Philosophial Claims
Deio Krause
Antonio M. N. Coelho y
Nuleo deEpistemologia e Logia
DepartamentodeFilosoa
UniversidadeFederal deSantaCatarina
(february2002)
Abstrat
The standard ways lassial logi and mathematis deal with the
oneptofindisernibility(indistinguishability),withspeialempha-
sis to theonept of indisernibility inastruture are onsidered.
Theaimistoemphasizethatinassertingthat`two'objetsofaer-
taindomain(generally, anonemptyset)are thesame objet,one
beomesommittedtotheaxiomsofsettheory,sinethestrutures
where theseonepts areexpressedaregenerallytakenassettheo-
retial onstruts. Someof theonsequenesof thesepointstothe
philosophialdisussiononidentityandindisernibilityinquantum
theoryaretheninvestigated.
1 Introdution
ThesignianeofLeibniz'sPrinipleoftheIdentityofIndisernibles(PII)in
quantummehanishasbeenwidelydisussedinthephilosophialliterature,mainly
in onnetion with the problem of whether or not it is violated in the quantum
domain.Elementary`idential'quantum objetsinthesamestateofmotionwould
beexamplesofentitieswhihdiersolonumero,havingalltheirintrinsiproperties
in ommon and being sothat in ertain situations even spatio-temporal loation
annotbeusedtodistinguishamongthem(forinstanewhentheyarein`entangled
states').Thiswould onstituteamotiveforPIItobefalse.
1
Leibniz'sprinipleisusuallywritten inaseondorderlanguageas
8F(Fa$Fb)!a=b (1)
where a and b denote individuals and F is a variable ranging over properties of
individuals. The onverseof this priniple is known asthe Priniple of the Indis-
ernibilityofIdentials(II),namely,
a=b!8F(Fa$Fb) (2)
Partially supported by the Programa Multidisiplinar - CNPq. E-mail:
dkrausefh.ufs.br, URL:www.fh.ufs.br/dkrause
y
E-mail:aoelhofh.ufs.br
1 SeeFrenh(1989)foramoregeneraldisussion.
identity(being thesameobjet, in symbols,`a=b') isdened bymeans ofindis-
dernibility(agreementwithrespetto allproperties).
2
In the quantum realm the disussion onerning the domain of the variable
F aquires a partiular importane, for it raises the problem of what should be
onsidered as alegitimateproperty of aquantum objet,and in onsideringsome
of the possibilities, at least three forms of PII havebeendistinguished elsewhere:
PII(1),theweakestform,statesthatitisnotpossiblefortwoindividualstohaveall
propertiesandrelationsinommon;PII(2)exludesthespatio-temporalproperties,
whilePII(3)inludeonlymonadi,non-relationalproperties,andmuhdisputehas
beenpresentedon the predominaneof one of these forms of PII overthe others
(seeFrenh(1989)and(1999)formoredetails).
In this paper we shall onsider aspets of this disussion from a perspetive
whihdoesnothavebeenmuhonsideredinthephilosophialliterature,butwhih
weguessmaybeofimportaneforafullunderstandingoftheinvolvedproblems,in
partiular,thoserelatedtothephilosophialdisussionsregardingquantumphysis
and onepts like identity, indisernibility, and individuality. Our approah ould
besaidtobemodeltheoretiinaertainsense,sineweshallbeonernedmainly
with themathematialstrutures involvedin suhadisussion. Inshort,weshall
pushtheontexttomainly itslogialandmathematialonsequenesandimpliit
asumptions, withthehopethat it anbeusefulin philosophialissuesonerning
the foundations ofquantum theory. The reallingof some ofthe basidenitions
and ways ofapproahingidentity-likeonepts below hastheonlyaim of keeping
thepapermoreself-ontainedandxingsometerminology.
2 Charaterizing identity
The standard way of onsidering the onept of identity in mathematis is
linked toatradition whihgoesbakat leastto Leibniz.Frege took possessionof
Leibniz'sditumEadem sunt quorum unumpotestsubstitui alteri salva veriate to
motivate his `denition' of identity (equality) in Die Grundlagen der Arithmetik
(Frege 1884) but, as remarked by A. Churh, he made a onfusion between use
andmention,forwhat istobesubstitutedarenotthethingsthemselves,buttheir
names: \things are idential ifthename of oneanbesubstituted forthat of the
otherwithoutlossoftruth"(Churh1956,p.300).
Theonfusion betweenuseand mentionwasorretednine yearslaterin the
Vol.1ofFrege'sGrundgesetzederArithmetik (Frege1893)but, there,insteadofa
denition,whatwendisanaxiom,writtenbyChurhas(x=y)((F)[F(x)
F(y)℄).
3
Previouslyin the Begrisshrift (1879),Frege used axiomswhih anbe
written as(foranyaandb):
(F1)a=a
(F2)a=b!(F(a)!F(b))
It should be realizedthat,bythat time, therewasnotaleardistintionbe-
tweenrstandhigherorderlanguages,andtodayweknowthatFrege'sfoundational
systemenompasses`more'thanrstordermathematis.Theaxiommentionedby
Churh makespart of Whitehead and Russell's denition of identity of Prinipia
2 Thereisnogeneralagreementonerningtheterminologyusedhere,forsomeauthorsprefer
toall`LeibnizLaw'theexpression(F2)below.
3 Frege'soriginalnotationwasbasedinhisBegrisshrift(Frege1879),andwillnotbeused
here.
esseniallythoseusedinrstorderlanguageswhenidentityistakenasanelement
of the underlying logibut, forexpressing the generalase(involvingformulasin
general), F(a) must beread asaformula whatever,while F(b) omes from F(a)
from thesubstitution ofsomefreeourrenesofabyb,providedb isfreeforain
F(a)(aandb beingtermswhatever;seeMendelson1997,p. 95).
