MASTER THESIS Applying Mathematics in the Natural Sciences An Unreasonably Effective Method Lisanne Coenen MasterPhilosophyofNaturalSciences July2015 LeidenUniversity FacultyofHumanities InstituteforPhilosophy Supervisor: VictorGijsbers Abstract Inthisthesistheunreasonableeffectivenessofmathematicsinthenaturalsciencesis discussed.Iwillshowthatthisisadeepphilosophicalproblemforwhichnoeasysolu- tionisavailable.Ahistoricalanalysisoftheroleofmathematicsinscienceshowsthat basicmathematics,anabstractionfromempiricalobservation,evolvedintocomplex mathematics,ahumaninventioncompletelydetachedfromitsempiricalroots.The conclusionofthisanalysisisthattheapplicabilityofmathematicscannotbeexplained byadheringtotheempiricalrootsofmathematics.Thisposesaphilosophicalproblem: howcansomethingthatisanthropocentricdescribeandpredicttheintricateworkings ofnaturalphenomenasoaccurately? Thisquestionismymainresearchquestion andisalsothoroughlydiscussedbyMarkSteiner(1998).Heplacesemphasisonthe predictivepowerofmathematicsinthenaturalsciencesandIwillshowthatSteiner’s mainargument, thatanthropocentricelementsinmathematicsplayacrucial, and unreasonableeffective,roleinthediscoveryofnewphysicaltheoriesisavalidobser- vationinneedofanexplanation.ThemappingaccountsofPincock(2004)andBueno andColyvan(2011)arediscussed,whoattempttorendertheanthropocentricelements inmathematicsintelligible.Theybothturnouttobeincompleteandtherefore,Ihave providedanimprovedinferentialmappingaccountthatisabletorenderpartsofthe anthropocentricinfluencesinmathematicsintelligible.Howeverthesuccessfuluseof tractabilityassumptionscannotbeexplainedbythismappingaccount.Thisleadsto theconclusionthattheworldlooks’user-friendly’,becauseouranthropocentricas- sumptionsresultincorrectknowledgeaboutthenaturalworld.Therefore,onecannot refrainfromametaphysicaldiscussionabouttherelationbetweenmathematics,mind andworld. Idiscussseveralmetaphysicalaccounts,ofwhichthemostreasonable isthesimpleexplanationthatwejust’seewhatwelookfor’. Apriceneedstobe paidhowever;completeknowledgeabouttheworldarounduswillneverbepossible. Moreover,itremainsmysteriousthatweareabletocontrolnaturalphenomenain suchadetailedway,whilstonlyhavingknowledgeofasmallpartofit. Thefinal chaptermentionsthechangingroleofmathematicsinscienceinthelast30years, whereadvancementsintheoreticalphysicsincreasedtheimportanceofmathematical methods,whereasadvancementsincomputersciencedecreasedthisrole.Iconclude thatnowmorethanever,itisimportanttoreflectontheroleofmathematicsinthe scientificmethod. Contents 1 Introduction 5 2 Theriseofcomplexmathematics 9 2.1 Theempiricaloriginsofmathematics: basicvs. complexmathe- matics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 MathematicsinancientGreece . . . . . . . . . . . . . . . . . . . . 11 2.3 Fromancienttomodernscience;thebirthofcomplexmathematics 12 2.4 Thecomplexificationofmathematics . . . . . . . . . . . . . . . . 15 3 Whatisunreasonableabouttheeffectivenessofmathematics? 19 3.1 Mathematics in the 20th century: the ’big three’ and Wigner’s puzzle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 FourcommonexplanationsofWigner’spuzzle . . . . . . . . . . 22 4 Steiner’santhropocentricargument 25 4.1 Steiner’santhropocentricclaim . . . . . . . . . . . . . . . . . . . 25 4.2 Themysteryofquantization . . . . . . . . . . . . . . . . . . . . . 28 4.3 ThepredictionofthepositronbyDirac. . . . . . . . . . . . . . . . 31 4.4 TwocriticismsofSteiner . . . . . . . . . . . . . . . . . . . . . . . 33 5 Howmappingaccountssolvepartofthemystery 35 5.1 Pincock’smappingaccount . . . . . . . . . . . . . . . . . . . . . 35 5.2 Theinferentialconceptionoftheapplicationofmathematics . . 37 5.3 Improvingtheinferentialmappingaccount . . . . . . . . . . . . 39 5.4 Whydoesmathematicsonlyworkinthenaturalsciences? . . . 44 6 Metaphysicalconsiderations 47 6.1 MathematicalPlatonism . . . . . . . . . . . . . . . . . . . . . . . 48 6.2 Embodiedmathematics: AKantianapproach . . . . . . . . . . . 49 6.3 Dowejustseewhatwelookfor? . . . . . . . . . . . . . . . . . . . 51 3 6.4 Themultiple-worldshypothesis. . . . . . . . . . . . . . . . . . . 