(02)NEWCOMBGREENLEAF-STANANDERRETTBISHOP
FORWARD:ErrettBishop,whosingle-handedly(following
Brouwer)developedtherevolutionarylogicalalternative,
'Constructive'mathematics,analgorithmicbasisforthe
wholeofmathematicsdenyingthelawoftheexcluded
middle,wasoneofStan'sgreatfriends,ahighlycherished
intimatesincetheirstudentdaysattheUofChicago.Ever
(intellectually)rebuffedbythemathematicalcommunitywho
littleunderstoodhim,Bishop'sworkwasdeterminedlykept
alive,indeedthrived,owinginwholetoStan'seffortsand
dedicationduringthe70'sattheNewMexicoState
University.NewcombGreenleaf,nowacommitted
'Constructivist',givesusanabsorbingaccountofthisperiod
togetherwithbackgroundastohiscloserelationshipwith
Stanandhispathtothe'Constructivist'light.
Completinghis1961PhDatPrincetonunderSergeLang,the
topic'LocalZerosofGlobalForms',Newcmovedontobea
PeirceInstructoratHarvard,arrivingin1964attheUof
RochesteraspartofLeonardGillman'sprogrammetobuild
themathdepartmentintoaworldclassoutfit.MeetingStan
in1965,theydevelopedacloserelationship,Stan's
influencescontinuingtothepresent.Threeyearsin,Newc
departedfortheUofTexas,wherehegraduallycameunder
thespellofbishop'swork,albeittheonlymathematicianina
departmentof100todoso.Sevenyearsinthewilderness
NewcleftforBouldertoworkwiththeTibetanteacher
ChogyamTrungpa,subsequenttowhich,afterashortstayat
acomputergraphicsfirm,hejoinedthecomputerscience
departmentatColumbia.Todayheenjoysacongenial
teachingpositionatGoddardCollege,finallyfinding'my
dreamteachingjob'.Hemightalsobeworkingonhisbook,
workingtitle'BibleorCookbook?AnAlgorithmicPrimerto
theBookofMath'.
In1992Newcpublishedin'ConstructivityinComputer
Science'averyinteresting'BringingMathematicsEducation
IntotheAlgorithmicAge'.SignalyourinterestandIwillsend
ittoyou.Furtherhecogentlyputsforwardtheconstructivist
viewpointinasegmentonYOUTUBE:'NondualMathematics:
ATragedyinThreeActs'.
StanTennenbaumandErrettBishop
NewcombGreenleaf
1.StanatRochester
AfterthreeintenseyearsasaPeirceInstructoratHarvard,whereIfelt
overmyheadandoutofmyleague,Ijoinedthemathematics
departmentoftheUniversityofRochesterin1964,ayearbeforeStan
arrived.Ithrived:confidencereturnedasresearchpickedupand
teachingripened.Butmymarriagedidnotthriveandin1965I
returnedtoRochesterforthefallsemesterasasinglefather,justasStan
andCarolmovedintotheneighborhood.Iwasinitiallydrawnintothe
TennenbaumorbitmorethroughCarol,wholeaptinandbecamea
sourceofmaternalwarmthformytwosons,9and10,makingsurethat
wedinedwiththematleastonceaweek.Thethreeofusstillremember
theremarkabledinner-tableconversationswhichStanfacilitatedina
waythatbroughteveryonein,fromsecondgraderstovisiting
luminaries.
Stanhadanimmediateeffectonthewholemathdepartment,enlivening
itinmanydimensions.Hecarriedwithhimtheintellectualspiritofthe
UniversityofChicagointheheydayofRobertHutchins,alongwitha
beguilingmixofmelancholyandjoy.Hewaspassionatelyinterestedin
ideas,andpassionatelyinterestedinpeople.Hedisplayedagenerosity
thatmadehimmanyfriends,andanuncompromisingidealismthat
couldgethimindeeptrouble.Forasizablegroupofstudentshewasa
majorlifelonginfluence,vividlyrecalled50yearslater.