Ifarstorderlanguagehasonlyanitenumberofprediates,thenidentityan
bedened.FordoingthatitsuÆes towriteasaformulaA(x;y)theonjuntion
ofallpossiblesubstitutionsintheprediates,inasensethatthereisanexhaustion
ofalltheprimitiveprediatesofthetheory(Hilbert&Akermann1999,pp.107).
Then identity is dened by suh a formula. For instane, suppose that the only
primitiveprediatesare thebinary prediateP and theunaryprediate Q. Then
A(x;y)should bethefollowingformula(exeptbythequantiers)
(P(x;z)$P(y;z))^(P(z;x)$P(z;y))^(Q(x)$Q(y)) (3)
whih`simulates'identityinthesensethatxandysharealltheprimitiveprediates
ofthelanguage.Arelevantaseisthatofsettheory,whereidentity(x=y)anbe
dened by
8z(z2x$z2y)^8z(x2z$y2z):
4
(4)
Furthermore,ifthelanguagehasonlybinaryrelationsymbolsinanitenum-
ber,then weould postulate that allthese relationsare reexive, symmetri and
transitive,sothat F2followsfromthese postulates.Butsomeareisinneedhere,
for suh anhypothesis mayonitwith theintendedsemantisfortherelational
symbols.Forinstane, ifmembershipisamongtheprimitivebinaryprediates,we
annot simplypostulatethat 2is to be interpretedas areexive,symmetri and
transitive relation,for reexivity (x 2x) andsymmetry (x 2y !y 2x) donot
hold due to the axiom of foundation, while transitivity holds in general only for
transitivesets.ButF1 andF2entailthat =is anequivalenerelation (Mendelson
op. it.,p.95).
5
The`rstorder'axiomsF1andF2abovehaveinterestingonsequenes,mainly
ifweonsidersemantis.Firstly,letusreallthatF1expressestheintuitiveideathat
`everyobjetis idential to itself', and isknownasthe ReexiveLaw ofIdentity,
while F2 is the Substitutivity Law.If weintuitively think of identity asreferring
to something an objet has to itself and with nothing more (the onsideration
of whether or not this `something' is a legitimate `property' shall be mentioned
below),thenweouldexpetthattherightsemantialinterpretationoftheidentity
prediate`='shouldbethediagonal ofthedomainofdisourse,namely,theset
D
=fhx;yi:x;y2D^x=yg (5)
where D standsfor the domain of theinterpretation (Churh 1956, p. 283).The
symbolof identity in thedeniens is thenaveset-theoretialidentity,for thein-
terpretation of suh arst order languageis made in a set theoretial struture,
4 SeeforinstaneFraenkel,Bar-Hillel&Levy1973,p.27.
5 Froma philosophialpointofview,weshoulddistinguish amonga `metaphysial'idea of
identity (beingthe very same objet,there areno two, et.) and the onepts webuilt inthe
mathematialframeworkwehaveatourdisposaltoexpressthisidea.Asweshallsee,thereisnot
totalagreementamongthem.
in order:
(1) Byusing F1 and F2 only, we annot distinguish between `individuals' of the
domain and ertain equivalene lassesof these individuals. In short, F1 and F2
donot`haraterize'thediagonalwithoutambiguity.Sinethisresultisimportant
andit isnotusuallymentionedinthephilosophialdisussionsonthesubjet,let
usseeinbriefsomeaspetsofsuhalaim.SupposethatA=hD;iisamodelfor
ourrstorderlanguageLinthestandardsense,whereisthedenotationfuntion
denedasusual,that is,suhthat,foreveryindividualonstantofthelanguage,
(), whih we denote by D
, is an element of D; furthermore, for every n-ary
prediate R , (R ) = R D
is asubset of the set D n
, and for any n-ary funtional
symbolf,(f)isamappingfromD n
toD.
Aswehaveseenabove,therelationwhihinterpretstheprimitivesymbolofiden-
tityis in partiular an equivalene relation (sine it must be reexiveby F1 and
symmetri and transitive, as it an be proven without diÆulty {Mendelson lo.
it.).Letusdenoteby
D
suharelation.IfA 0
=hD 0
; 0
iisanotherinterpretation
for ourlanguagesuh that itsdomain D 0
is thequotient set of D bythe relation
D
(that is,D 0
=D=
D
),and suh thatthe relationwhih interpretstheequal-
itysymbolin this new interpretation isdenoted by
D
0,let f : D !D 0
be the
anonimapping,denedasfollows,whihassoiatetoeveryx2Ditsequivalene
lass f(x) 2 D 0
(that is, the equivalene lass f(x) to whih x belongs); thus, f
satises:
(i)f(x)
D 0
f(y)ifandonlyifx
D y
(ii) Foreveryn-ary prediateletterR ofthe language,if(R )=R D
and 0
(R )=
R D
0
,thenR D
0
(f(x
1
);:::;f(x
n
))ifandonlyifR D
(x
1
;:::;x
n )
(iii)Foreveryn-aryfuntionalsymbolg ofthelanguage,if(g)=g D
and 0
(g)=
g D
0
,theng D
0
(f(x
1
);:::;f(x
n ))
D 0
f(g D
(x
1
;:::;x
n ))
(iv)For everyindividualonstant,wehavethat D
0
D 0
f( D
)
Then,wean provethatthestruturesAandA 0
areelementarilyequivalent,that
is,whateversenteneofL issuhthat (instandardnotation):
Aj= i A 0
j= (6)
The details may be found in Hodges 1983 (pp. 68), but what the elementary
equivalenebetweenthestruturessaysisthatthatwhatholdsforx
1
;:::;x
n inA,
holdsalsoforf(x
1
);:::;f(x
n )inA
0
.AsshowninMendelson (op.it., p.100),the
existeneofsuh anf dependsessentiallyofthevalidityofF2.
