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The Nature of Reality

THOUGHT EXPERIMENTS

Math: Discovered, Invented, or Both?By Mario Livio on Mon, 13 Apr 2015

Themiracleoftheappropriatenessofthelanguageofmathematicstotheformulationofthelawsofphysicsisawonderfulgiftwhichweneitherunderstandnordeserve.

EugeneWignerwrotethesewordsinhis1960articleTheUnreasonableEffectivenessofMathematicsintheNaturalSciences(http://math.northwestern.edu/~theojf/FreshmanSeminar2014/Wigner1960.pdf).TheNobelprizewinningphysicistsreportstillcapturestheuncannyabilityofmathematicsnotonlytodescribeandexplain,buttopredictphenomenainthephysicalworld.

Credit: Flickr user Barney Livingston (https://www.flickr.com/photos/barnoid/), adapted under aCreative Commons license (https://creativecommons.org/licenses/by-nc-sa/2.0/).

Howisitpossiblethatallthephenomenaobservedinclassicalelectricityandmagnetismcanbeexplainedbymeansofjustfourmathematicalequations?Moreover,physicistJamesClerkMaxwell(afterwhomthosefourequationsofelectromagnetismarenamed)showedin1864thattheequationspredictedthatvaryingelectricormagneticfieldsshouldgeneratecertainpropagatingwaves.Thesewavesthefamiliarelectromagneticwaves(whichincludelight,radiowaves,xrays,etc.)wereeventuallydetectedbytheGermanphysicistHeinrichHertzinaseriesofexperimentsconductedinthelate1880s.

4/16/2015 Math:Discovered,Invented,orBoth?TheNatureofRealityTheNatureofReality|PBS

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Andifthatisnotenough,themodernmathematicaltheorywhichdescribeshowlightandmatterinteract,knownasquantumelectrodynamics(QED),isevenmoreastonishing.In2010agroupofphysicistsatHarvardUniversitydeterminedthemagneticmomentoftheelectron(whichmeasureshowstronglytheelectroninteractswithamagneticfield)toaprecisionoflessthanonepartinatrillion.CalculationsoftheelectronsmagneticmomentbasedonQEDreachedaboutthesameprecisionandthetworesultsagree!Whatisitthatgivesmathematicssuchincrediblepower?

Thepuzzleofthepowerofmathematicsisinfactevenmorecomplexthantheaboveexamplesfromelectromagnetismmightsuggest.Thereareactuallytwofacetstotheunreasonableeffectiveness,onethatIcallactiveandanotherthatIdubpassive.Theactivefacetreferstothefactthatwhenscientistsattempttolighttheirwaythroughthelabyrinthofnaturalphenomena,theyusemathematicsastheirtorch.Inotherwords,atleastsomeofthelawsofnatureareformulatedindirectlyapplicablemathematicalterms.Themathematicalentities,relations,andequationsusedinthoselawsweredevelopedforaspecificapplication.Newton,forinstance,formulatedthebranchofmathematicsknownascalculusbecauseheneededthistoolforcapturingmotionandchange,breakingthemupintotinyframebyframesequences.Similarly,stringtheoriststodayoftendevelopthemathematicalmachinerytheyneed.

Passiveeffectiveness,ontheotherhand,referstocasesinwhichmathematiciansdevelopedabstractbranchesofmathematicswithabsolutelynoapplicationsinmindyetdecades,orsometimescenturieslater,physicistsdiscoveredthatthosetheoriesprovidednecessarymathematicalunderpinningsforphysicalphenomena.Examplesofpassiveeffectivenessabound.MathematicianBernhardRiemann,forexample,discussedinthe1850snewtypesofgeometriesthatyouwouldencounteronsurfacescurvedlikeasphereorasaddle(insteadoftheflatplanegeometrythatwelearninschool).Then,whenEinsteinformulatedhistheoryofGeneralRelativity(in1915),Riemannsgeometriesturnedouttobepreciselythetoolheneeded!

Atthecoreofthismathmysteryliesanotherargumentthatmathematicians,philosophers,and,mostrecently,cognitivescientistshavehadforalongtime:Ismathaninventionofthehumanbrain?Ordoesmathexistinsomeabstractworld,withhumansmerelydiscoveringitstruths?Thedebateaboutthisquestioncontinuestoragetoday.

