Master Thesis

61 Pages·2015·0.32 MB·English

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
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6.4 Themultiple-worldshypothesis. . . . . . . . . . . . . . . . . . . 52
7 Conclusionandoutlook 55
References 58
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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
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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
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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
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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.
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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
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