ThisintuitivelysaysthateveryelementofthedomainofA 0
(whihisanequivalene
lass, hene, aolletion of elements of the domain) ats as an individual of the
domain of A. So, from the point of viewof the languageL, it is not possibleto
know whether we are onsidering an element of D or an equivalene lass in D 0
(a ertain olletion of elements of D). For uniquely haraterizing the diagonal
D
,weneedseondordervariables,thatis,variableswhihrunoverolletionsof
objetsof D or,alternatively(in extensionalontexts), overtheirproperties. But
eveninthisasetherearedetailstobepaidattentionto,asweshallremarkbelow,
for the use of seond-ordervariables is notenough; we need to onsider also the
meta-mathematialframeworkwheretheplayisating.
6
6 Ofourse, to mentioninbrief,inthe semantisfor higher-orderlogis (Henkin style)we
mayfoundmodelsforalanguagesothatevenLeibnizLawbeingtrue(seeequation8below),a
andbmaybenot`theverysameobjet';seeRobbin1969,p.144,exerise1.
identitymaybethefollowingone.SupposewehaveLeibnizLaw(1)+(2)inmind
and tryto aptureit asloserasweanin arstorder language.
7
So, weshould
betemptedtowrite somethinglikethefollowingshema,sineweannotquantify
overprediates:foreveryF denotingaprediateofindividuals,wehave:
a=b$(F(a)$F(b)): (7)
Butsupposenowthat wehaveadomainD withardinality
0
and that ourrst
order language has a ountable number of monadi prediates, so asthat a and
b name two elements of D, whih we denote by (a) and (b), aording to the
abovenotation.But,asimpliedbytheaxiomsoftheset theory(supposeZermelo-
Fraenkel),whih istheambienttheorywhere theinterpretationis supposed tobe
dened, (a) = (b) if and only if for every subset X of the domain, (a) 2 X
if and only if (b) 2 X. Sine in standard semantis F(a) is true if and only if
(a) 2 F D
(the extension of F), then the right side of the shema (7) refersto
at most
0
subsetsofD, whileweknowthat D (withsuh aardinality)has2
0
subsets. Hene,even if(7) holds, this fat doesnotensure that (a) and(b) are
the very same element of D, for neessarily there is a subset Y D whih is
not the extension of any prediate of the languageunder the interpretation suh
that (a) 2Y and (b)2= Y. Thisis essentiallywhythere mayexist models fora
seondorderlanguagewhereLeibnizLawistruebutwhereaandb(onsideragain
the onjuntion of (1) and (2)) have not the same referent, as remarked above.
Denumerablerstorderlanguageshavelimitationsalsofrom thispointofview.
Inregardingrstorder languages,westillremark that theyarealsonotgen-
erally adequate forstudying in full the relationshipsbetweenobjets ofa ertain
domain,exeptifthisdomainisnite,whihisnottheasewiththemostinterest-
ingmathematialtheories,whihareusuallyommittedtomathematialonstruts
likereal numbersand the like, that is, with innitedomains. Furthermore,it fol-
lowsfrom the theLowenheim-Skolemtheorems that if astruture hasan innite
domain,thenitannotbeharaterizedbyarstordertheory(thepartiularase
involvingidentitywasmentionedabove).OfourseLowenheim-Skolemtheoremsdo
notholdin higherorder logis,butotherkindsofproblemsarisein suhontexts,
asweshallmentionbelow.
3 The ase of higher-order languages
InhigherorderlogisthedenitionofidentityusuallyhasmotivationinWhite-
headand Russell'sonepresentedin Prinipia Mathematia (1925).Thedenition
maybewrittenas follows:
a=b=
df
8F(Fa!Fb): (8)
We remark that in the deniens that what appears is the material implia-
tion, andnotabionditional, asin the earliermentionedLeibnizLaw.Intuitively
speaking,asremarkedbyBoolos&Jerey(1989,p. 200,wheretheformaldetails
are alsogiven),the motiveis that(8) \isvalidbeauseamong thepropertiesof a
isthepropertyofbeingidential with a[theprediateI
a
dened above℄;then,ifb
7 Thatis,wemaybetryingtowritedownthe`metaphysial'oneptionofidentitymentioned
atnote4.
remark ould betaken asquite obvious bysomeone, but sometimes the property
`being idential with a' is put in doubt asa legitimate (relational) attribute of a
in theontextof thedierent formsof PII (f.setion 1;see Frenh 1989). Then,
in onsideringthesesituations,weshouldpayattentiontothewaywehavestated
thedenitionofidentity;withoutthis`problemati'property(namely,theproperty
of `being idential with a') in the range of X, the denition needs the biondi-
tional.Thisremarkwillhaveonsequenesinwhatfollows,mainlyinregardingthe
onnetionwithnon-rigidstruturesmentionedbelow.
4 Charaterizing indisernibility
Cantor's informal haraterization of the onept of set states that \by an
aggregatewearetounderstandanyolletionintoawholeofdeniteandseparate
objets of ourintuition orof our thought" (Cantor 1955, p. 85).
8
The axiomati
versionsofsettheory(Zermelo-Fraenkel,vonNeumann-Bernays-Godel,et),despite
theirdierenes,in ertainsense`aept'thisharaterization,and henelassial
mathematistoo.TheimportantpointtobeemphasizedhereisthatfortheAxiom
ofExtensionalitytohold,itisneessaryto haveariterionfortwoelementsbeing
the sameobjet.In otherwords,set theoriesdisplayatheory ofidentityforboth
theelementsofaset and forthesets themselves(usually givenbytheunderlying
logiplus theAxiomofExtensionality).
