Explaining Beauty in Mathematics: An Aesthetic Theory of Mathematics

224 Pages·2013·1.36 MB·English

by  Ulianov Montano

Synthese Library 370
Ulianov Montano
Explaining
Beauty in
Mathematics: An
Aesthetic Theory
of Mathematics
Explaining Beauty in Mathematics: An Aesthetic
Theory of Mathematics
SYNTHESE LIBRARY
STUDIESIN EPISTEMOLOGY,
LOGIC, METHODOLOGY,AND PHILOSOPHY OFSCIENCE
Editors-in-Chief:
VINCENTF.HENDRICKS,UniversityofCopenhagen,Denmark
JOHNSYMONS,UniversityofTexasatElPaso,U.S.A.
HonoraryEditor:
JAAKKOHINTIKKA,BostonUniversity,U.S.A.
Editors:
DIRKVANDALEN,UniversityofUtrecht,TheNetherlands
THEOA.F.KUIPERS,UniversityofGroningen,TheNetherlands
TEDDYSEIDENFELD,CarnegieMellonUniversity,U.S.A.
PATRICKSUPPES,StanfordUniversity,California,U.S.A.
JANWOLEN´SKI,JagiellonianUniversity,Kraków,Poland
VOLUME 370
Forfurthervolumes:
http://www.springer.com/series/6607
Ulianov Montano
Explaining Beauty in
Mathematics: An Aesthetic
Theory of Mathematics
123
UlianovMontano
MexicoCity,Mexico
ISBN978-3-319-03451-5 ISBN978-3-319-03452-2(eBook)
DOI10.1007/978-3-319-03452-2
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013956239
©SpringerInternationalPublishingSwitzerland2014
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Contents
PartI Antecedents
1 OnNon-literalApproaches ................................................ 3
1.1 TheTwoCultures,ShaftesburyandHutchenson................... 3
1.1.1 UnsoundDivide............................................. 4
1.1.2 Shaftesbury.................................................. 5
1.1.3 Hutchenson.................................................. 6
1.2 TheEpistemicCharacterofMathematics........................... 8
1.2.1 Rota’sInterpretationofMathematicalBeauty............ 9
1.2.2 Rota’sProblems............................................. 11
1.3 TheRationalCharacterofMathematics............................. 15
1.3.1 RationalReinterpretations.................................. 16
2 Beautiful,Literally.......................................................... 21
2.1 MetaphorandAestheticTerms ...................................... 21
2.2 McAllister’sApproach............................................... 23
2.2.1 McAllister’sAestheticTheory............................. 24
2.2.2 Aesthetic Properties, Aesthetic Criteria
andAestheticCanon........................................ 25
2.2.3 TheAestheticCanon........................................ 26
2.2.4 TheAestheticInduction.................................... 27
2.2.5 AestheticInductioninMathematics....................... 29
2.2.6 McAllisterinSummary..................................... 30
3 Ugly,Literally................................................................ 33
3.1 UglinessinMathematics............................................. 33
3.2 Computer-AssistedProofs ........................................... 36
3.3 PhenomenologicalFactors........................................... 38
3.3.1 IntentionalObjects.......................................... 39
3.3.2 IntentionalProofs........................................... 40
3.3.3 WhytheConflict?........................................... 42
v
vi Contents
4 ProblemsoftheAestheticInduction...................................... 45
4.1 TwoKindsofProblems .............................................. 45
4.1.1 Anomalies................................................... 45
4.2 TheoreticalTensions.................................................. 51
4.2.1 Induction..................................................... 52
4.2.2 BeautyandAesthetic ....................................... 54
4.2.3 Objectivism/ProjectivismInconsistency................... 55
4.2.4 TheoryandModelling...................................... 55
5 NaturalizingtheAestheticInduction..................................... 57
5.1 The Mere-Exposure Effect and the Nature
oftheAestheticInduction............................................ 57
5.1.1 NaturalizingPreferenceEvolution......................... 58
5.1.2 NaturalizingtheAestheticInduction...................... 60
5.1.3 Preference,AffectionandEmotion........................ 60
5.1.4 ConstraintsofAffectivePlasticity......................... 62
5.2 ConceptualizingPreferenceEvolution.............................. 63
5.2.1 TheAestheticCanonasaSystem.......................... 64
5.2.2 CriticalAdequacy........................................... 65
