More  Problematic  that  Ever:  the  Julius  Caesar  ObjectionFraser  MacBride

Published  in    in  Identity  and  Modality,  MacBride  (ed.)  OUP  2006,  174-­‐202.

1. Introduction2. What  is  the  ‘Caesar  Problem’?

2.1 Frege2.2 Dimensions  of  the  Caesar  problem2.3 The  Epistemological  Problem2.4 The  Metaphysical  Problem2.5 The  Meaning  Problem

3. Two  Solutions3.1 The  Supervaluationist  Solution3.2 The  Neo-­‐Fregean  Solution

4. Conclusion

1.  IntroductionIs   it   possible   that   Julius   Caesar   was   not   only   a   person   but   also   a   number?   Might   the  conqueror  of  Gaulbeneath  his  material  guisehave  been  the  bearer  of  numerical  properties?  Common  sense  appears  to  inform  us  otherwise.  Caesar  was  a  man  and  no  doubt  added  up  sums.  But  he  could  hardly  have  been  the  result  of  a  subtraction.  And  what  appears  to  be  true  of  Caesar  also  appears  to  be  true  of  other  concrete  items.  They  enjoy  the  benePits  of  being  spatially   related   and   causally   interacting   with   their   neighbours.   By   contrast,   numbers  appear  quite  different  kinds  of  thing.  They  stand  in  equations  and  Pigure  in  the  ancestral  of  the  successor  relation  to  zero.   It  seems  that   there   is  a  Moorean  fact,  basic   to  our  ordinary  ways   of   thinking,   any   adequate   philosophy   of  mathematics  must   accommodate.     The   fact  must   be   ensured   that   no   concrete   item   is   a   number.   A   theory   encounters   the   ‘Caesar’  problem  when  it  fails  to  meet  this  adequacy  constraint.  

These  remarks  serve  only  to  introduce  an  aspect  of  the  Caesar  problem.  The  Caesar  problem  demands  our  attention  alongside  other  fundamental  issues  that  ask  for  elucidation  and  justiPication  of  the  basic  structure  of  our  conceptual  schemefor  example,  the  problem  of  universals  or  the  problem  of  change.  And,  like  them,  the  Caesar  problem  does  not  engage  a  single   subject   matter   easily   isolated.   It   engages   a   variety   of   logical,   semantical   and  metaphysical  questions  and  their  interrelations.  The  failure  to  recognise  the  rich  and  varied  

nature  of   the  Caesar  problem  has  resulted   in  a   failure   to  appreciate   the  sort  of  difPiculties  that  confront  a  satisfactory  resolution.  It  may  also  have  resulted  in  the  relative  neglect  of  the  Caesar  problem,  a  problem  that  deserves  a   rightful  place  alongside  other  more  venerable,  indeed  ancient,  philosophical  concerns.

2.  What  is  the  ‘Caesar  Problem’?2.1  Frege.  Frege  brought  the  ‘Caesar  problem’  to  the  attention  of  analytic  philosophy  in  the  course   of   his   attempt   to   introduce   abstract   objects   (directions,   numbers)   into   ordinary  discourse.  Frege  began  his  discussion  by   forging  the  now  familiar  connection  between  the  notions  of  object  and  identity.  Self-­‐standing  objects  (persons,  mountains)  may  be  identiPied  and  then  re-­‐identiPied  on  different  occasions  and  from  different  perspectives.  So  expressions  that  are  used  to  refer  to  objects  must  have  associated  with  them  a  class  of  statements  (Frege  calls  them  “recognition  statements”)  that  settle  identity  criteria  for  these  objects:

“If   for   us   the   symbol   a   is   to   denote   an   object,   then   we   must   have   a   criterion   which  determines  in  every  case  whether  b  is  the  same  as  a,  even  if  it  is  not  always  in  our  power  to  apply  this  criterion.”  (Frege  [1953],  §62).