As weshallmakeexpliitbelow,ifthetheoryadmits theexisteneofentities
whiharenotsets,butthatmaybeelementsofaset,thatis,theUrelemente,then
weanonsiderthemasindistinguishable,asFraenkeldid,butthisistobetakenin
thesensethatwhateverpermutationofUrelementeinduesanautomorphismofthe
universe(thisexamplewillberealledagainbelow).Whatwewouldliketoremark
is that even in this ase the Urelemente are subjeted to the theory of identity
imposed by the underlying logi, and henethey annot be taken as `legitimate'
solo numero indisernibleobjets. Independentlyif settheory is treatedasarst
order theory, asithasbeenusualeversineSkolem,orifit isaxiomatizedhaving
(say)aseond orderlogiinitsbases,theinvolvedtheoryofidentityisessentially
thatonedesribedintheprevioussetions.So,beingaandbeitherUrelementeor
sets, theremakessense to assertthat either a=b ora6=b holds.Inthelast ase,
there existsaset (whih inextensionalontextsanbeassoiatedto a`property')
that separates a from b, that is, a set X suh that a belongs to X but b does
not.Hene,theintuitiveoneptofindividualitymakessenseto allobjetsofthe
domain, independently of whether or not we an atually realize the distintion
betweenthem.
But,eveninsuha(atleastinpriniple)`distinguishable'framework(perhaps
it would be better to say `land of individuals'), there are ways of making sense
to the idea of indistinguishability, or indisernibility. The important point to be
remarkedisthat theseproedures provideonlyawayofsaving the appearenes in
the sense of providing a way of pretendingthat someentities an be treated asif
theywereindistinguishable.But`lassial'logiandmathematispassawayofthe
onsideration of ab ovo indistinguishable objets in a way that Cantor's original
8 ItisworthnothingthatCantor'soriginal`denition'ofsetatpage204ofhisGesammelte
Abhandlungen,ofwhihtheabovedenitionisarestatement,doesnotdisplaysuhaommitment
withindividualitytoutourt,foritreads:\asetisamanywhihanbethoughtasone".
waysofonsideringthingsasindiserniblestartsfromof individualsofakind,for
instane, elements ofaset (here, by aset weunderstand themathematialentity
`dened'bytheaxiomsofasettheorylikeZermelo-Fraenkel)andthenwepostulate
someonditions oroperationson these objetssothat theylook asindisernible;
the neessities of these moves are, in ouropinion, due to the assumptions of the
underlyingmathematiallanguage(inludinglogi),whihisinessenea`language
ofindividuals',aswehavesaid.Justtohaveanideaofthispoint,letusmakeashort
referenetosomeofthestandardwaysofonsideringindistinguishableentities.
4.0.1 Ramsey's indisernibles
Ramsey's indisernibles play an important role in model theory and in set
theory(Bell&Mahover1977,p.218).Buttheyarenot`genuine'indiserniblesat
all.Letusrealltheirdenition,whihwillprovidethegroundstounderstandthis
fat.SupposethatAisastrutureforarstorderlanguageL.LetX beasubsetof
thedomainofAand<astritlinearorderonX (whihdoesnotneessarilybelong
to thestruture). Ifa
0
;:::;a
n andb
0
;:::;b
n
aretwostritly resentsequenesof
elementsofX(intheorder<),wesaythathX;<iisasetofRamsey'sindisernibles
in Aifandonlyif
Aj=(a
0
;:::;a
n
) i Aj=(b
0
;:::;b
n
); (9)
forwhateverformula(x
0
;:::;x
n )ofL.
In partiular, if X has exatly two distint elements, say X = fa;bg, then
hX;fha;bigi is set of Ramsey's indisernibles in A, for there exists just only one
stritly resentsequene ofelementsof X. Thisshowsthat Ramsey'sindiserni-
bility is rather distint from indisernibility solo numero but, as it will be made
learbelow,underertainonditionsaandbanbeonsideredasindiserniblesin
astruture.
4.0.2 Weyl'sstrategy
Inonsidering `aggregatesof individuals'fordisussionsin thefoundations of
quantum theory, Hermann Weyl has examined the ase where the elements of a
ertainolletionmaybeinertain`states'butonlythequantityofthemineahof
these stateswouldbeknown(Weyl1949,App.B).Aordingto him,thisiswhat
happensin quantum physis.Then,Weylhastakenaset S (let usemphasizethis
fat)withnelements,sayx
p1
;:::;x
pn
,endowedwithanequivalenerelation.The
elementsC
1
;:::;C
k
of thequotientset S=weretaken tostand forthe `states',
andsinethateahoneofthemhasaardinaln
i
,i=1;:::;k,`theimportantthing'
tobeonsideredisthatthere isaertainquantityofelementsofS ineah`state',
whihisahievedbyonsideringtheordereddeomposition n
1
++n
k
=n.Then
Weyl hassuggested to takefor grantthis deomposition, and `to forget' that the
involvedobjetswhiharebeingountedareelementsofaset, hene`individuals'
of akind(hedidn't usethesewords,buttheonsequenes arepreiselythese). In
doingthat,Weylsuggestedthatwearrivedatasituationwhere
\::: no artiial dierenes between elements are introdued by their
labelspandmerelytheintrinsidierenesofstatearemadeuseof:::".