52 7 Conclusionandoutlook 55 References 58 4 Chapter 1 Introduction Scienceisanextremelypowerfultool;bothinitsabilitytodescribetheworld and as the starting point for many innovations and novel technologies. The scientificmethodreliesheavilyonmathematicswhichisusedtoquantifythe phenomenaintheworld. Oldmathematicalstructuresarere-inventedandnew mathematicalstructuresareputinplacetoquantifyobservationsandtheories and every time it became apparent that the mathematical toolbox perfectly fittedontothephysicaldescriptionoftheworld. Thisraisedsuspicionabout thetruestatusofmathematicsandEugeneWigner,inhisfamousarticle’the unreasonableeffectivenessofmathematicsinthenaturalsciences’described thissuspicion(Wigner,1960). Howcanitbe,heasked, that the mathematical formulation of the physicist’s often crude experienceleadsinanuncannynumberofcasestoanamazingly accuratedescriptionofalargeclassofphenomena. Thisshowsthat themathematicallanguagehasmoretocommenditthanbeingthe onlylanguagewhichwecanspeak;itshowsthatitis,inaveryreal sense,thecorrectlanguage. (Wigner,1960,pp. 5-6) What Wigner makes clear is that the view that mathematics is merely a tool forscientistscannotbethewholestory. Heaskedwhythelanguageofmath- ematicsisabletodescribeandpredictnaturalphenomena-andwhyitdoes thatsoaccurately. Aresultthatexemplifiesthisincredibleaccuracyofgeneral mathematicalmethodsisthedeterminationofthetheoreticalvalueofthegyro- magneticratiog,aconstantthatwasimportantfordeterminingthemagnetic momentofanelectron. Themagneticmomentofanelectronfollowstheequa- tionµ = g(eh/2mc)S,wheregwasdeterminedbythebythenacceptedDirac equationandshouldequal2accordingtothatequation. However,experiments 5 showedadeviationfromthisvalue,whichwasstrangesincetheDiracequation predictedresultswithanaccuracywaybetterthanthisdeviation. Theattempts tosolvethisanomalyresultedinthenewfieldofrelativisticquantumelectrody- namics,inwhichstateoftheartmathematicswasusedtodescribethebehavior ofsmallsystems. Afterlong,deviouscalculationsandveryprecisenewmea- surementsduringthe70sand80s,atomicphysicistsfoundthefollowingvalues forg(Gross,1988,p. 8372): g =2 (1,000159652459 0,000000000123) theory · ± g =2 (1,000159652459 0,000000000004) experiment · ± Thisresulthasevenbeenimproveduponsincethefirstexperiments,whichled toarelativestandarduncertaintyof7,6 10 13in2006(Odom,Hanneke,D’Urso, � · &Gabrielse,2006). Youcannotbutwonderhowthisimpressiveresultcame aboutandwhattheapparentlystrongrelationisbetweenthepuremathematical structureunderlyingquantumelectrodynamicsandthenaturalworld.Itdoesn’t seemlikeanapproximationanymore,whenthevalueisaccurateuptothirteen decimals. It is cases like these that led Wigner to state that the applicability of mathematics is a "wonderful gift which we neither understand nor deserve" (p. 9). With his article he articulated more clear and more pressing than ever themysteriousapplicabilityofmathematicsinthenaturalsciences. However asBochner(1966)andColyvan(2001)bothstate,thisphilosophicalproblem hasnotreceivedandisnotreceivingenoughattention. Thetopicwasnever discussedindepthinphilosophicalandscientificcirclesandthefewauthors thatdodiscussitconcludewithphraseslike’asuggestionismade’,’itremains anopenquestion’and’moreworkneedstobedone’. Wigner’s question will be the main research question in this thesis. Is the use of mathematics in the natural sciences truly unreasonably effective or can it be rendered intelligible? I will approach the problem from three differentperspectives: Ahistoricalperspectivethatshowsthedevelopmentof mathematicsovertimeanditsconnectiontonaturalscience,amethodological perspective that shows how mathematics enters the scientific practice and a metaphysical perspective that questions our definition of mathematics. The structureofmythesisfollowslargelythesethreeperspectives. Chapter 2 focuses on the historical approach and discusses how math- ematicsandsciencegotintertwined. Itfocusesmainlyontheperiodduring the scientific revolution, wherein the merge of mechanics and mathematics 6 initiatedtheclosecollaborationbetweenmathematicsandscienceingeneral. Fresnel’stheoryoftotalreflectionisthoroughlydiscussed,sinceitisbelieved thatthisisthefirsttimeinsciencethat’morecameoutofmathematicsthan wasputinbyit’. Twoconclusionsfollowfromthissection: mathematicsaswe knowittodayisdetachedfromitsempiricaloriginsandhistoryhasshownthat mathematicsdevelopedwithoutanapplicationinsightcouldneverthelessbe usefulfordescribingnaturalphenomena. Chapter3discussesthescope,relevanceandvalidityofthemainresearch question,becausealthoughmanyphilosophershaverecognizedthatthereis somethingstrangehere,veryfewhaveactuallytakenupthetaskofdefiningand