Personally,Ibelievethatbyaskingsimplywhethermathematicsisdiscoveredorinvented,weforgetthepossibilitythatmathematicsisanintricatecombinationofinventionsanddiscoveries.Indeed,Ipositthathumansinventthemathematicalconceptsnumbers,shapes,sets,lines,andsoonbyabstractingthemfromtheworldaroundthem.Theythengoontodiscoverthecomplexconnectionsamongtheconceptsthattheyhadinventedthesearethesocalledtheoremsofmathematics.

ImustadmitthatIdonotknowthefull,compellinganswertothequestionofwhatisitthatgivesmathematicsitsstupendouspowers.Thatremainsamystery.

GoDeeperEditorspicksforfurtherreading

4/16/2015 Math:Discovered,Invented,orBoth?TheNatureofRealityTheNatureofReality|PBS

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NOVA:TheGreatMathMystery(http://www.pbs.org/wgbh/nova/physics/greatmathmystery.html)Ismathinventedbyhumans,orisitthelanguageoftheuniverse?NOVAtakesonthisquestioninanewfilmpremieringApril15,2015at9pmonmostPBSstations.

NOVA:DescribingNaturewithMath(http://www.pbs.org/wgbh/nova/physics/describingnaturemath.html)Howdoscientistsusemathematicstodefinereality?Andwhy?PeterTysoninvestigatestwomillenniaofmathematicaldiscovery.

TheWashingtonPost:TheStructureofEverything(http://www.washingtonpost.com/wpdyn/content/article/2009/02/05/AR2009020502876.html)LearnmoreabouttheunreasonableeffectivenessofmathematicsinthisreviewofMarioLiviosbookIsGodaMathematician?

Mario Livio

Dr.MarioLivioisaninternationallyknownastrophysicistattheSpaceTelescopeScienceInstitute,theinstitutewhichconductsthescientificprogramoftheHubbleSpaceTelescope,andwillconductthescientificprogramoftheupcomingJamesWebbSpaceTelescope.HeisaFellowoftheAmericanAssociationfortheAdvancementofScience.Hehaspublishedmorethan400scientificpapersontopicsrangingfromdarkenergyandcosmologytoblackholesandextrasolarplanets.Dr.Livioisalsotheauthoroffivepopularsciencebooks,including"TheGoldenRatio"(forwhichhereceivedthe"PeanoPrize"andthe"InternationalPythagorasPrize")and"IsGodAMathematician?"Livio'srecentbook,"BrilliantBlunders,"wasontheBestsellersListoftheNewYorkTimes,andwasselectedbytheWashingtonPostasoneofthe"2013BestBooksoftheYear."Otherpostsfromthiscontributor(http://www.pbs.org/wgbh/nova/blogs/physics/author/mariolivio/)

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JessTauber 2daysagoSeveralyearsagoIdiscoveredthatatomicshellstructures(bothelectronicandnuclear)appearedtodependonmathematicalcombinatoricsthatcamerightoutofthePascalTriangle.Forexample,inthesimpleharmonicoscillatormodelofshellstructureinthespherical