(op. it.,p. 239)
Of oursethereisagaphere.Weyl'salternativeofonsidering asetendowed
with an equivalene relation does not work in haraterizing indistinguishability
withtheaboveordereddeomposition,butwean'tforgetthatSwastakenasaset
rightfromthestartafter all,sothattheirelementsaredistinguishable entities,at
leastinpriniple,asitresultsfromCantor'sseond `denition'referredtoabovein
thetext,soasfromwhateverstandardaxiomatisettheorywemayuse(ZF,NBG,
et.)forformallydesribingallofthis,asseenabove.So,forarrivingatWeyl'son-
lusionweshould agreethat the(at leastinpriniple) `identiable'harateristis
oftheelementsofSweremaskedbythemathematialtrikofabstratingthefat
that theyare elements of aset, heneindividuatable entities, andthat what was
takenintoaountwasonlytheirroleaselementsofaertainequivalenelass.Of
oursethismayworkformathematialpurposes,but fromthephilosophialpoint
ofviewitseemstousthatsomethingmoreisinneedforharaterizing`legitimate'
indisernibilityrightfromthestart(forfurtherdetailsonerningWeyl'sapproah
in thisontext,seeKrause1991).
4.0.3 PermutationalSymmetries
The roleof permutational invarianein physial ontextsanbesummed up
by the words of Paul Weingartner, who gives us an idea of what permutational
symmetriesmean:
\[p℄ermutationalsymmetry meansthat `dierentindividual' partiles of
the`samesort'aretreatedasidential.Thusthelawsandtherespetive
physial world(universe)desribedbythese lawsremainthesameifwe
interhangeanytwoeletrons."(Weingartner1996)
Later,in talkingaboutpermutationinvariane,hesaysthat
\[p℄ermutation hange, i.e., interhange of elementary partiles of the
samesort doesnothange lawsbut also {aordingtothe usualunder-
standingof aphysialsystem(this systemmaybethewhole universe){
doesnothangethissystem.(:::)Thatmeansthatelementarypartiles
of the same kind are treated asindistinguishable although numerially
dierent."(ibid.,p.80).
This invariane of physial laws enters in the ontext of the appliations of
group theory to the problem of the onstitution of physial objets (Castellani
1998).
9
Bymeansofinvarianeunderertainsymmetrygroups,partilesarelassi-
edinategories,or`kinds';asrealledbyCastellani(op. it.),thepioneeringwork
ofE. Wignerin 1939(Wigner1939)entailsthat eah`elementarypartile'isasso-
iated withanirreduiblerepresentationof aertainsymmetrygroup,then being
haraterizedby aertain number of invariantproperties. But, as also remarked
byCastellani,\thatweobtainin thiswayisnomorethanalass of objets"(her
emphasis), that is, grouptheorygivesus nomorethanthe dierenesamong the
kinds of partiles, providing no morethan their lassiation. Groupstoo, let us
reall,aresets endowedwithaertainstruture.
In short,despite permutational symmetriesare useful, they onstitutealso a
kind of`trik'in Weyl's sense mentionedabove:webeginwith individuals(mem-
bersof aset)and, by permutingthem, weprovide away ofexpressing that their
permutationannotberegardedasobservable,henenotgivingadierentphysial
law.Then wereason as if weare dealingwith indistinguishableentities.Suh an
indistinguishability may savethe appearenes, but we should do not forget that,
9 Forageneralhistorialviewontheroleofgrouptheoryinphysis,seealsoBueno&Frenh
1999.
beginning.
Weouldontinuegivingexamplesofhowstandardmathematisprovidestools
fordealingwithindistinguishability,butwethink thatthese fewaseshaveexem-
plied our point: all of them are triks, useful and relevant in ertain situations,
but stilltriks.Inthenextsetionweshallonsideranalternativeway,but atthe
expenseofrestritingthedisourseto aertainstruture.
5 Indisernibility in a struture
In mathematial ontexts, it is possible to haraterize a notion of indistin-
guishabilityalso byonsidering theideaofinvariane underautomorphisms. But,
indoingso,webeameommittedtoindisernibilityrelativetoaertainstruture;
letus givesomedetails onthispoint,whihwehopewill illuminatesomeaspets
underlyingthephilosophialdisussiononindividuation.Inthisdiretion,wefound
Fraenkel,Bar-HillelandLevysayingthat
\(:::) there isno harateristi whih distinguishes oneindividual from
another (:::) in mathematial terms one would say that every permu-
tation of the individuals an be extended to an automorphism of the
universeofelements."(Fraenkeletal.1973,p. 59)
TheindividualstheyrefertoaretheUrelementeofZFU(theZermelo-Fraenkel
with Urelemente set theory).Theabovequotationrefersto theroleplayedbythe
Urelemente inFraenkel'sproof oftheonsistenyoftheAxiomofChoiewiththe
remainingaxiomsofZFU(exludingtheAxiomofFoundation).Asitiswellknown,
insuhaproofFraenkeladmittedtheexisteneofadenumerableinniteolletion
ofUrelemente (Fraenkel1922).
The onept of invariane under automorphisms wasalso noted by the Por-
tuguese mathematiianJoseSebasti~ao eSilvain 1944 asbeing adequatefor har-
aterizingindisernibility;ashesaid,
\if an element [of a ertain struture℄ is not individuatable, and hene
logially disernible from others (as the number i is indisernible from
i by means of the usual primitive notions), it seems that there is an
automorphismofthesystemwhiharriesthiselementinanyoneofthe
others."(Sebasti~aoeSilva1944,p.281) 10
A wayof understanding howthe aboveintuitivenotionof inditinguishability
ats is to relate it with another important onept, namely, that oneof absolute
denibility.Letusprovidethegroundsforseeingthat.