ForthenextthreeyearsStanwasthecolleaguefromwhomIlearned
themost,aboutmathematics,aboutlife,andparticularlyaboutteaching,
whichhetookbothveryseriouslyandveryplayfully.FromStanI
learnedtocomeoutfrombehindtheauthorityofTheProfessorandbe
real.Now,teachingatGoddard,IwouldsaythatStantaughtmeto
modelimperfection,whichempowersstudentstomakemistakesand
admitignorancewithoutfear,andthustoexplorecourageouslyand
joyfully.Stancouldembodyimperfectionperfectly.AsIwritethisI
realizehowmuchhepreparedmetoteachatGoddard.
Stanhadanotherimportantinfluenceonme:heintroducedmetothe
“constructivemathematics”ofErrettBishop,whichultimatelychanged
mylifeprofoundly.StanandErretthadbeencloseasgraduatestudents
atChicago,soStanpaidattentionin1966whenErrettbecamea
polarizingfigureintheworldofmathematics.Stanwasfascinatedby
therevolutioninmathematicalthoughtthatBishopproposed,andby
theuncomprehendingreactionofthemathematicscommunity,andwe
oftenspokeofit.ButIremainedwiththeuncomprehendingmajority,
andmyconversiontoBishop’sprogramcameseveralyearslaterafter
bothStanandIweregonefromRochester.BelowI’lldescribethe
significantrolethatStanlaterplayedinBishop’srevolution.Butfirst,
sinceBishopissolittleknowntoday,I’llintroducehimandhisfailed
revolution,inwhichStanplayedasignificantpart.
2.Errett
In1966ErrettBishop,atage38,wasatthepeakofhismathematical
powers,abrilliantstarinthemathematicalfirmament.Buthehadlong
beenawarethathenaturallythoughtdifferentlyaboutmathematics,
basingitmoresecurelyoncomputation.Hehadgonealongwiththe
established“classical”mathematicsbecausemathematicsisasocial
activity,anditwastheonlygameintown.Heknewthatapredecessor,
theDutchmathematicianL.E.J.Brouwer(1881-1966)hadsimilar
understandingsandhadfailedtocreateaviablemathematicalworld
withthem.SeveralyearsearlierBishophadavisionofhowtofix
Brouwer,andoveraprodigioustwo-yearperiodhesucceededin
bringingafullconstructivemathematicsintobeing,succeedingwhere
Brouwerhadfailed.
Brouwerhad,howeversucceededinreforminglogicsothat
mathematicscouldbebasedoncomputation,andBishopusedthe
“Intuitionist”logicBrouwercreated,BelowI’llshowindetailwhy
numericalmeaningrequiresamodificationofclassicallogic.
Erretthadbeeninvitedtogiveamajoraddressatthe1966meetingof
theInternationalCongressofMathematicians.Itwasexpectedthathe
wouldspeakofhisdeeptechnicalresearchin“functionalgebras”or
“severalcomplexvariables”.Instead,hegavearatherelementarytalk
entitledAConstructivizationofAbstractMathematicalAnalysis,a
passionatebutproperlymutedcallforarevolutioninthewaythatwe
normallydoandthinkaboutmathematics,“classical”mathematics.
Afriendwhoattendedthetalkdescribedhowitwasreceived.Errett
wasaclearexpositorwithawarmpersonality,andduringthetalkthere
wasmuchsmilingandnodding.Butafterthetalkthesmilesgradually
changedtofrownsandnoddingtoheadscratching,andthenextdayit
justdidn’tmakemuchsense,exceptforafew.Itwasapatternthatwas
tohauntBishopthereafter,atoxicmixofrecognitionand
incomprehension.
Bishopstartedwithveryhighexpectations.In1967Foundationsof
ConstructiveAnalysis(FCA)appearedwithChapter1calledA
ConstructivistManifesto.Inhis1970reviewintheAMSBulletin,Gabriel
Stolzenbergasserted:
He[Bishop]isnotjokingwhenhesuggeststhatclassical
mathematics,aspresentlypracticed,willprobablyceasetoexist
asanindependentdisciplineoncetheimplicationsand
advantagesoftheconstructivistprogramarerealized.Aftermore
thantwoyearsofgrapplingwiththismathematics,comparingit
withtheclassicalsystem,andlookingbackintothehistorical
originsofeach,Ifullyagreewiththisprediction.