5.2.3 RobustnessofCriticalAdequacy.......................... 65
5.2.4 DynamicsoftheNewModel............................... 66
5.3 TheNewModel:ConstrainedAestheticInduction................. 68
5.3.1 ConstrainedAestheticInduction(CAI) ................... 68
5.3.2 ProblemsAddressed........................................ 68
PartII AnAestheticsofMathematics
6 IntroductiontoaNaturalisticAestheticTheory ........................ 73
6.1 TheMarkoftheAesthetic ........................................... 76
6.1.1 PhasesofanAesthetic-Process ............................ 79
7 AestheticExperience........................................................ 85
7.1 CharacterizingAestheticExperience................................ 85
7.2 AestheticIntentionalObjects ........................................ 87
7.3 MathematicalIntentionalObjects ................................... 88
7.4 ANotionofAestheticMathematicalIntentionalObject........... 89
7.4.1 DimensionsandProperties................................. 89
7.4.2 ActivitiesandRelations .................................... 92
7.4.3 A Model of Aesthetic Mathematical
IntentionalObjects.......................................... 96
7.4.4 AestheticForm.............................................. 97
7.5 TypesofExperience.................................................. 98
7.5.1 BasicAppreciationResponse.............................. 98
7.5.2 Non“Inductive”Response ................................. 100
Contents vii
7.5.3 PerformativeAppreciationResponses..................... 101
7.5.4 AdaptiveAppreciationResponse.......................... 107
7.6 ThePleasure-Relation................................................ 111
7.6.1 Formalization................................................ 112
8 AestheticValue .............................................................. 117
8.1 ExtensionofAestheticValue ........................................ 118
8.2 TypicalPositiveAestheticCriterion................................. 121
8.3 Dynamics.............................................................. 122
8.3.1 ValueandAestheticCanon................................. 122
8.4 AestheticCanonEvolution........................................... 124
8.4.1 NaturalisticEvolutionRuleII(NERII).................... 124
8.4.2 PositiveValue............................................... 126
8.4.3 NegativeValue.............................................. 128
8.4.4 AestheticExperienceandtheApplicationof
ValueRepositories.......................................... 129
9 AestheticJudgementI:Concept .......................................... 131
9.1 OnAestheticTerms................................................... 131
9.1.1 Sibley........................................................ 131
9.1.2 Hungerland.................................................. 133
9.1.3 Kivy.......................................................... 134
9.1.4 DeClercq.................................................... 135
9.1.5 Zangwill..................................................... 135
9.2 ANewModelofAestheticTerms................................... 136
9.2.1 Metaphor .................................................... 137
9.2.2 MetaphoricalTermsandAestheticTerms................. 138
9.2.3 ExperienceandResponseSpaces.......................... 139
9.2.4 ANewModelofAestheticTerms......................... 140
9.2.5 GenuineAestheticTerms................................... 141
9.2.6 DefinitionsofAestheticTerm,Description
andJudgement .............................................. 145
9.2.7 Always andOnly Aesthetic Terms;The
SpectrumofAestheticTerms .............................. 146
10 AestheticJudgementII:Functions ....................................... 149
10.1 ConceptandFunction ................................................ 149
10.1.1 EmphasesandPerspectives ................................ 150
10.2 Articulation............................................................ 150
10.2.1 DefinitionofArticulation................................... 152
10.3 BroadcastingFunction................................................ 154
10.3.1 LocallyTerminalAestheticTerms......................... 156
10.4 UnifyingApproaches................................................. 157
11 MathematicalAestheticJudgements ..................................... 161
viii Contents