Recognition   statements   determine   when   it   is   appropriate   to   label   and   then   re-­‐label   an  object   on   a   different   occasion   with   the   same   expression.   If   abstract   objects   are   to   be  introduced   into   ordinary   discourse   then   a   class   of   recognition   statements   must   also   be  supplied  for  them.  

Frege   attempted   to   meet   this   constraint   by   stipulating   identity   criteria   for   the  abstract  objects  he  proposed   to   introduce   ([1953],   §§62-­‐5).  Numbers  are   stipulated   to  be  identical  just  if  their  associated  concepts  are  1-­‐1  correspondent:    

(HP)  Nx:Fx  =  Nx:Gx  iff  F  1-­‐1  G

Directions  are  stipulated  to  be  identical  just  if  their  associated  lines  are  parallel:

(D=)  Dir  (a)  =  Dir  (b)  iff  a  is  parallel  to  b.

However,  Frege  soon  came  to  doubt  whether   these  dePinitions  supply  genuine  recognition  statements.  He  Pirst  concentrated  his  attention  upon  the  case  of  (D=):

“In  the  proposition  ‘the  direction  of  a   is  identical  with  the  direction  of  b’  the  direction  of  a  plays  the  part  of  an  object,  and  our  dePinition  affords  us  a  means  of  recognising  this  object  as  the  same  again,  in  case  it  should  happen  to  crop  up  in  some  other  guise,  as  the  direction  

of  b.   But   this  means  does   not   provide   for   all   cases.   It  will   not,   for   instance,   decide   for   us  whether   England   is   the   same   as   the   direction   of   the   Earth’s   axisif   I   may   be   forgiven   an  example   that   looks   nonsensical.   Naturally,   no   one   is   going   to   confuse   England   with   the  direction  of  the  Earth’s  axisbut  that  is  no  thanks  to  our  dePinition  of  direction.”  ([1953],  §66)

Frege  then  went  on  to  question  (HP)  “for  the  same  reasons”  ([1953],  §68).  Frege   appears   to   have   deliberated   in   the   following   manner.   Genuine   recognition  

statements  distinguish  an  object  of  a  given  kind  K   from  all  other  objects.  This  means   that  they  must  determine  the  truth-­‐values  of  two  sorts  of  identity  statementpure  and  impure.  A  pure   identity   statement   says   that   an   object   of   kind   K   is   identical   to   another   object   also  explicitly  stated  to  be  of  kind  K  (“Kx  =  Ky”).  By  contrast,  an  impure  identity  statement  says  that  an  object  of  kind  K  is  identical  to  another  object  that  may  not  be  described  in  K  terms  (“Kx   =   q”).   (D=)   and   (HP)   notably   determine   truth-­‐values   for   pure   identity   statements  concerning,  respectively,  directions  and  numbers.  They  each  provide  a  rule  for  determining  the   truth-­‐value  of   identity  statements   in  which,   respectively,  dual  occurrences  of  direction  and  number  terms  Plank  the  identity  sign  (“Dir  (a)  =  Dir  (b)”,  “Nx:Fx  =  Nx:Gx”)  by  appeal  to  the  obtaining  of  a  familiar  equivalence  relation  (“a   is  parallel  to  b”,  “F  1-­‐1  G).  But  (D=)  and  (HP)   fail   to  settle   truth-­‐values   for   impure   identity  statements   (“Dir   (a)  =   the  Earth’s  axis”,  “Nx:Fx  =  Caesar”)  in  which  there  are  only  single  occurrences  of  direction  and  number  terms.  They  simply  say  nothing  about  statements  of  this  form.  As  a  consequence  (D=)  and  (HP)  fail  to  supply  identity  criteria  for  the  objects  they  are  designed  to  introduce.  

2.2  Dimensions  of   the  Caesar  problem.  These   rudimentary   rePlections  do  not,  however,  reveal   the  depth  and  breadth  of   issues   the  Caesar  problem  raises.  Beneath   the   superPicial  simplicity   of   Frege’s   reasoning   lies   a   welter   of   distinct   worries.   These   concerns   may   be  arranged  along  three  distinct  dimensions:

A) Epistemology:  do   the   identity  criteria  supplied   for  directions  and  numbers  provide  any  creditable  warrant  for  a  familiar  piece  of  knowledge,  namely  that  directions  and  numbers  are  distinct  from  such  objects  as  Caesar?  