Bishop’sprediction,secondedbyStolzenberg,couldhardlyhavebeen
fartheroffthemarkWhilehewasinvitedtospeakatallthemajor
universitiesandmanyconferences,herarelyfeltthathewas
understood,andveryfewjoinedhiscause.His Ph. D. students couldn’t
get good academic jobs and his few disciples had difficulties getting their
papers published. Bishop’scampaignwonone—andonlyone—
significantvictory,whichtookplaceinthemathdepartmentofNew
MexicoStateUniversity,andwasentirelytheworkofStan
Tennenbaum.
TherewasonegroupthatfoundBishop’sapproachtomathnatural:
computerscientists.IexperiencedthiswhenIspent8yearsteachingCS
atColumbia(backinthedayswhentherewasashortageofPh.Dsin
CS).ButErrettwasfixatedonmathematiciansandwroteonlyforthem,
sothatevenincomputerscienceheisnotwellknown.Bishophidthe
algorithmicinspirationofhisvisiontomakehismathematicslook
ordinary.ThecomputersciencepioneerD.E.Knuthsaidof,FCA:
Theinterestingthingaboutthisbookisthatitreadsessentially
likeordinarymathematics,yetitisentirelyalgorithmicinnature
ifyoulookbetweenthelines.
Bymakinghisapproachlooknormaltomathematicianswhodidn’tcare,
Bishoptendedtohideitsalgorithmicnaturefromfromthecomputer
scientistswhoappreciatedit.
3.StanandErrett
Here’saquotefromFredRichman’swonderfulessayConfessionsofa
formalistPlatonistintuitionist.
WhenIreturnedtoNewMexicoStatefromasabbaticalleaveat
FloridaAtlanticUniversity,peoplethereweretalkingaboutErrett
Bishop'sbookFoundationsofconstructiveanalysis.Stanley
Tennenbaumhadvisitedtheprevioussemesterandconducteda
verypopularseminaronthesubject.
Thiswasremarkableatmorethanonelevel.Tostartwith,constructive
mathematicsmakesmostmathematiciansuneasy,foritquestions
Aristotle’sLawofExcludedMiddle(LEM),whichassertsthatevery
meaningfulpropositioniseithertrueorfalse(althoughwemaynot
knowwhich).Thisisasacredcowthatgenerallyoperatesata
subconsciouslevel.Ifyouchallengeit,youmayencounterintense
opposition,asinthisfamousquotefromDavidHilbert:
DeprivingamathematicianoftheuseofTertiumnondaturis
tantamounttodenyingaboxertheuseofhisfistsoran
astronomerhistelescope.
“Tertiumnondatur”referstoAristotle’sLawofExcludedMiddle.
Hilbert,theleadingmathematicianofthetime,wasattackingBrouwer.
HehadtrieddoingmathwithoutLEM,butfounditdifficult.Aswe’ll
discussbelow,itisdifficulttoswitchfromclassicaltoconstructivelogic
thougheasytogotheotherway.Thesymbolismoftheboxer’s
manhoodrepresentedbyhisfistsandtheastronomererectinghis
telescopeisalmostembarrassinglyFreudian.
Stannotonlyattractedalivelyquorumtohisseminar,hetookthecore
ofthatgroupsodeeplyintoBishop’sthinkingthattheyturnedintoa
researchgroupinconstructivemathematicsthatremainedprolificfor
overadecade.AfterStanleft,FredRichmanreturnedandjoinedthe
group,soonbecomingitsintellectualsparkplug.FredandDouglas
Bridges,aBritwhomovedtoNewZealand,stoodoutinthesmallband
ofmathematiciansaroundtheglobewhotriedtorealizeBishop’svision.
AndwithoutthesupportoftheNMSUgroup,evenBridgesmighthave
chosenadifferentdirection.Thiswasatotallyuniqueevent.Exceptfor
NMSU,therewasnoplacewhereBishop’svisionlived,onlylonely
constructivistsinclassicaldepartments.
3.LogicversusArithmetic
AtthispointI’mgoingtoleavethestoriesofStanandErretttointerject
asimpleexplanationofthecentralissue,theincompatibilityoftwocore
mathematicalstructures:classicallogicwithLEM,andnumberswith
theirarithmetic.Allthenumbersweconsiderwillbepositiveintegers.
