1Phenomenology of Incompleteness: from Formal Deductions to Mathematics and Physics

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1Phenomenology of Incompleteness: from Formal Deductions to Mathematics and Physics Francis Bailly & Giuseppe Longo Physique, CNRS, Meudon & LIENS, CNRS–ENS et CREA, Paris Summary. This paper is divided into two parts. The first proposes a philosophical frame and it “uses” for this a recent book on a phenomenological approach to the foundations of mathematics. Godel's 1931 theorem and his subsequent philosophical reflections have a major role in discussing this perspective and we will develop our views along the lines of the book (and further on). The first part will also hint to the connections with some results in Mathematical physics, in particular with Poincare's unpredictability (three-body) theorem, as an opening towards the rest of the paper. As a matter of fact, the second part deals with the “incompleteness” phenomenon in Quantum physics, a wording due to Einstein in a famous joint paper of 1935, still now an issue under discussion for many. Similarities and differences w.r. to the logical notion of incompleteness will be highlighted. A constructivist approach to knowledge, both in mathematics and in physics, underlies our attempted “unified” understanding of these apparently unrelated theoretical issues1. Part I. Revisiting “Phenomenology, Logic, and the Philosophy of Mathematics”2 Constructivism is the most common philosophical attitude in the mathematics (and practice) of Computing and this in contrast with the prevailing debate in mathematical circles still ranging from Platonism to Formalism.