B) Metaphysics:  do  the  identity  criteria  supplied  for  directions  and  numbers  determine  whether   the   things   that   are   directions   or   numbers  might   also   be   such   objects   as  Caesar?

C) Meaning:   do   the   identity   criteria   supplied   for   the   novel   expressions   putatively  denoting  directions  and  numbers  bestow  upon   them  the  distinctive  signiPicance  of  singular  terms?

Whether  these  different  dimensions  of  concern  may  ultimately  be  distinguished  will  depend  upon  a  variety  of  prior  theoretical  choices.  For  example,  in  the  context  of  a  robust  realism—say,  a  posteriori  realism—that  draws  a  clear  separation  between  ontological  and   linguistic  issues,   the   metaphysical   and   meaning   dimensions   are   held   apart.   In   the   context   of   a  minimalist  approach  to  ontology,  these  different  dimensions  of  the  Caesar  problem  will,  by  contrast,  be  intimately  related.  In  advance  of  settling  anterior  realist  or  anti-­‐realist  choices  the  Caesar  problem  is  likely  to  resist  canonical  speciPication.  The  Caesar  problem  also  raises  different  issues  in  the  cases  Frege  considers  (number  and  direction).  Here  attention  will  be  given  primarily  to  the  numerical  case.

2.3  The  Epistemological  Caesar  Problem.  According  to  one  epistemological  version  of  the  Caesar  problem  suggested  by  Frege’s  remarks,  we  already  know  that  Caesar  is  not  a  number.  So  adequate  identity  criteria  for  numbers  should  settle  that  Caesar  is  not  a  number.  But  (HP)  leaves  it  open  whether  Caesar  is  or  isn’t  a  number.  Therefore,  (HP)  fails  to  supply  adequate  identity  criteria  for  the  entities  they  respectively  purport  to  introduce.  

Whilst  this  epistemological  concern  possesses  considerable  prima  facie  force  a  number  of  questionable  assumptions  are  made  that  require  (in  this  context)  to  be  legitimated:

i) Common  Sense:  we  ordinarily  know  that  Caesar  is  not  a  number.ii) Discrimination:  adequate   identity   criteria   for   numbers   should   enable   a   subject  

who   grasps   them   to   discriminate   numbers   from   all   other   kinds   of   object  (however  presented).  

First,   let   it   be   granted   that  we  do   ordinarily   know   that   Caesar   is   not   a   number   (Common  Sense).  According  to  the  version  of  the  Caesar  problem  delineated,  putative  identity  criteria  for  numbers   that   fail   to   conPirm   that  ordinary   item  of   knowledge  are   inadequate.  But   the  question  needs   to  be  asked:   in  what   sense   inadequate?  The  guiding   thought  must  be   that  identity   criteria   that   fail   to   meet   this   constraint   fail   to   isolate   the   subject   matterthe  numberswith   which   ordinary   discourse   deals.   In   other   words,   this   version   of   the   Caesar  problem   assumes   that   if   (HP)   is   to   be   used   to   talk   about   an   intended   range   of   familiar  numbers   these   identity   criteria   must   provide   for   all   the   discriminations   that   we   would  ordinarily  make  between  these  and  other  objects  (Discrimination).  

However,  Discrimination  appears  too  strong  an  assumption  to  make.  It  may  reasonably  be  supposed  that  the  ability  to  refer  to  a  given  kind  of  object  presupposes  some  capacity  to  recognise  items  of  that  kind.  But  it  would  also  be  unreasonable  to  claim  that  a  novice  who  only  possesses  limited  recognitional  skills  could  never  light  upon  the  same  subject  matter  as  

a  thinker  with  more  rePined  discriminatory  capacities.  It  therefore  remains  open  that  (HP)  provides   a   rudimentary   but   nevertheless   genuine   basis   for   identifying   and   re-­‐identifying  numbers.  They  allow  us  to  identify  and  distinguish  between,  respectively,  numbers  (that  are  given   as   such)   even   though   they   fail   to   underwrite   the   sort   of   discriminatory   abilities  required   to   identify   and   distinguish   between   numbers   and   others   objects   differently  presented.