Thefollowingpassage,takenfromRogers(1967),helpsinxingthemainidea;
ashesays,
\[i℄sthereanabsolutenotionofdenibility?Althoughlogiians,in their
onernwithpartiularformalsystems,havelargelyignoredit,anatural
notionforabsolutedenibilityhasbeenurrentinmathematisforsome
time.Thisisthenotionofinvarianeunderautomorphisms(:::)Wesay
that V V [where V is the domain of a struture℄ is invariant under
all automorphismsiff(V)=V foreveryautomorphismf [oftheabove
struture℄. It is lear that if V is to be \denable" (in some sense) in
a given struture, it must beinvariantunder all automorphisms of the
10 WethankProf.NewtondaCostaforpointingusthisreferene.
verselyitanbearguedthat theinvariantsubsetsofV arejustthesets
whih are determined (in somesense) by the struture, and henethat
theyshouldbealled`denable'."
So,letusstateadenition.If
A=hD;fR
i g
i2I
;ff
j g
j2J
i (10)
isastruture,letaandbbeelementsofD(thedomainofthestruture).Thenwe
saythataandbareA-distinguishable (ordistinguishableinthestrututeA)ifand
onlyifthereexistsasub-olletionXD suhthat:
(i)X isinvariantunderautomorphismsofA,thatis,f(X)=X foreveryautomor-
phismf ofA.
(ii)a2X ifand onlyifb2=X,
otherwise,wesaythataandbareA-indistinguishable.
11
Inthestandardextensionalsettheoretialtradition,wemay(roughlyspeaking)
identify a property of ertain objetswith a olletion of suh objets, preisely
the olletion of the objets whih havethat property. In suh a framework, we
maylink theabovedenition ofA-distinguishabilitywith theaforesaidoneptof
absolute denibility; the idea is to say that two elements are A-indistinguishable
in agivenstrutureifandonlyiftheysharealltheabsolutelydenableproperties
of this struture, that is(in set theoretial terms),whenthey belong to thesame
olletionsofelementsofthedomain thatareinvariantunderautomorphisms.
Asitiseasytosee,weanrestatetheabovedenition inthefollowingequiv-
alent way: a and b are A-indistinguishable i there exists an automorphism f
of the struture A suh that f(a) = b. Indeed, if f(a) = b for some automor-
phism f of Aand if X D isinvariantunder automorphisms of A,then a 2X
i b 2 X, for f(a) = b and f 1
(b) = a. So, a and b are A-indistinguishable.
Conversely, if f(a) = b does not hold for any automorphism of A, then X =
fg(a) : GisanautomorphismofAg is sub-olletion of the domain whih is in-
variant under automorphisms of A suh that a 2 X and b 2= X. So a and b are
A-distinguishable.
Of ourse this kindof indistinguishability is notequivalent to identity in all
strutures(here,byidentityweunderstandthediagonalofthedomain),butitisin
agreementwiththeintuitionofthemathematiianandlariessomephilosophial
aspetsinvolvedinthistopi.Letustryto larifyalittlebitthislastidea.
Wesaythat astruture Aisrigid ifand onlyifitsonlyautomorphismisthe
identityfuntion. It is lear that in a rigid struture, everysubset of the domain
is invariantunder automorphisms. So, given a and b in the domain, with a 6= b,
thenaandbareA-distinguishable,sinea2fagbutb2=fag.Furthermore,ifAis
astruturewhere A-indistinguishabilityand identityoinide,thatis, aandb are
A-indistinguishableifandonlyifa=b,thenAisrigid.Theproofiseasytostate.
Supposethat f isanautomorphism ofAwhihisnottheidentityfuntion.Then
there exists an a in the domain suh that f(a) =b 6=a. But, sineb 6= aand in
suh anstruture by hypothesis identity andA-indistinguishability oinide,then
there exists a subolletion X of the domain suh that: (i) X is invariant under
automorphisms; (ii) a 2 X but b 2= X. But this is aontradition, for being X
invariantunderautomorphismsanda2X,weshouldhavef(a)=b2X.
11 Thatis,aandbareA-indistinguishableiforeverysub-olletionXD,ifXisinvariant
underautomorphismsofA,thenz2Xib2X.
property\tobe identialwith a"for haraterizingaand fordistinguishingit(in
suhstrutures)fromtheotherobjetsofthedomain,byusingthejustmentioned
oneptofdistinguishabilityin astruture.Letusgivesomeexamples.
(i) If A = hA;<i is a well ordered struture, then A is rigid. Really, if f is an
automorphism of A whih is not the identity funtion, then there exists a least
elementa 2A suh that f(a) 6=a. Of ourse, wean't havef(a) =x <a, sine
foreveryx<a,wehavef(x)=x.Thenf(a)>abut, sinef issurjetive,there
exists b 2A suh that f(b) = a. Butneither b =a, for f(a) 6= a, norb <a, for
f(x)=xforeveryx<a.Hene b>aandf(a)>f(b),whihontraditsf being
anautomorphism.
Thisshowsthat everyordinalisarigidstruture.Frequently,whenwearedealing
with aertain olletionof n(in priniple) indistinguishable objetsand weneed
tomakethendistinguishablebysomemotive,asforinstanefortalkingofthem, 12
generally what we dois to assoiatean ordinalto theolletionof these objets,
whihorrespondstosaysomethinglike\let0;1;2;:::;n 1besuhobjets".For
instane, inonsideringtwo`idential'(in thephysiist'sjargon)eletrons,thenin
ordertowritedownthe funtion forthejointsystemweusuallylabelthemwith
`names', saypartile#1andpartile#2(see Teller1995,pp.21).Ournotionof
indistinguishability in a struture, at least in priniple, makes lear what we are
doing when we label the partiles: we assoiate the onsidered objets with the
elementsof thedomain of arigid struture. Thispointwill bedisussed againat
theendofthepaper.
(ii) Let A = hZ;+i the additive group of the integers. Then A is of ourse not
rigid, for f :Z !Z dened by f(x)= x,for everyx 2Z, is an automorphism,
theonlyonewhihisnottheidentityfuntion.So,foreveryx2Z,x and x are
A-indistinguishable.
