R. Daniel Mauldin The Scottish Book Mathematics from The Scottish Café, with Selected Problems from The New Scottish Book Second Edition TheScottishCafé,birthplaceoftheScottishBookproblems,asitappearedinapostcardfrom theearly1970s.TheScottishCaféisontheright,withtheCaféRomaontheleft.Accordingto StanisławUlam,thisscenehaschangedlittlefromtheperiodprecedingWorldWarII R. Daniel Mauldin The Scottish Book Mathematics from the Scottish Café with Selected Problems from The New Scottish Book Second Edition R.DanielMauldin DepartmentofMathematics UniversityofNorthTexas SanDiego,CA,USA ISBN978-3-319-22896-9 ISBN978-3-319-22897-6 (eBook) DOI10.1007/978-3-319-22897-6 LibraryofCongressControlNumber:2015947275 MathematicsSubjectClassification(2010):01A60. SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland1981,2015 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. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Preface to the Second Edition TheScottishBook,wrappedinthemistsofthepast,isalegendinthemathematics world.Itsfascinatingstoryandthelegendaryfigureswhoformulatedtheproblems inthebookcontinuetoholdourattention.Itrepresentsthebestofcafémathematics, an informal, free-wheeling style of mathematical conversation and interaction that seems almost lost today. For the solution to some of the problems, prizes were offered, ranging from the famous live goose, a bottle of whiskey of measure > 0, onekiloofbacontoonesmallbeer. One can imagine the atmosphere at the Scottish Café while the problems were being formulated. We should take to heart the playfulness and enjoyment that is ondisplay,agoodmodelforencouragingandstimulatingmathematicsatanyage. One can also see that a few of the problems must have been stated after spending some time drinking tea or perhaps a brandy or two. In his book Adventures of a Mathematician,Ulamdescribeswhatasession(onelastingatleastseventeenhours) atthecafémightbelikeandsketchessomeofthecentralfigures.DuringtheyearsI knewandworkedwithUlam,helovedthecafémodeofdiscussion.Gian-CarloRota givesadetaileddescriptionofthisinhisarticle‘TheLostCafé’,inFromCardinals to Chaos, edited by N. G. Copper (Cambridge University Press, 1989). The entire schoolofmathematicsinLwówiswonderfullypresentedbyRomanDudainPearls FromaLostCity(AmericanMathematicalSociety,2014),atranslationofhis2008 Polishversion.Thetragedythatbefellsomanyofthefiguresaroundthebookhas beentracedmanytimes,includingthearticlebyJoannaDianeCaytas,‘Survivalof theScottishBook:APhoenixfromtheHolocaustofPolishMathematics’,available on the internet. There is even a collection of poems about the book by Susana H. Case,TheScottishCafé(SlaperingHotpress,2002). The problems and the ideas behind continue to effect mathematics today. In the 35 years since the first edition of the book, many more problems have been solved or partially solved. But even today, quite a few remain unsolved. In view of this, I decided to gather new commentaries and update some of the old ones. The appendices of this edition include a list of the unsolved and partially solved problems together with those that have no commentary, a list of unsolved prize problems,alistofproblemsposedbyeachauthor,andalistofproblemsbysubject v vi PrefacetotheSecondEdition area. Besides correcting many errors in the first edition, for clarification a few changesinUlam’stranslationoftheoriginalScottishBookhavebeenmade. Inadditiontosomeofthelecturesgivenatthe1979ScottishBookconference, thiseditionalsoincludesabriefhistoryofWrocław’sNewScottishBookandsome selectedproblemsfromit. This edition would not be possible without the generous contributions and suggestionsofthecommentators.Iofferthemmyheartfeltgratitude.Iwanttothank Kirby Baker, Larry Lindsay, Bill Bernard, and Sue DeMeritt for their assistance in getting this project underway. I thank Al Hales, Joe Buhler, and Jan Mycielski fortheircounsel.IthankAllenMann,ChristopherTominich,andBenjaminLevitt at Birkhäuser for their support during the preparation of this new edition. Finally, IthankDiana,mywife,forsupportingmethroughoutthislongproject. Itismysincerehopethatthiscollectionwillbringtothereader,asithastome, manyhoursofenjoymentandanimageofwhatmusthavebeenamostwonderful place,TheScottishCafé. SanDiego,CA,USA R.DanielMauldin Preface to the First Edition Once while working on a problem, someone was kind enough to point out that my problem was in the “Scottish Book.” I had not heard of the Scottish Book and certainly did not realize that this book had no connection with Scotland. But, since that introduction, I’ve become more and more aware of the magic of the mathematiciansandthemathematicsinvolvedinitsbirth. TheScottishBookoffersauniqueopportunitytocommunicatewiththemen(no womenwereonthescene,asIunderstandit)andideasofatimeandplace,Lwów, Poland,whichhavehadanenormousinfluenceonthedevelopmentofmathematics. The history of the