Bishop,andBrouwerbeforehim,sawnumbersasobjectsthatcanbe
added,multiplied,divided,subtractedbyexpressingtheminstandard
decimalnotation.Inhisfarewelladdressof1973,Bishopputitlikethis:
TheConstructivistThesis.Everyintegercanbeconvertedin
principletodecimalformbyafinite,purelyroutine,process.
Bishopdidn’twanttosoundlikeacomputerscientist,sohealwayssaid
“finite,purelyroutine,process”insteadof“algorithm”,
I’dprefertoputtheThesisinastylethatBishopoftenused,whereone
asks,Whatmustbedonetoconstructanumber?andanswerswith:
TheConstructivistThesis-AlgorithmicStyle.Toconstructa
numberyoumustprovidedataalongwithanalgorithmwhichwill
convert—atleastinprinciple—thedataintoadecimal
representation,afinitesequenceofthedigits0through9.The
simplestcaseiswhenthedataitselfisadecimalrepresentation,
whichthealgorithmjustpassesalong.
“Inprinciple”meansthatwhileyoucandescribethealgorithm,itmight
notberealistictostartitrunningandwaitfortheresult.Inpracticethe
algorithmisoftenobviousandnotmentioned,asin:Letnbethe
numberofprimenumberslessthan1010.
Ineitherformulationthepointisthatnumberscanbeputindecimal
form,whichallowsustodoarithmeticwiththem(providedwe
rememberourtablesandalgorithms).We’renowgoingtoverifywhat
Brouwerfirstshowed:theconflictbetweenclassicallogicand
arithmetic.LEMintroducesnumbersthatyoucan’tdoarithmeticwith.
Theorem.LEMandtheConstructivistThesisareInconsistent.
Brouwer’sProof:
BrouwerusesLEMtoconstructanumberqthatcannotbeconverted
todecimalnotationbecauseitencodesignorance:
• Takeyourfavoriteunsolvedmathematicalproblem.Thereare
zillionsofthem.I’llpicktheRiemannHypothesisaboutthezeros
ofthezetafunction.
• IftheRiemannHypothesisistrue,thenletq=1.
• IftheRiemannHypothesisisfalse,thenletq=0.
• LEMsaysthattheRiemannHypothesisiseithertrueorfalse.
• Soeitherq=0orq=1.
• Since0and1arenumbers,qisanumberineithercase.
• Proofbycases:qisanumber,period.
• Butwehavenoalgorithmtocomputethedecimalrepresentation
ofq.NoalgorithmtodetermineiftheRiemannHypothesisis
true.
• Contradiction,comingfromtheThesisandLEM.
Itmaybeworthnotingthatourconstructionofqdoesconstruct
something,justnotanumber.Itcanbedescribedasanon-empty
subsetof{0,1}withatmostoneelement.Knowingthatthestatement
“qisempty”isfalseisofnohelpinconstructinganelementofq.Close,
butnocigar.
ThiscompletestheproofthattheConstructivistThesisandLEMare
contradictory:youcan’thavebothofthem.IfyouacceptLEM,youmust
admitnumberswhichdonotappearinthetables.Thusthesplit.
Brouwer’sintuitionistsandBishop’sconstructivistschosetheThesis,
whichgivesaclearunderstandingofnumbers.Anewlogicfordoing
constructivemathematicswasneeded.Brouwerdefinedtheproper
logic,calledintuitionist,forBrouwer’sphilosophy.Butthedefinition
lackedacompellingstructurecomparabletothetruthtablesthatgive
shapetoclassicallogic.Inthe1930sabeautifulfoundationwasfound
forintuitionistlogic,calledNaturalDeduction,whichdescribesthe
logicalconnectiveintermsofrulesforintroductionandelimination.
MostmathematicianshavechosentoremainwithLEM,andacceptthat
itadmitsnumberswithwhichwecannotcompute,andtotrytoavoid
suchnumbersonanadhocbasis.
AthispointI’mgoingtoreturntoStaninLasCrucesandconsidersome
qualitiesofStanthatenabledhimtosucceed,sometimeswith
correspondingqualitiesofErrettthatledtohisfailure.