(iii)LetV betheZermelo-Fraenkel(ZF) universe.Then,bytheisomorphismthe-
orem(Jeh 1997,p. 74), A=hV;2iis rigid, where2 is themembership relation.
Thisagreeswiththeideathatinthestandardmathematis(thatis,thatonebuilt
inZF),identityandindistinguishabilityoinide,aswehavesaidabove,foridentity
andA-indistinguishabilityoinide.
(iv) Let U be the Zermelo-Fraenkel with Urelemente(ZFU) universe. Then, A =
hU;2i, where 2is still membership, is not rigid.Really, whateverpermutation of
the Urelemente indues an automorphism of A (Jeh op. it., pp. 198-9). As we
have said above, this fat is in the ore of Fraenkel's proof of the onsisteny of
the negation of theAxiom of Choie (with the remainingaxioms of ZFU, exept
Foundation),andinaertainsensejustiesthestandardviewthattheUrelemente,
despitenotidential,areindistinguishable.
13
Furthermore,wesaythat astrutureAis trivially embedded in arigidstru-
tureBwhenB anbeobtainedfromAbyaddingto Anewobjets,relationsand
funtions,andtheseaddedelementsalonearesuÆientformakingBarigidstru-
ture.Forinstane,everyAanbetriviallyembeddedinarigidstruturebyadding
to A allthe singletons ofthe elements ofits domain. Weremark that the adding
12 ThisrelationshipwiththeneedsoflanguagewasemphasizedforinstanebyG.Toraldodi
Frania,whosaidthat`objetuation'isa primitiveatofunderstanding.Aordingto him,we
divide upthe worldin`objets'(individuals)to speakofthem;seeToraldodiFrania1981,p.
222.
13 Ifpressed,alogiianouldsaythattheUrelementearesetsofthesamerankinanadequate
modelofsettheory.
`to beidential witha' foreahelementaof thedomain ofA. These notionswill
beusedbelow.
On the other side, we say that a struture A is non-trivially embedded in a
rigid struture B when B an be obtained from A by adding to A new objets,
relationsandfuntions buttheseaddedobjetsalone arenotsuÆientformaking
B arigid struture.An exampleofanon-trivialembeddingin arigid struture is
that of hZ;i in iZ;+;i, sine hZ;i is not rigid (for eah h 2 Z, f
k
: Z 7! Z
dened byf
k
(x)=x+kisanautomorphismofhZ;i).
6 Physis and metaphysial alternatives
The development of quantum statistis hasbroughtthe laim that quantum
partiles annot be regarded as on a par with `marosopi' objets like roks
and people. Aordingto somesholars,theyare, in somesense,`non-individuals'
(Shrodinger 1998, Weyl 1949, App. B). A way of making sense to this idea is
to onsider that the onept of identity does not makesense to them, as expli-
itly emphasizedbyShrodinger (Shrodinger 1952,pp. 17-8;seeFrenh &Krause
forthoming).Butthealternativeviewhasalsobeendefended,bysayingthat, on
theontrary,quantumpartilesan,infat,beregardedasindividuals,albeitwith
verydierentpropertiesandbehaviourfromthelassialones.AsputbyS.Frenh,
thisgivesrisetoaveryinterestingsituation(notexploredhere)aordingtowhih
ourfundamentalmetaphysisisunderdeterminedbythephysis(Frenh1999; see
Frenh&Krause op.it.). Inonsideringthese views,weandistinguishbetween
twomain linesofthoughtinwhat onernsindistinguishablepartilesandanalyse
themfromtheperspetiveofwhatwehavesaidintheprevioussetions.So,letus
sumupthem asfollows:
(A) The rst hypothesis says that elementary partiles of the same kind are in-
distinguishableinanontologialsense,beingexamplesoflegitimatevague objets.
Jonathan Lowe,forinstane, sustainssuhaview, whih hasbeendisussedelse-
where (see Frenh and Krause 1995 for the referenes). Aording to this view,
partiles in asuperposition state annot be distinguished, even in priniple. This
ts well the idea of non-individuals, put by several authors like those mentioned
above,andapparentlyisinaordanewiththebasisuppositionsofquantumeld
theory(seeFrenhandKrause1995a).
(B) Elementarypartilesofthesamekindare indistinguishablefrom anepistemo-
logialpointofviewonly.Thisideamayberelatedtosomekindofhiddenvariables
theory,forwemaysupposethattheprogressofphysis,logi,et.,willtelluswhat
shouldbetherelevantpropertiesthatservetodistinguishamongthem(Sant'Anna
2000).ThisviewmayalsobemergedwithsomeaspetsofvanFraassen'sonstru-
tiveempiriism,despitevanFraassendoesnotendorsehiddenvariables.Aording
tohim,despite`idential'elementarypartilesareindistinguishablefromthepoint
of viewof all the mehanisms provided by quantum theory, even so they anbe
distinguished:\[identialpartilesinthesamestateofmotion℄areertainlyquali-
tativelythesamein allrespetsrepresentable inquantummehanialmodels{yet
still numerially distint" (vanFraassen 1991, p. 376). This strange possibility is
related to theanswersvan Fraassenoersto this dilemma:either the prinipleof
theidentityofindiserniblesisviolatedorquantummehanisisnotomplete.As
it is well known, both possibilities raise a luster of philosophial problems. van
FraassenmaintainsPII,sinehedistinguishes betweenquantum dynamialstates
erator represents,andareembedded in thetheory, whoseevolutionis governedby
dynamiallaws.Inotherwords,dynamialstatesaredesribedwithintheformalism
of quantum mehanis.Experimentalevents,on theontrary,are extra-theoretial
entities whih respet the waysthe probabilities alulations are performed. The
same oneptual distintion is made by distinguishing between state attributions
and value attributions of aphysial system.Theformeris atheoretial onstrut,
and part of the problemsinvolved in theory onstrution depends upona proper
representationofthestates,whilethelatterwouldbesomething`meta-theoretial'.