These  rePlections  invite  a  rePinement  of  the  epistemological  concern.  It  is  certainly  true  that   there   is   a   distinction   to   be   drawn   between   succeedingon   however   a   rudimentary  basisin   directing   thought   and   talk   upon   a   given   range   of   objects,   and,   being   able   to   draw  sophisticated,  Pine-­‐grained  distinctions  amongst  the  objects  in  question.  But  the  possibility  of  directing  thought  and  talk  upon  a  subject  matter  presupposes  a  basic  grasp  of  the  sort  of  objects  under  consideration.  But  someone  who  has  only  grasped  (HP)  has  no  grasp  of   the  sorts  of  objects   that   are  under   consideration.  They  don’t   even  know  whether  persons  are  numbers.  So  they  can  hardly  claim  to  be  able  to  think  or  talk  about  numbers.

However,   this   response   is   also   contestable.   Imagine   a   child   otherwise   ignorant   of  arithmetic  who  is  taught  (HP).  For  all   that  has  been  said  so  far  this  child  may—within  the  arithmetical   language   game—go   on   to   reliably   distinguish   between   different   numbers  (presented   as   such).   According   to   the   result   called   Frege’s   Theorem,   (HP)   entails   the  fundamental  truths  of  arithmetic  (the  Peano  Postulates)  (Wright  [1983],  pp.  158-­‐69,  Boolos  [1987]).   So,   continuing   the   fantasy,  we   can   also   imagine   that   the   child   goes   onto   develop  basic  arithmetical  skills  (addition,  multiplication  etc.).  This  means  that  the  child  may  mingle  at  school  with  peers  who  are  taught  arithmetic  in  the  ordinary  fashion  and  come  home  with  good   test   results   in  maths.  Do  we   really  want   to   say   that   this   computationally   competent  child   does   not   succeed   in   talking   about   numbers?   Does   the   fact   that   he   expresses  incomprehension  when  asked  whether  Caesar   is  a  number   immediately   settle   that  his   test  results   are  nothing  more   than  a   sham?  Would   it   not  be  more  appropriate   to   say   that   this  child   trains   his   thought   upon   numbers   well   enough   but   lacks   an   additional   piece   of  metaphysical  knowledge  (that  Caesar  is  not  a  number)?  

Another   assumption  made   by   the   epistemological   version   of   the   Caesar   problem   also  merits  scrutiny.  According  to  Common  Sense  we  already  know  Caesar  is  not  a  number.  But  do  we   know   this?   If  we  do,   it   is   certainly   not   in   virtue   of   grasping   Frege’slet’s   face   it,   pretty  decentguesses  at  identity  criteria  for  numbers  and  directions.  So  how  else  might  we  know  this?   It   may   seem   that   there   is   no   pressing   need   to   answer   this   question.   For   there   is   a  tendency  in  contemporary  philosophy  to  assign  to  common  sense  an  epistemologically  and  theoretically  innocent  nature.  As  a  result  the  verdicts  of  common  sense  are  simply  taken  for  granted.  However,   it   is   important   to  bear   in  mind   the  possibility   that   common   sense  may  

itself  be  corrupt,  nothing  other  than  the  consequence  of  signiPicantalbeit  prolonged  and  low  level  theoretical  labour.  As  Russell  once  remarked  upon  the  common  sense  understanding  of  such  notions  as  ‘thing’  and  ‘object’:

“the   thing   was   invented   by   prehistoric   metaphysicians   to   whom   common-­‐sense   is  due.”  (Russell  [1911],  p.  148).