solving’Wigner’spuzzle’.Thisdisinterestednessisnotstrange,sincemathemat- icsissuchanormalpartofourlivesthatitsusefulnessseemsunproblematicand notworthofphilosophicalattention. Iwilltakesometime,therefore,todiscuss Wigner’sarticleindetailandtoshowthattherelationbetweenmathematics, scienceandthenaturalworldisnotsounproblematicasitlooks,alongtheway rejectingsomeofthe’easywayout’solutionstoWigner’spuzzle. Chapter4isthestartofthemethodologicalapproachandisconcerned withoneofthemostimportantresponsestoWigner’sarticle: MarkSteiner’s book The Applicability of Mathematics as a Philosophical Problem. I will discuss Steiner’santhropocentricargumentanddiscusshistwomostimportantexam- plesthatdefendthisargument: thequantizationprocedureandtheprediction of the positron by Dirac. His methodological approach leads to the insight thattheanthropocentricelementspresentinmathematics,suchasthebeauty ofequations,playacrucialroleinthedevelopmentofnewphysicaltheories and moreover, that this role is unaccounted for and the mathematics there- fore unreasonably effective. AlthoughI grantthat thereareanthropocentric elementspresentinthemathematicalmethods,Iquestionhisconclusionthat theseanthropocentricelementsareunintelligibleinthescientificmethod. This question,whethertheanthropocentricelementsinmathematicscanberendered intelligibleisthereforethemainquestionofChapter5.HereIinvestigateseveral mappingaccountsthatshowhowmathematicsisusedinthenaturalsciences. I rejectPincock’smappingaccountandacceptpartofBueno&Colyvan’sinferen- tialmappingaccountinwhichinferentialrelationsbetweenexperimentsand mathematicalconclusionsplayanimportantrole. However,Iwillshowthat theirmappingaccountisnotcompletesinceitdoesnottakeintoaccountthe firstandmostimportantstepintheprocess: theuseoftractabilityassumptions tomaketheempiricalsituationmathematicallytractable.Iarguethattractability assumptionsareusedtohandletheempiricalsituationmathematically,andthat 7 theseassumptionsareanthropocentricandcannotbemadeintelligiblebyin- vokinginferentialrelations. Iprovideanimprovedmappingaccount,inwhich theseinfluencesaredisplayedandwheretheroleofexperimentsbecomemore clear. ItisconcludedthatpartofWigner’sandSteiner’sproblemscanbesolved by adopting this improved mapping account though the role of tractability assumptionsinthescientificmethodisstillunaccountedfor. Thisleadstotherealizationthattheworldlooks’user-friendly’andthat ananswerhastobefoundtothequestionwhatmathematicsreallyisandwhere itcomesfrom. ThesearemetaphysicalquestionsandChapter6willtherefore be concerned with a metaphysical approach. Here I provide metaphysical solutions to the problem of the applicability of mathematics in the natural scienceswithoutpretendingtobefullyexhaustive. Platonismisreviewed,a solutionfromcognitivesciencediscussed,thesimplesolutionthatwejust’see whatwelookfor’proposedandaninsightfromtheoreticalphysicsgiven. All thesesolutionsquestionthewayIhavedefinedmathematicsanditsrelationto thehumanmindandthenaturalworld. 8 Chapter 2 The rise of complex mathematics Itisnotimmediatelyapparentthatthereisaproblemwiththeeffectiveness ofmathematics. Manyscientistsneverconsideredtherelationbetweenmathe- maticsandscienceasproblematicandhavetakenitseffectivenessforgranted. Theonesthatdidpuzzleoverthecloseconnectionbetweenmathematicsand scienceallagreethatthereissomethingstrangeabouttherelationship-among themAlbertEinstein: Atthispointanenigmapresentsitselfwhichinallageshasagitated inquiringminds. Howcanitbethatmathematics,beingafteralla productofhumanthoughtwhichisindependentofexperience,is soadmirablyappropriatetotheobjectsofreality? Ishumanreason, then,withoutexperience,merelybytakingthought,abletofathom thepropertiesofrealthings. (Einstein,1922,p. 15) Indeedwhenwelookatthesuccessesofscience,italmostseemsmiraculous howwellthemathematicalpredictionsmatchtheoutcomeofexperiments. One explanationoftheapplicabilityofmathematicscouldbethatourmostprofound and complex mathematical concepts can be led back to simple abstractions fromNatureandthatthisisthereasonthatmathematicsasweknowitisso useful. MacLane(1990)defendsthisstance,claimingthatthisisthesolution toWigner’spuzzle. Idon’tagreewithhim,orwithanyotherdefenderofthis claimandIwillshowthatmathematicsismorethananabstractionfromNature bylookingatitshistoricalnarrative. Whatistheoriginofmathematicsandhow diditgetintertwinedwithscience? Thischapterhastheaimofshowingthat 9