BSHMChristmasMeeting

7December2019

DepartmentofComputerScience,UniversityofWarwick,Coventry,CV47AL

PROGRAMME&ABSTRACTS

Programme

09.30CoffeeandRegistration

09.55Welcome(MarkMcCartney,President)

10.00HelenRoss(Stirling):Dicuilandtriangularnumbers

10.40SteveRuss(Warwick):VisionsintheNight:Bolzano'sAnticipationsofContinuity

11.20Coffee

11.40JaneWess(Independent):FromNewtontoNewcomen:MathematicsandTechnology1687-1800

12.20BSHMAGMandlunch

13.50Shortmembertalk:TroyAstarte(Newcastle):OntheDifficultyofDescribingDifficultThings

14.10Shortmembertalk:CatalinIorga(ENTC,romania):KnownandUnknownInAl-Kashi'sMathematics

14.30RobinWilson(Open):Huntingandcountingtrees:theworldofCayleyandSylvester

15.10Tea

15.30ChrisPritchard(Independent):Fromcollectingcoinstosearchingthearchives:Personalreflectionsonbecomingahistorianofmathematics

16.10MartinCampbell-Kelly(Warwick):VictorianDataProcessing

17.15Finish

OrganisedjointlywiththeDepartmentsofComputerScienceandMathematics,UniversityofWarwick

ABSTRACTS

HelenRoss(Stirling):Dicuil(9thcentury)ontriangularandsquarenumbers

DicuilwasanIrishmonkwhotaughtattheCarolingianschoolofLouisthePious.HewroteaComputusorAstronomicalTreatiseinLatininabout814-16,whichcontainsachapterontriangularandsquarenumbers.Dicuildescribestwomethodsforcalculatingtriangularnumbers:thesummationofthenaturalnumbers,andthemorecomplexmethodofmultiplication,equivalenttotheformulan(n+1)/2.Healsostatesthatasquarenumberisequaltotwiceatriangularnumberminusthegeneratingnumber,equivalentton2=[2n(n+1)/2]–n.Heregardedthemultiplicationformulaasnovel.ItwasinfactdescribedinthethirdcenturyADbytheGreekauthorsDiophantusandIamblichus.Itwasalsoknownasasolutiontoothermathematicalproblemsasearlyas300BC.ItreappearedintheWestinthesixteenthcentury.Dicuilthusfillsagapinourmedievalknowledge.SteveRuss(Warwick):VisionsintheNight:Bolzano'sAnticipationsofContinuity

MuchofthemathematicalworkofBernardBolzano(1781-1848)presentsachallengetohistorians.Howshouldwebestintegrateintothemainstreamofhistoricalnarrativewhatappeartobeoriginalandwell-documentedinsightswhichwereunknown,orunrecognised,intheirowntime,butwhichwererediscovereddecadeslater?Threeexamplesfromtheearly19Cwillbereviewed:neighbourhooddefinitionsofgeometriccontinua(line,surface,solid),theconstructionofanon-differentiablebutcontinuousfunction,andtheconceptof'measurablenumber'whichjustifiedtheso-called'axiomofcontinuity'andidentifiedwhatlaterbecameknownasrealnumbers.Risingtothischallengeforhistory,andrenderingaccuratelytheresultsofBolzano'sthinking,canneverthelessbeaninspiration(ifnotaninfluence)forlatermathematicians.TwoexamplesfromBolzano'sworkoninfinitecollectionswillbeoffered.Finally,someobservationswillbeattemptedontheroleofcontextinassessingwhatconstitutesananticipation.JaneWess(Independent):FromNewtontoNewcomen:MathematicsandTechnology1687-1800

Thistalkpresentsasmallcontributiontoalargeprojectinvolvingaboutfiftyhistoriansofmathematicsglobally.Itwillformachapterinvolumefourofasix-volumesetontheSocialHistoryofMathematics.Mysmallpartinthisis‘MathematicsandTechnology1687to1800’.

The18thcenturywasatimeofdevelopingindustrialisationandimperialism,whichwerechangingthenatureofthephysicalandculturallandscapeinEurope.Forbothpurposesmathematicswasincreasinglyappliedto‘technology’,awordimplyingtheuseoftoolsandmachines.Thetalkwillexploretechnologiestowhichthenewcalculuswasapplied,andtechnologieswhichinvolvedlargenumbersofpeoplebecomingmathematicallyliterateforthefirsttime.

Thetopicscoveredhavebeendividedintothosewhichservedthepurposesofindustrialisationandthosewhichservedimperialism.Undertheformercamelandmanagement,construction,watersupply,transportandpower.Underthelattercamenavigation,shipdesign,ballistics,andalcoholomtery.Thetalkwilltakefourexamplesfromthesetopics,arguingthatthenewcalculuswasnoteffectiveinmanyreal

situations.Ontheotherhandthenumberofpeoplecompetentatmathematicalmanipulationincreasedconsiderably.