4.Pluralism
ItwasonesecrettoStan’ssuccessinLasCrucesthat,unlikeclassical
andconstructivemathematicians,hedidnotchoosebetweenarithmetic
andlogic.HewasequallyathometalkingmathwithErrettBishopor
withKurtGödel.Stanwasatruepluralist,whereasmost
mathematiciansaremonistswhobelievethatthereisonlyonetrue
mathematics.
Bishopcouldsometimessoundpluralistbutathearthewasamonist
whotriedtowinconvertsbyshowingwhatwaswrongwithclassical
mathematics.Butwithinitsowncontextthereisnothingwrongwith
classicalmath,itisvalid.InADefenceofMathematicalPluralism(2005)
theBritishmathematicianE.B.Davieswrote:
Weapproachthephilosophyofmathematicsviaadiscussionof
thedifferencesbetweenclassicalmathematicsandconstructive
mathematics,arguingthateachisavalidactivitywithinitsown
context.
Stan’spluralismmeantthathedidnotapproachtheseminarwithany
antagonismortension.Whatadifferencethatmade.
AndwhatacontrasttoBishop,whowasinvitedtogivetheColloquium
LecturesatthesummermeetingsoftheAMSin1973.Amemorial
volume,ErrettBishop:ReflectionsonHimandHisResearch,
publishedbytheAMSin1984,containedthetextofhisColloquium
Lectures.Hereishowitopens:
DuringthepasttenyearsIhavegivenanumberoflecturesonthe
subjectofconstructivemathematics.Mygeneralimpressionis
thatIhavefailedtocommunicatearealfeelingforthe
philosophicalissuesinvolved.SinceIamheretoday,Istillhave
hopesofbeingabletodoso.Partofthedifficultyisthefearof
seemingtobetoonegativisticandgeneratingtoomuchhostility.
Constructivismisareactiontocertainallegedabusesofclassical
mathematics.Unpalatableasitmaybetohavethoseabuses
examined,thereisnootherwaytounderstandthemotivationsof
theconstructivists.
ThevolumealsocontainsRemembrancesbyNerode,Metakides,and
ConstablewhichdocumentthedifficultiesthatErretthadwiththe
mathematicalcommunity,andthecontrastingreceptivityofcomputer
scientists.
5.Logic
Mathematiciansarealwaystrainedtouseclassicallogicwith
unrestricteduseoftheLEM(andequivalentslikedoublenegation
elimination).Onesecrettobeingagoodmathematician(andaquick
one)istoputtheuseoflogiconasubconsciouslevel.Itcanbevery
hardindeedtochangesubconsciouspatterns,asanymeditatorcan
affirm.AsFredRichmanputitinInterviewwithaConstructive
Mathematician:
Catchingwhenthelawofexcludedmiddleisusedismuchmore
difficult.It'sbeenmyexperiencethatmostmathematicians
cannotdoit.That'sbecausethelawofexcludedmiddleisan
ingrainedhabitataverylowlevel.
Ittookmealongtimetoreformmylogicalthinkingsothatitwas
naturallyconstructiveandthentomakeitagainquickand
subconscious.
ForBishopintuitionistlogicseemedtocomenaturally.Perhapshe
nevermadeexcludedmiddlesuchaningrainedhabit.Inanycasehe
wasnotstrongongivingthosetryingtolearntothinkconstructivelya
placetolandwhentheyletgoofclassicallogic.Hereallydidn’twantto
talkaboutlogic,perhapsbecauseconstructivemathematicianswere
oftentoldthattheywerenolongerdoingmath,butweredoinglogic!
Whilethetruthisthatclassicalandconstructiveuselogicexactlythe
sameway,toprovetheorems.Theyjustusedifferentlogics.
Classicallogicismadelegitimatebytruthtables.Intuitionistlogicis
legitimizedbyNaturalDeductionwithitsintroductionandelimination
rules.ButBishopneverpresentedthemasacompletesystem.He
seeminglydidnotwanttogiveanyoneachancetosaythathewasdoing
logic.Heprovidednosafelandingforthosewholetgoofclassicallogic.
WithhisrichbackgroundasalogicianStanmanagedtoprovideitinhis
NMSUseminar,butIdon’tknowifheusedNaturalDeduction.