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thesimpleharmonicoscillatormodelofshellstructureinthesphericalatomicnucleus,magicnumbersofnucleons(whereshellsarefilled,analogouslytonoblegasesinelectronicstructure)are2,8,20,40,70,112,168theseareexactlydoubledtetrahedralnumbers1,4,10,20,35,56,84...Variationsonthisthemealsohelpdeterminetheharmonicoscillatormagicnumbersforellipsoidallydeformednuclei.Differencesbetweenthesphericalmagicsaresinglecopiesofdoubledtriangularnumbers,andtheoscillatorratioofasphereis1:1.Whentheoscillatorratiois2:1thenTWOcopiesofeachdoubletriangularnumbersumtogiveeachnextdeformedmagicinsuccession.Whentheratiois3:1,thenTHREEcopiessosum.Ontheotherhand,foroblatelydeformednuclei,therelationisdivisionalratherthanmultiplicational.Forthesphere,adoubletriangularnumberdifferenceexistsforeverymagicnumber.Butfortheoblatenucleusofoscillatorratio1:2,thatdifferenceisbetweenEVERYOTHERmagic,andforratio1:3betweenEVERYTHIRDmagicandsoon.Exceptionsexistwhenyouhaven'tyetaccumulatedenoughmagics(equaltothedenominatoroftheoscillatorratio),andthenthemagicsaresimpledoubletriangularnumbersuntilyoureachthatpoint.Noexceptions.Morecomplicated'realistic'modelsofthenucleusincludeaspinorbitinteractionaswellasmodificationstothepotentialwellthenucleonsfeel.ButevenheretherearePascalrelationships.Socalled'intruder'levels(thehighestspinorbitalpartialsfromthenexthighershell)lowertheirenergies(becauseofthespinorbitcoupling)enoughtodropintothepreviousshell,addingtheirnumberofnucleonstothemagics(so40+10(g9/2)=50,70+12(h11/2)=82,112+14(i13/2)=126,168+16(j15/2)=184andsoon.TheDEPTHofsuchpenetration(thetotalcountjustbeforeorafteroneaddstheintrudernucleons)turnsout(forneutrons)tobeexactlydoubledtriangularnumbers.So402=38,withg9/2startingjustafter,and502=48,wherethe10nucleonsterminate.706=64,withh11/2startingafter,and826=76,wherethe12nucleonsterminate.Andsoon.TherearemanyotherPascalmathematicalrelationshipspresentinnuclearshellstructure.Thereasontheyworkseemstobethatintheharmonicoscillatornucleuseachperiodiscomposedofsumsofeveryotherorbital,segregatedbyparity.Sos=2,p=6,ds=12,fp=20,gds=30,hfp=42,igds=56,jhfp=72...Notethatthesearealldoubledtriangularnumbers.Further,theSIZESoftheadditionsgivenbyintrudersaren'tarbitrary,butareexactlythoseneededtoincreasethetotalsizeoftheperiodtheyaddtotothenextdoubletriangularnumber20+10=30,30+12=42,42+14=56andsoon.Thephenomenonpreservesdoubletriangularity!

Intheelectronicstructure,ifwereformattheperiodictablesothatthesblockelementsareontherightedge(the'leftstep'tableofCharlesJanet

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blockelementsareontherightedge(the'leftstep'tableofCharlesJanet(discoveredinthelate1920's,whichorganizesthingsthewayaphysicistwould,ratherthanachemist),thenthealkalineearths(butalsoincludinghelium)terminateeachperiod.Ifyoulookattheatomicnumbersofeveryothers2elementtheyareALLtetrahedralnumbers(4,20,56,*120).Andalltheintermediateatomicnumbersarethearithmeticmeansofthese(so(0+4)/2=2,(4+20)/2=12,(20+56)/2=38,(56+120)=88...This'triad'relationshipwasdiscoveredoriginallyintheearly19thcenturybutbasedonatomicweightsratherthanatomicnumbers,andwasthebasisoftheperiodicrelation'sdiscovery(mostfamouslybyMendeleevbuthewasn'ttheonlyone).Becauseofthesenumericalrelationshipstheentireperiodicsystemcanbereconceptualizedinthreedimensionsasaperfecttetrahedroncomposedofclosepackedspheres,eachsphererepresentingoneelement.AswiththenuclearsystemtherearemanyotherPascalrelationshipsinshellstructure.Eachelectronicperiodisahalf/doublesquarenumberinsize:2,8,18,32,50...ButintheJanettabletheyareallpairedforsize,whichgivessumsofdualsasfullsquares4,16,36,64...AndifyousumTHESEyougetthetetrahedralnumbers4,20,56,120.Thequestioniswhatarewetomakeofallthisisitmere'numerology'orcanwegainsomeadvancetoourunderstandingofNaturethroughsuchstudies?

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SamuelBass 10hoursago>JessTauberLookuptetryonics.Soundslikeyou'vegotapieceofthepuzzlefiguredout.

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PaulStamison 9hoursagoIjustsawtheprogramandwasIeversoexcitedasIlovemathandIalwaysvieweditjustliketheprogramdid.IwilltrytogettheDVDandwillshowittomyafterschoolprogramforsomebrilliantgrade7and8students.Thankyou.