Scottish Book as detailed in the following lectures by Ulam, Kac, and Zygmund provides amazing insights into the mathematical environment ofLwówbeforeWorldWarII. Therearemanycollectionsofproblems,butthissethasbecomeworld-renowned. Perhaps,aprimaryreasonforthisrenownisthattheproblemsareclearlyandsimply formulated,accessibletothegeneralmathematicalcommunity,andyetstrikeatthe heartoftheconceptsinvolved. It is my pleasure and honor to edit this version of the Scottish Book, which includesacollectionofsomeofthetalksgivenatthe“‘ScottishBookConference”’ heldatNorthTexasStateUniversityinMayof1979.Thepurposeoftheconference was to examine the history, development, and influence of the Scottish Book. As John Oxtoby toasted at the conference, there was a “‘condensation of Poles”’ at thisconference.AmongthemweresomeoftheoriginalcontributorstotheScottish Book,ProfessorsUlam,Kac,andZygmund.Theireditedtalksappearhere,together with the talk given by one close to them in spirit and collaboration, Professor Paul Erdós. Also presented here is a talk by a member of a younger generation, Professor Andrzej Granas, in which one problem of Schauder is discussed with its many-faceted implications and connections. It should come as no surprise that theconferencewasheldinTexas;themathematicalsimilaritiesbetweentheTexas schoolandthePolishschoolhavelongbeennoted,beginningwiththefactthatthe firstAmericantopublishinFundamentaMathematicaeisR.L.Moore. From a glance at the problems, one sees that they cover a wide range of mathematics. I think this simply reflects the wide interests of the unusual group, vii viii PrefacetotheFirstEdition which assembled the collection. The problems are concentrated in the areas of summability theory, functional and real analysis, group theory, point set topology, measuretheory,settheory,andprobability.Itislikewiseeasytoconfirmthatsome ofthecontributorstotheBookwere,asR.H.Bingtoasted,the“‘leadinglights”’in thesefields. Ihaveattemptedtoobtainanappropriatecommentaryforeachoftheproblems, although quite a few of the problems remain without comment. Some of these, as well as a number of problems with comments, remain unsolved to this day. For othersIsimplyfailedtogetanappropriateexpertcomment(Iwouldbegratefulfor contributionsfromreadersofthisedition). Following a problem there may appear the word Addendum. This indicates a comment that was entered into the Scottish Book during the time when the problemswerebeingcollectedinLwów.Latercommentaries,remarks,andsolution toproblemswhicharepresentedhereforthefirsttimefollowtheoriginaladdenda. The problems and original addenda appear here essentially as they have in the two earlier English-language editions of the Scottish Book, both edited (and one produced) by Stanisław Ulam. The first, in 1957, was a mimeographed version of Ulam’s own translation from the original languages in which the problems were inscribed in the Book (mostly Polish), which he distributed on personal request fromhisprofessionalbaseatLosAlamosNationalLaboratory.By1977,thevolume of requests addressed to both Professor Ulam and the Los Alamos Laboratory’s library made it only reasonable to prepare a somewhat more formal edition. This edition again presented only the translated problems and their contemporary addenda, and has been distributed by the Los Alamos laboratory since then. The recent reconcentration and expansion of interest in the Book, including the 1979 conference, had made a place for a new edition, including a collection of at least someoftheworkwhichhasbeenstimulatedbytheScottishBookproblemsinthe yearssincetheywerefirstcollected. This project enjoyed the aid of several individuals and institutions, beginning with the encouragement of Stan Ulam and Gian-Carlo Rota. I sincerely thank all of the commentators for their generosity in providing the commentaries and suggestions. It is obvious that a major contributor to this edition is Jan Mycielski. His encouragement and constant flow of comments and references kept life in the project. Bill Beyer provided many significant comments on the formulation of the problems. TheScottishBookconference,whichwasheldinDentoninMay1979,focused our efforts. It was my hope that some of the spirit of that time and place would be recaptured at the conference. Perhaps, it was through the contributions of the speakersincludingR.D.AndersonandD.A.Martin,bothofwhomtracedsomeof themostoutstandingworkintheirfieldsbacktotheScottishBook. PrefacetotheFirstEdition ix TheNationalScienceFoundation,throughgrantMCS-79-0971andNorthTexas State University, provided funding for the conference. A number of commentaries werewrittenundertheauspicesofaFacultyResearchgrantfromNorthTexasState University. I sincerely thank Lynn Holick for the superb typing, and the people at BirkhäuserBostonfortheirhelpinbringingtheprojecttofruition. SanDiego,CA,USA R.DanielMauldin 1981