This distintion is interesting and raisesfurther importantphilosophial insights,
for it seems that we should regard events as something `outside' the theory, so
thatthedistintnesspropertiesofthepartilesouldbeahievedonlyinthemeta-
theoretial realm. So, PII ould be saved in onsidering the role played by the
metalanguageofquantum physis.Theintuitionsheregoestotheonsiderationof
someresultsrelatedtoSkolem'srelativism,whileaertainset theoretialonept
an havedierent meanings in dierent models. In other words,maybe the issue
of absoluteness should bepursued alsoin onnetion to empirialsienes, but of
oursethedetailsofsuhanideaneedtobefurtherdeveloped.
Despite weare avoiding to enter into the detailed philosophial dispute over
the positions (A) and (B), in aeptingthat bothof these views anbedefended
withstrongarguments,our`moremathematial'frameworkgivenabovepermitsus
to onsiderthemasfollows.
First,wemaysaythatthosewhosustainposition(A)shouldalsoaeptthat
the mathematialstruture of quantum theory annot be embedded, nottrivially
(reallthedenition givenat theendof thelastsetion),in arigidstruture.Po-
sition(B), ontheontrary,anbeonsidered aviewaordingto whih quantum
mehanisan be, not trivially, embedded in arigid struture B. That is, in this
ase we are aepting that the objets of the domain (elementary partiles, say)
may bedistinguishedfrom oneeahother,but notmerelybyposingnewrelations
orfuntionsthatbythemselvesalone areresponsibleforthedistinguishability(dis-
ernibility).
We an motivate these views by onsidering the following passage,in whih
Shrodinger gives an intuitive aount of Maxwell-Boltzmann and Bose-Einstein
statistis:
\Threeshoolboys,Tom,Dik,andHarry,deserveareward.Theteaher
hastworewardstodistributeamongthem.Beforedoingso,hewishesto
realize for himself how many dierent distributions are at all possible.
(:::) Itis astatistialquestion:(:::) dierentkindsof rewardwill illus-
tratethe(:::)kindsofstatistis.
(a) The two rewardsare twomemorial oins with portraits of Newton
and Shakespeare respetively. The teaher may give Newton either to
Tom or to Dik or to Harry. Thus there are three times three, that is
nine,dierentdistributions(lassial statistis).
(b) Thetworewardsaretwoshilling-piees (whih, for ourpurpose, we
mustregardasindivisiblequantities).Theyanbegiventotwodierent
boys,thethirdgoingwithout.Inadditiontothesethreepossibilitiesthere
arethreemore:eitherTomorDikorHarryreeivestwoshillings.Thus
there aresixdierentdistributions (Bose-Einsteinstatistis).
(:::)therewards representthepartiles(:::)Memorialoinsareindivid-
ualsdistinguishedfromoneanother.Shillings,forallintendsofpurposes,
adierenewhetheryouhaveoneshilling,ortwo(:::).Thereisnopoint
in twoboysexhangingtheirshillings." (Shrodinger1998)
The ideaisthat ifweouldonstrut astruturerepresenting thissituation,
wherethepropertyofbeingarewardisoneofitsonstitutiverelations,thenthere
wouldbenoautomorphismofthisstruturearryingonememorialointoanother,
but, sine,asrewards,theshillingsare indistinguishable,there would be anauto-
morphism arryingoneshillingto another.This would be themeaningofthelast
senteneoftheabovequotation.Ontheotherside,althoughtheshillings,asmate-
rialobjets,anbedistinguishedfrom oneanother,this kindofdistinguishability
suÆes, alone, to identify eah shilling. It doesn'tinvolvethe property ofbeinga
reward. So,theembeddingof thestruture representingShrodinger'sexamplein
a rigid struture having among its onstitutive relations the property of being a
materialobjetisatrivialembedding inarigidstruture,and thisdoesn'tbother
those whosustain position (A) above.A suessful refutation of position (A), in
ouropinion,wouldrequireanontrivialembeddinginarigidstruture,thatis,the
onsideration ofnew relations that not alone, but ombined with the property of
beingareward,produetherigiditythatwillallowtheidentiationofeahshiling
(rememberthegivenexampleofhZ;+ibeingembeddedinhZ;+;<i).
Withintheontextofposition(A),themetaphysialviewaordingtowhih
elementarypartiles arenon-individuals anbesustained onlyin those strutures
that annotbemaderigidnottrivially.Thisinformallymeansthat there arethe-
oretial ommitments that,in some way, preventthe useof properties like`being
identialwitha' todistinguishafrom otherobjetsunderonsideration.
Whattheabovedisussionperhapshasontributedtoshowisthat thephilo-
sophialdisussion on the logialfoundations of physismust onsider the power
andthelimitationsoftheunderlyinglogialandmathematialapparatususedinthe
disussions.Sometimestheproblemsannotberightlyanalysedonlyattheinformal
level,asthestandardphilosophialdisussionsusuallydo,outoftheonsiderations
of the logi and mathematial axioms.A `more mathematial' onsiderationmay
illuminatesome ofthe problems,as theaseof theonepts oftrivialand of non
trivialembeddinginarigidstruturedoes,aordingtotheaboveguidelines.This
perhapshelps philosophers to preise their intuitions; forinstane, in sayingthat
`tobeidentialwitha'isnotalegitimaterelationalproperty,whatreallysomeone
meansis notrulingoutthe possibilityof makingastruturerigidby addingto it
thesingletonsoftheelementsofitsdomain.
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