PartIII Applications
12 CaseAnalysisI:Beauty .................................................... 167
12.1 Knowledge,Beautyand“IsolatedPearls” .......................... 168
12.2 Case1,Beauty:y Dex .............................................. 169
12.2.1 Experience................................................... 169
12.2.2 Value......................................................... 172
12.3 Dynamics.............................................................. 173
12.4 Judgement............................................................. 174
12.5 TheProcessinSummary............................................. 177
13 CaseAnalysisII:Elegance................................................. 179
13.1 Experience ............................................................ 179
13.1.1 Step-SeriesInterpretation .................................. 182
13.1.2 Single-ObjectInterpretation................................ 183
13.1.3 AffectiveResponseinStep-SeriesInterpretation......... 186
13.1.4 AffectiveResponseintheSingle-ObjectInterpretation.. 188
13.2 Value................................................................... 189
13.3 Judgement............................................................. 192
13.4 TheProcessinSummary............................................. 195
14 CaseAnalysisIII:Ugliness,Revisited.................................... 197
14.1 TheProof.............................................................. 197
14.2 Experience ............................................................ 199
14.3 Value................................................................... 201
14.3.1 Scenario1:ChangebyAestheticInduction............... 201
14.3.2 Scenario2:ChangeintheNatureofOurExperience .... 201
PartIV ClosingRemarks
15 IssuesofMathematicalBeauty,Revisited................................ 207
Conclusion......................................................................... 211
Bibliography....................................................................... 217
Introduction
Mathematicians often and naturally evaluate certain pieces of mathematics using
words like beautiful, elegant, or even ugly.1 Such evaluations are prevalent; they
appearindiscussions,inteachingcontextsandintheliterature.However,rigorous
investigation of them, of mathematical beauty in general, is much less common.
Among the few authors that address this issue is James McAllister, who has
extensivelydocumented[62,64]thatwhenmathematicianstalkaboutmathematical
beauty they are being “serious” mathematicians: McAllister shows that beauty in
sciencedoesnotconfineitselftoanecdotesorpersonalidiosyncrasies,butratherit
hadplayeda rolein shapingthe developmentandprogressofscience [62,63]. To
interpretaestheticjudgementsinmathematicswhiletakingtheirseriouslyaesthetic
character is no simple matter, however. Addressing aesthetic phenomena is no
simple endeavour in itself, and if we take into account that aesthetic phenomena
are traditionally associated with things such as art and expression and considered
alientoscience,thedifficultyofthetaskisonlyenhanced.Thebeliefthataesthetic
matters and mathematics are alien to each other seems to be confirmed to the
mathematicallylaypersoneverytimemathematiciansqualifyanobscureformalism
as “beautiful”. That formalism is to the lay person as impenetrable as any other
“non-beautiful”pieceofmathematics.Laypeoplecansurelyfindsuchevaluations
strange.Theaimofthisbookistodeveloparigorousandcomprehensivetheoryof
beautyin mathematicscapableofexplainingthisandotherproblemswhiletaking
intoaccountthe role ofbeautyin the developmentof mathematicsandexplaining
howaestheticjudgementsinmathematicsaregenuinelyaesthetic.Perhapsacaveat
1Many other words can be found in aesthetic evaluations in mathematics. For example: neat,
handsome, harmonious, charming, pristine, tidy, simple, clumsy, horrible, nasty, crude, rough,
dirty, etc. However, some of those terms, clumsy, for instance, can be argued to be more
metaphoricalthanaesthetic.Others,simple,forinstance,canbearguedtobeliterallydescriptive
butnotnecessarilyaesthetic.Termssuchasbeautiful,elegantoruglyaremuchmoreobviously
aesthetic.Theissueofmetaphoricalandcontroversiallyaesthetictermsshallbeaddressedatlength
lateron,but,forthesakeofclarity,inmostofthisbookIshallrefertothemoreobvious,andin
factmostfrequentlyused,aestheticterms.
ix