MoliehiMarake 16hoursagoForsolong,Iwonderedwhymostprinciplesandstudiesinlifearebetterunderstoodwhenexpressedandexplainedinmathematicalterms,butthispiecehasjustgivensomeperspectivetomyquestions.Ipersonallyconsidermathematicsaninnatepartofman,justaslanguageis.Itisatoolwithinusthatnaturallyexists,atoolweresorttoindireneedtoanswersomeofthephenomenaaroundus.

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4/16/2015 Math:Discovered,Invented,orBoth?TheNatureofRealityTheNatureofReality|PBS

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answersomeofthephenomenaaroundus.u15269559

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GeorgeGantz adayagoWithallduerespect,mathematicsisaconprehensiveandintegratedwholearisingatthepointofconsciousawarenessofthefirstdistinctionofsomethingfromnothingthroughapriorilawsoflogic.ThisisthethesisofmyessayTheHoleattheCenterofCreation,oneof200submittedinthe2015FQXiessaycontestonthistopic.AnyoneseriouslyinterestedinthewideanddisparateviewsonthisissueshouldvisitFQXi.org.Publiccommentsandessayratingsarewelcome!

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JessTauber 2daysagoOneotherpoint.OnawhimItookFibonaccinumbersandmappedthemasATOMICNUMBERS.Itturnedout,foratleastallknownelements,theymappedasFIRST/LEFTMOSTelementsintheorbitalhalfrows.Thisisrelevantbecauseelectronsfilllobessinglybeforetheystarttodoubleup,sotheleftorbitalhalfhasonlysinglyfilledlobesandtherightorbitalhalfstartsdoublinglobes.AND,theoddFibnumberswerealwaysmappedtothesinglyfilledfirst/lefthalforbital,andtheevenFibnumberstothedoublefillingsecond/righthalforbital.At144shellmodelspredictthatthismappingschemewillbeseverelyoutofalignment,butthenagainwe'llprobablyneverseesuchheavyelements.Interestingly,therelatedLucasnumbers2,1,3,4,7,11,18,29,47,76,123....tendtomaptoLAST/rightmostpositionsinorbitalhalfrows.Thisworksupto18,butafterwardsmisaligns.EvensoNatureseemstohavebehavioralfixes:29,copper,and47,silver,areinthesamecolumn(coinagemetals)withelectronicconfigurationsthatare'anomalous'.Bypositioninthetablethey*should*haved9,s2butinsteadhaved10,s1,whereoneelectronhasbeenabstractedfromafullsshell(renderingithalffilledbutthatisokfortheLucastrend)andfillingthedshell,againokintheLucassystem).76,osmium,hasthecorrectelectronicconfigurationforitspositionwithintheperiodictable,butacts'asif'itwerexenoninsomeofitscompounds,thatisanoblegaswithafullorbital.Apparentlyd6isreinterpretedasifitwerep6.

NigelReading|ASYNSIS 2daysagoBothInvented&Discovered...I'mwithMario.Theintelligenceisinthegeometryitself,ifit'sselfconsistent,itcanalsobeselforganising.My@Quoraanswerto"DotheFibonacciSequence,Pi,andtheGolden

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My@Quoraanswerto"DotheFibonacciSequence,Pi,andtheGoldenRatiodoanythingtoprovethattheuniversewasintelligentlydesigned?"http://t.co/iq7EyNKbgj

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RussAbbott 3daysagoAsyousaid,Newtoninventedthecalculusinordertoexplainthephenomenahesaw.Inotherwords,mathematicsissimplyawaytoexpressregularities.Thequestionmaybewhythereareregularitiesintheuniverse.Butifoneassumesthatthereare,thenmathematicsissimplyawaytoexpressthemoncewestarttoseethem.It'slessamysterythanthispostmakesit.

Putanotherway,ifweencounterandbegintorecognizearegularityforwhichnomathematicsexists,weinventthemathematicsnecessarytoformalizewhatwesee.It'snotthatmathematicsisindependentoftheregularitieswewanttoexpress.It'sour(invented)languagesforexpressingthem.

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MichaelKeating 10hoursago>RussAbbottIagree.Itisthedescriptionofrelationshipsbetweenobjectsorphenomena.Certainratiosappearrepeatedlyinnaturebecausetherulesaffectallormostthingsequally.Thisseemsintuitivetome.Isitpossiblethatadifferentbasenumberingsystemcoulddemystifyoruntanglecomplicatedequations?

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