TroyAstarte(Newcastle):OntheDifficultyofDescribingDifficultThings

Inthe1960s,afullformaldescriptionwasseenasacrucialandunavoidablepartofcreatinganewprogramminglanguage.Akeypartofthatwasathoroughandrigorousdescriptionofthesemantics.However,inthedecadessince,thefocusonprovidingthishassomewhatdiminished.Whywasformalsemanticsonceseenassocritical?Whydiditnotsucceedinthewayshoped?MyPhDwasspentresearchingtheearlyhistoryofprogramminglanguagesemantics,withaparticularfocusontheIBMLaboratoryViennaunderHeinzZemanek,andtheProgrammingResearchGroupatOxfordUniversityunderChristopherStrachey.Itcouldalsobeseenasanhistoryofmodel-based(ratherthanalgebraicoraxiomatic)semantics.Inthistalk,Iwillpresentthekeyfindingsofmyresearch,asawaytowhetmyaudience'sappetiteformythesis,andarguethatformaldescriptionwasacrucialpartoftheformationoftheoreticalandformalcomputerscienceintheEuropeantradition.CatalinIorga(EdmondNicolauTechnicalCollege,Romania):KnownandUnknownInAl-Kashi'sMathematics

ThispaperisfocusedonthemagnificentmathematicalworkofJamshidAl-Kashi,oneofthemostimportantscholarsofIslam.

Helivedinthe15thcenturyandwasagreatmathematicianandastronomer.Hisremarkablemathematicalbookis“TheKeytoArithmetic”(MiftahAl-Hisab)whichremaineduntranslatedandunknowninWesternEuropeuntiltheendof19thcentury.ThelawofcosinesisknowninFranceasAl-Kashi’stheorem(Theoremed’Al-Kashi)andhiscontributiontodecimalfractionsissosignificantthatformanyyearshewasconsideredastheirinventor.Al-Kashiobtainedaccuratevaluesof2πandsin1oinbothsexagesimalsanddecimals.Hisaimwastocalculateavaluewhichwasaccurateenoughtoallowthecomputationoftheboundariesoftheuniverse.Al-KashialsodiscoveredaveryinterestingalgorithmforcalculatingthenthrootswhichisaspecialexampleofthetechniquesgivencenturieslaterbyRuffiniandAbel.Thepropertiesofbinomialcoefficientswerediscussedinhis’’TheKeytoArithmetic”ofc.1425.

ThepaperalsocomprisesmanyothermathematicaltechniquesandmethodsusedbyAl-Kashi,oneoftheoffspringsofHouseofWisdom(BaytAl-Hikmah)ofBaghdad.RobinWilson(Open):Huntingandcountingtrees:theworldofCayleyandSylvester

Wheredidtheword‘graph’(inconnectionwithgraphtheory)comefrom?Howmanyparaffinsaretherewithagivennumberofcarbonatoms?InthisillustratedtalkIshalloutlinesomecontributionsofArthurCayleyandJamesJosephSylvester,withparticularreferencetotheenumerationoftreesandchemicalmoleculesbetweentheyears1857and1889.Nopreviousknowledgeofgraphtheoryisassumed.

ChrisPritchard(Independent):Fromcollectingcoinstosearchingthearchives:Personalreflectionsonbecomingahistorianofmathematics

Asomewhatself-indulgentlookathowsomeonewithabentformathematicsandacuriosityaboutthepastmadethatjourneytowardshistoricalresearch,withafewwell-knowncharactersmakinganappearanceontheway,includingArchimedes,Brahmagupta,Cardano,PeterGuthrieTait,FrancisGaltonandGeorgeDarwin.MartinCampbell-Kelly(Warwick):VictorianDataProcessing

Large-scaledataprocessingdidnotbeginwithaccountingmachinesandcomputers--itbeganinthe1860swiththefirstindustrial-scaleoffices.Theseofficesemployedhundredsorthousandsofclerkstoprocesscountlessthousandsoftransactionsperday,entirelybyhand.Althoughtheseofficesdidtheirdataprocessingwithnothingmoresophisticatedthanapenandledger,theydevelopedastonishinglycomplexandrobustsystemsperfectlyadaptedtowhatcouldbedonewiththemostprimitivetechnology.ThistalkwilltakeyouonanillustratedtourofsomemajorVictorianoffices,includingtheBankersClearingHouse,theCensusOffice,thePrudentialAssuranceCompany,theCentralTelegraphOffice,andthePostOfficeSavingBank.Thecentralmessageofthetalkisthatwhiletechnologyevolves,informationprocessingsystemsandstructuresareextraordinarilypersistentandsometimeshaverootsthatgoback150years.

TheNewYorkClearingHouse,c.1864.Theimageshowsportersandtellersofthe54NewYorkbanksexchangingchecks--eachexchangetookabout10seconds.