Mathematician for all seasons. Recollections and notes, Vol.1: 1887-1945 PDF
Preview Mathematician for all seasons. Recollections and notes, Vol.1: 1887-1945
Vita 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 LibraryofCongressControlNumber:2015954183 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Covercredit:PhotoofHugoSteinhaus.CourtesyofHugoSteinhausCenterArchive,WrocławUniversity ofTechnology Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com) 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
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