Moreimportantly,Stanhadanuncannysenseforthetroublespotsof
hisstudents.Hewouldhavebeenveryawarethatthetransitionfrom
classicallogictointuitionistlogicisadifficultone,akindof
psychoanalysisinwhichsubconsciouspatternsareexposed,modified,
andmadehabitualagain.
6.Meaning
Onewaytodescribehowtheworldofmathematicschangeswhenone
movesfromaclassicaltoanalgorithmicframework,istofocusonthe
subtleshiftsinthemeaningofsuchcentralconceptsas:
truth,falsity,number,set,element,equality,identity,
infiniteset,higherinfinity,function,proof,existence.
Theproblemisthatthemeaningsofthesetermsarehighlyinterlocked,
inawaythattendstokeepmeaningstable.Soifyoulearnanew
meaningforequalitybutothertermsstaythesame,thenewmeaning
forequalitywillbeoutvotedbytheoldmeanings,andwillnolonger
makesense.Youneedtolearnenoughconstructivemeaningssothat
theycanformacoherentareaofunderstandingfromwhichyoucan
extend.We’lllookathowthemeaningchangesforsomeoftheseterms,
forsetandelement,equalityandidentity.
Classicalmathematicsstartswithadomainofprimitiveelementswith
distinctidentities.Thesebasicelementsaregroupedintosets,whichin
turncanbeelementsofothersets.Theclearfactisthattheelements
existfirstandarecollectedintosets.InBishop’sapproachtosets,itis
thesetwhichcreatesthepossibilityofelements.Toconstructasetyou
mustdescribewhatmustbedonetoconstructanarbitraryelement.
Thesetprecedesitselements.Ofcoursethenumber3hadbeen
aroundforalongtimebeforeanyoneconsideredasetℕofall
numbers,butonlyexistedasanelementofℕafterthelatterwas
constructed.
Let’sconsiderequalityandidentity.Classicalmathtendstoconflate
them,andtousetheequalsignwhenthingsareidentical.Andidentity
isaglobalpredicate:anytwomathematicalobjectsareeitherthesame
ornot.
Constructively,equalityisalwaysaconvention,asBishopproudly
proclaimed.Andthereisnouniversalidentityrelation.Partofthe
constructionofasetisprovidingadefinitionofequality,andthe
constructionmustincludeproofsthattheproposeddefinitionis
reflexive,symmetric,andtransitive.
Ifthereisnouniversalidentityrelation,whatitmeansifthesame
symbolisusedmorethanonce,asinreflexivity
x=x.
Bishopsuggestedthathereliedontherepeatabilityofmathematical
constructions,orintentionalidentity.Mathematicalconstructions,like
scientificexperiments,mustberepeatable,andinthereflexiveequation
thesecondxisarepeat,acopy,ofthefirst.Toprovetheproposed
equalityrelationisreflexiveyoumustshowthatifyoucarryoutany
constructionofanelement,andthenrepeattheconstruction,the
proposedrelationwilldeclarethattheoriginalandthecopyareequal.
Thisformallyeliminatesthepossibilitythatyoucouldconstructasetin
whichequalityisdeterminedbyflippingacoin.
Stanwouldhavegonethroughthechangesinmeaningwithhisseminar,
helpingthemtoputenoughconstructivemeaningtogethersothata
coherentworldcameintofocus.Itwouldhavebeenajointexploration
throughtrickyterrainratherthanbeingcalledfromonhigh.
Ifirstexperiencedthesechangesaround1971when,afterseveralyears
ofwrestlingwithBishop’swork,Ilearnedtoseemathematics
constructively,orasIwouldsaynow,algorithmically.Myconversion
wouldneverhaveoccurredsaveforthepatientmentoringofafriend,
GabrielStolzenberg,withadeepunderstandingofBishop.Attheend,I
foundthattheappearanceofmymathematicalworldhadchanged.It
wasstillunquestionably“mathematics”thatIobserved,butithada
differenttexture,morealive,lessremote.