  • his subsequent philosophical

  • active measure

  • fulfilled mathematical

  • husserl's phenomenology

  • who wants

  • built up

  • godel's philosophy

  • up godel's realism

  • distinction between

  • constructive principles


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1PhenomenologyofIncompleteness:fromFormalDeductionstoMathematicsandPhysicsFrancisBailly&GiuseppeLongoPhysique,CNRS,Meudon&LIENS,CNRS–ENSetCREA,Parishttp://www.di.ens.fr/users/longoSummary.Thispaperisdividedintotwoparts.Thefirstproposesaphilosophicalframeandit“uses”forthisarecentbookonaphenomenologicalapproachtothefoundationsofmathematics.Go¨del’s1931theoremandhissubsequentphilosophicalreflectionshaveamajorroleindiscussingthisperspectiveandwewilldevelopourviewsalongthelinesofthebook(andfurtheron).ThefirstpartwillalsohinttotheconnectionswithsomeresultsinMathematicalphysics,inparticularwithPoincare´’sunpredictability(three-body)theorem,asanopeningtowardstherestofthepaper.Asamatteroffact,thesecondpartdealswiththe“incompleteness”phenomenoninQuantumphysics,awordingduetoEinsteininafamousjointpaperof1935,stillnowanissueunderdiscussionformany.Similaritiesanddifferencesw.r.tothelogicalnotionofincompletenesswillbehighlighted.Aconstructivistapproachtoknowledge,bothinmathematicsandinphysics,underliesourattempted“unified”understandingoftheseapparentlyunrelatedtheoreticalissues1.PartI.Revisiting“Phenomenology,Logic,andthePhilosophyofMathematics2Constructivismisthemostcommonphilosophicalattitudeinthemathematics(andpractice)ofComputingandthisincontrastwiththeprevailingdebateinmathematicalcirclesstillrangingfromPlatonismtoFormalism.But,whatdowemean,today,by“conceptualconstruction”,inthebroadestsense?Phe-nomenologymayprovideonepossibleanswertothis,byadeeplyrenewedun-derstandingofWeyl’s(andBrower’s)ideas,inaperspectiveclosetoHusserl’sphilosophy.Tieszen’sbookproposesacriticalaccountofmodernviewsinthefoundationsofmathematics,whichisofdirectconcernforthelogicianand1ToappearinDeduction,Computation,Experiment(Lupacchinied.),Springer,.80022PartIisalargelyexpandedversionofareviewof[Tieszen05],whichappearedinMetascience,areviewjournalinPhilosophyofScience,15.3,615-619,Springer,.6002
2FrancisBailly&GiuseppeLongothetheoreticianinnaturalscienceswhowantstoreflectontheconstructiveprinciplesofthemathematicalintelligibilityoftheworld.Wewillrefertothisbooktogofurtherandmotivateabroadeningofthenotionof“construc-tion”asgiveninformaldeductionsandarithmeticalcomputations,ineitherclassicalorintuitionisticframes.Bythisbroadening,wewillunderstandtheincompletenessphenomenonasa“gap”betweenmathematicalconstructionprinciplesandformalproofprinciples,followingandfurtherdevelloppingsomeideashintedin[BaillyLongo06].Tieszen’sperspectiveisoriginal,asthePhilosophyofmathematicshasbeenlargelydominatedbyacontrapositionbetweenOntologismandNomi-nalism,asrecalledabove.Thisseparatedthefoundationalanalysisofmath-ematicsbothfromourlifeworldandfromotherscientificdomains,includingphysicswheremathematicshasaconstitutiverole.Bycorrelatingfounda-tionalissuesinmathematicsandphysics,alongthelinesof[BaillyLongo06],wewilltrytorecomposethefoundationalbreak,atleastasfortheissueofincompleteness.PartI.IThefirstpartofTieszen’bookisdedicatedtoanintroductiontoaHusser-lianperspectiveinthefoundationsofmathematics.Itisinterestingperse,asabroadsurveyofHusserl’sphenomenology.ThisismadepossiblebytherelevancethatHusserlhimselfgavetoLogicandmathematicsinhisphiloso-phyofknowledge:writingsonLogicandArithmeticareamongtheearliestofHusserl’sandtherelatedissuesaccompanyhislifelongwork.Theconstitutionofidealobjects,inHusserlianterms,isbasedonacleardistinctionbetweenthetranscendentalperspectiveandpsychologism.Itisthehumansubjectwhomakessciencepossible,yetthecommonendeavourofthehistoricalcommunityshouldnotbeconfusedwiththeindividualanalysis:epistemologyisageneticanalysis,providedthathistoryisnotunderstoodintheusuallimitedsense,explainsHusserlinthe“OriginofGeometry”(1933).Therearedifferenttypesandlevelsofconsciousness,whichallowthehistoricaldynamicsofknowledge:scienceisbuiltupfromthelifeworldexperienceofhumansubjectsonthebasisofactiveabstraction,idealization,reflection,formalization.Theobjectivityofknowledgeisaconstructedone,aresultoftheinteractionbyanactivesubject,beginningwith“kinaesthesia”,inalivingbody,ineverydayworldoflife.Meaningisnotthepassiveinterpretationofindependentsigns,butitisconstructedinthisinteraction,itistheresultofa“friction”andofstructuringofthisveryworldbyourattemptstogivesensetoit;meaningistheresultofanaction.Ofcourse,wedaretoadd,thismustbeunderstoodinabroadsense:QuantumMechanicsforexampleseemstoowelittletokinaesthesia.Yet,itisaparadigmaticcasewheremeaningistheresultofactiveconsciousness,beginningwiththepreparationoftheexperimentorofthetechnicalcontextforinsight:weareconsciousofaquantumobjectasaconstitutedphenomenon.Ourlivedbodyisjustexpandedbyinstruments
TitleSuppressedDuetoExcessiveLength3which,inturn,resultfromatheoreticalcommitment:thisistherichestformofinteractioninthesensementionedabove,withnomeaningfulobjectwithoutactiveknowingsubject.A(conscious)intentionalprocessisattheoriginofthisformofknowledge.AsTieszenexplains,consciousness,forHusserl,isconsciousnessofsome-thing.Itcouldbeanidealobjectofmathematics.Thelaterbeingtheresultofaformationofsensefoundedonunderlyingactsandcontents,whichmakepossibletheidealconstruction.WewouldliketoexemplifybyconsideringEu-clid’sactiononspacebyrulerandcompass.Thisactionorganisesfiguresinspacebyrotations,translationsandreflections,toputitinmodernterms.AdialoguewiththeGodsforsure,butalsoactivemeasureofground.But,howtodefineandmeasuresurfaces,atechniquethat,initsmathematicalgeneral-ity,isthekeyGreekinvention?Inordertoconceiveexactmetricsurfacesonehastoconceivelineswithoutthickness;thereisnowaytogiveamathematicalsoundnotionofsurface,withoutfirstproposing,withEuclid’sclarity,lineswith0thickenss.Then,asaconsequenceofintersectinglines,oneobtainsdi-mensionlesspoints,asEucliddefinesthem(aremarkbyWittgenstein).Theseextraordinarilyabstractconcepts,point,lineetc.aretheidealizedresultofapraxisofmeasureofsurfacesandaccesstotheworldbytranslatingandrotatingrulerandcompass,faraway,butgroundedonsensoryexperience.Rotations,translationsandsymmetriesare“principlesof(geometric)con-structions”,anotiontobeoftenusedinthesequel.InEuclid’sgeometrytheyareusedinproofsandtheydefinethegeometricobjectsasgivenbyinvariantpropertiesw.r.tothesetransformations.TieszenstressesseveraltimestherelevanceofinvariantsinHusserl’sfoun-dationalapproach.“Mathematicalobjectsareinvariantsthatpersistacrossacts”carriedoutindifferentcontexts.Thepracticalconstructionsofmathe-matics,inhumanspaceandtime,arealsostabilizedbylanguageand,then,bywriting,saysHusserl:theirconstitutedidealnatureisprimarilythere-sultoftheirinvariance,asconceptualconstructions,withrespecttosuitabletransformationsofcontext.Andthisisextremelymodern:invariantstructuresandtransformationswerefirstthefoundationalcoreofRiemann’sgeometry,inKlein’sapproach,thenofCategoryTheory.HusserlseemstoprecedetheunderlyingphilosophyofCategoryTheorybyhisanalysisofmathematicalknowledge.Theseinvariantsarethentheessencesand,thus,providetheonlypossibleontologyformathematics;theyaretheresultofdifferent“fulfilledmathematicalintentions”,asconstructions.Andtheseconstructionshaveahorizon,thespaceofthehistoricalpraxiswhichleavesastracethemoststableinvariantsofallourmentalpractices,thestructuresandobjectsofmathemat-ics.Then,underlinesTieszen,“truthiswithinthishorizon”asthereisno,forHusserl,absolutemathematicaltruthnorevidence.Yet,mathematicalthe-oriesarenotarbitrarycreations(considertheexampleofGreekgeometryabove),theyarenoconventionalgamesofsigns:wedonotsolveopenprob-lemsbyconvention,astheyaretheresultofameaningfulandmotivated