtoeducation(eventhoughhe,asaformer
Polish Legionnaire, might have been a beneficiary of them), general political
problems,totalitarianisminitshitlerianandcommunistmanifestations,andissues
ForewordtotheFirstPolishEdition(1992) vii
of anti-Semitism and Polish-Jewish relations. He said many things people did not
likebackthen,andthingstheydon’tliketoday.
I believe that especially today, when our reality is so different from that of
Steinhaus’s time, it is well worthwhile to acquaint oneself with his spirit of
contrariness and his sense of paradox, since these are ways of thinking that are
todayevenmoreusefulthaninpasttimes.
* * *
Steinhaus believed deeply in the potential for greatness and even perfectionof
the well-trainedhumanmind. He often referredto the so-called “Ulam Principle”
(named for the famous Polish mathematician Stanisław Ulam, who settled in the
USA) according to which “the mathematician will do it better”, meaning that if
twopeoplearegivenatasktocarryoutwithwhichneitherofthemisfamiliar,and
oneofthem isa mathematician,thenthatonewilldo itbetter. ForSteinhaus,this
principleextendedtopracticallyeveryareaoflifeandespeciallytothoseassociated
withquestionsrelatedtoeconomics.
A particular oft-reiterated claim of his was that people who make decisions
pertaining to large facets of public life—politics, the economy, etc.—should
understand,in order to avoid mistakes and resultant damage, that there are things
they don’t understand but which others do. But of course such understanding is
difficulttoattainandremainsrare.
Hugo Steinhaus represented what was best in that splendid flowering of the
Polish intelligentsia of the first half of the twentieth century, without which our
nationcouldneverhavesurvivedtoemergereborn.Thisconstitutedagreatimpetus
for good, triumphing over tanks, guns, and the secret police combined—a truly
Polishstrikeforce.
KazimierzDziewanowski1
1Kazimierz Dziewanowski (1930–1998), Polish writer, journalist, and diplomat. Polish ambas-
sadortoWashington1990–1993.
Introduction to the English Edition
Therearetwowell-knownromanticanecdotesconcerningHugoSteinhaus.Follow-
ing a period of military service in the early part of World War I, he was given a
desk job in Kraków. In the summer of 1916, he went on a “random walk” from
his Kraków residence at 9 Karmelicka Street to Planty Park, where he overheard
the words“Lebesgueintegral”spokenbyone of two youngmenseated ona park
bench—none other than the self-taught lovers of mathematics Stefan Banach and
Otto Nikodým. Later Steinhaus would create, with Banach, the famous Lwów
school of mathematics, one of the two prominent Polish mathematics schools—
the other was in Warsaw—flourishing in Poland between the wars. According to
the second anecdote, in the 1930sSteinhaus, Banach, and others used to frequent
the“ScottishCafé” inLwów,wheretheywouldengageinanimatedmathematical
discussions,usingthemarbletabletopstowriteon.1 Atsomepoint,Banach’swife
Łucjagavethemathickexercisebook,andthe“TheScottishBook”2 wasborn,in
final form a collection of mathematical problems contributed by mathematicians
since become legendary, with prizes for solutions noted, and including some
solutions.Itwasdestinedtohaveatremendousinfluenceonworldmathematics.
In addition to “discovering” Banach and collaborating with him, Steinhaus
pioneered the foundations of probability theory, anticipating Kolmogorov, and of
game theory, anticipating von Neumann. He is also well known for his work on
1The mathematical activity connected with “The Scottish Café” has inspired a cycle of poems
bySusanH.Case,publishedbySlaperingHolPress,2002.FromthereviewbyCharlesMartin:
“ThisseriesofpoemsislooselybasedupontheexperiencesofthemathematiciansofTheScottish
Café,wholivedandworkedinLvov[Lwów],Poland,now[in]Ukraine.Thereisnothememore
importantforpoetrytoaddressinourtime,whenthatlifeisimperiledbybarbarismsfromwithin
andwithout.Byrecallingwithcelebratoryjoythevigor,themessiness,thecourageofthatlifeas
itwasoncelivedinaterribletimebythepatronsofTheScottishCaféinLvov,thesepoemsdous
agreatservice.”
2AvailableinEnglishas:R.DanielMauldin(ed.),TheScottishBook,BirkhäuserBoston,Boston,
MA,1981.SteinhauscontributedtenproblemstoTheScottishBook,includingthelast,datedMay
31,1941,justdaysbeforetheNazisoccupiedLwów.
ix