7.Brouwer
I’mgoingtoclosewithanaccountofmyowndifficultywithBrouwer,
whichwasanearlybarriertoengagingwithBishop’sprogram.
whenIwasingraduateschoolIhadbeensocializedtobelievethat
Brouwerhadbeenagreatyoungmathematician,whoprovedthe
BrouwerFixedPointTheorem,oneofthefirstdeepresultsinthenew
fieldoftopology.Then,inlaterlife,hebegantoworrytoodeeplyabout
whatitallmeant,andbasicallywentcrazy.Brouwermadeaspeaking
touroftheUSAin1960,andhowwegraduatestudentslaughedatpoor
Brouwerwhowaspresentingacounter-exampletohisowngreatest
theorem.Itwasasadwarningtostayawayfromdangerousideasthat
couldthreatenyoursanity.ThatwarningwasstillfreshwhenStanand
IbegandiscussingBishop’sconstructivism,itwasanobstaclethat
slowedmedown.AndwhenStanwenttoLasCruces,that
understandingofBrouwer,whichwasstillverywidespreadwouldhave
beensomethingtoconfront.It’seasytoimaginehowskillfullyStan
wouldhavetakenthaton.
Brouwer’slifeandworkismuchbetterdocumentednow.Thestory
thatwehadacceptedorfabricatedaboutBrouwer’slifewascompletely
false.Hehadthedangerousideaswhenhewasyoung,andthey
appearedinhisPh.D.thesis.Thenherealizedthatnobodywouldpay
attentiontothemunlesshewasarecognizedmathematician,sohedid
standardmathematicsuntilhewasafamoustopologist,andthenhe
cameout.
I’mgoingtoclosewithBrouwer’scounter-exampletohisownFixed
PointTheorem.Thattheoremisclassicallyvalidinalldimensions,but
we’llonlyconsiderdimensionone,wherethetheoremisessentiallythe
IntermediateValueTheoremofelementarycalculus.Brouwer
constructedacontinuousfunctionfsuchthatf(0)isnegativeandf(1)
ispositive,andyoucannotlocateapointwhereftakesthe
intermediatevalue0,withoutsolvingsomeunsolvedproblem.Here’sa
graphofBrouwer’sfunction:
Youmayhaveguessedthatbetween0.3and0.7thefunctionfis
constant,takingavaluewe’llcallzeroeyverycloseto0,whichmight
bepositiveornegativeorexactlyzero.Ifzeroeyispositivethenthere
isauniquevalue,slightlylessthan0.3,negativeandthevalueisslightly
morethan0.7.Andifzeroeyiszero,thenfhasthevalue0from0.3to
0.7.Againourignoranceisencodedinzeroey.Considerourignorance
aboutthedecimalexpansionof"worksverynicelytoconstruct
zeroey.Weknowvirtuallynothingaboutit,butwedohaveverygood
algorithmsforcomputingit.Forourunsolvedproblem,let’saskif
3.14159…evercontains100consecutive7s.Startcomputingthe
decimalexpansionof",andwatchingourfor100consecutive7s,and
outputtingasequenceofrationalnumbers(fractions)todefinezeroey,
andalsokeepingtrackofhowmanydigitsof"you’velookedat.As
longasyouhaven’tfound100consecutive7s,youroutputis0.Butif
youfind100consecutive7s,endinginthedecimalplacen,thenchange
tooutputto± 1 10& wherethesigndependsonwhethernisoddor
even,andleavetheoutputthereforeverforever.Tosumup,whenyou
startcomputingthesequencethatdefineszeroey,youkeepgetting0
butthesequencealwaysretainsthepossibilityofswitchingtoan
infinitesimalpositiveornegativevalue.
ButallisnotlostfortheIntermediateValueTheorem.Brouwer’s
examplecapturestheessenceoffunctionsforwhichtheintermediate
valuetheoremdoesn’thold.Ifafunctionisneverconstant,thenthat
functiontakesallintermediatevalues.Mostfunctionsarenever
constant,unlesstheyareconstructedtobeconstantoversomeinterval.
Forinstancepolynomials,trigonometricfunctions,etc.The
IntermediateValueTheorembecomesmuchmoreinteresting
constructively.ItrytoimaginehowStanwouldhavepresentedthis
exampleinLasCruces.IwishI’dbeenthere.
ButIlosttouchwithStanafterourtimeinRochester.Therewere
occasionswhenIcouldhavecontactedhim,andregretthatIdidn’t.I
particularlywishwe’dspokenafterIfinallycametounderstandBishop.
Therewouldhavebeensomuchtotalkabout.