It  may  indeed  be  that  our  common  sense  understanding  delivers  the  verdict  that  Caesar  is  no  object.  But  it  remains  open  that  the  endorsement  of  Common  Sense  may  incur  signiPicant  epistemic  costs.

There  appear   to  be  two  possibilities  concerning  the  putative  knowledge  that  Caesar   is  no  number.  Either  this  knowledge  is  immediate  or  it  is  derived.  It  is  important  to  realise  that  the   former   option   is   far   from   plausible.   It   is   part   of   the   beguiling   nature   of   the   Caesar  problem   that   when   we   try   to   form   a   clear   and   distinct   idea   of   Caesar   we   do   not   Pind   it  explicitly  represented   there   that  he  cannot  be  a  number.  Nor  when  we   try   to   form  a  clear  and  distinct  idea  of  a  number  (say,  zero)  do  we  Pind  it  explicitly  represented  that  zero  cannot  be  a  person.  Rather  we  encounter  a  modest  silence  on   these  matters.  Caesar  has  personal  properties.  Zero  has  numerical  properties.  But  it  is  neither  explicitly  ruled  in  nor  ruled  out  that  a  Caesar  might  be  zero.  So  if  we  really  do  know  that  Caesar  is  not  a  number  then  there  must  be   some  argument   implicit   in  our  ordinary  understanding   that   shows   this   to  be   the  case.  Since  the  metaphysical  version  of  the  Caesar  problem  concerns  the  availability  of  just  such  an  argument  let  us  turn  our  attention  there.

2.4   The   Metaphysical   Caesar   problem.   According   to   this   development   of   Frege’s  reasoning,   it   is   impossible   for   radically   different   kinds   of   object   to   overlap.   (HP)   fails   to  preclude  the  possibility  that  the  same  objects  fall  under  radically  different  kinds  (persons,  numbers).  This  is  because  it  leaves  open  whether  a  range  of  identity  statements  concerning  objects   drawn   from   disparate   kinds   are   true   or   false   (for   example,   “Nx:Fx   =   Caesar”).  Therefore,   (HP)   fails   to  provide   adequate   identity   criteria   for   the  objects   it   is   intended   to  introduce.  

This   version   of   the   Caesar   problem   rests   upon   the   following   apparently   sane   and  sensible  assumption  

iii) Sortal   Exclusion:   such   radically   contrasting   kinds   of   objects   as   numbers   and  persons  cannot  overlap.

But  part  of  what  makes  the  metaphysical  Caesar  problem  so  problematic  is  that  it  is  far  from  

clear  what   legitimate  grounds  for  Sortal  Exclusion   there  might  be.   It   is   frequently  asserted  that  it  is  simply  “absurd”  to  suppose  otherwise  (Parsons  [1990],  pp.  308-­‐9).  However,  brute  intuition  has  proved  a  notoriously  unreliable  guide  in  theory  construction.  So  an  argument  for  Sortal  Exclusion  is  wanted.  One  tempting  strategy  is  to  argue  that  numbers  are  abstract  whereas  persons  are  concrete  and  thereby  obtain  Sortal  Exclusion  as  a  conclusion.  Waive  the  usual  concerns  about  whether  the  abstract-­‐concrete  distinction  is  in  good  enough  shape  to  distinguish  between  mutually   exclusive   classes  of   abstract   and   concrete   items   (Burgess  &  Rosen   [1997],   pp.   12-­‐25).   Just   suppose   for   current   purposes   that   concrete   objects   are  located  and  capable  of  entering  into  causal   interaction  whereas  abstract  objects  are  not.  It  still  does  not  follow  that  the  kinds  in  question  cannot  overlap  unless  it  is  also  presupposed  that  numbers  are  abstract  and  persons  are  concrete.  

Arguments   for   this   last   claim  may   appear   readily   forthcoming.   It  may   be   thought  that  there  are  more  numbers  than  concrete  objects.  Perhaps  there  are  inPinitely  many  of  the  former  whereas  only  Pinitely  many  of  the  latter.  So,  it  may  be  concluded,  numbers  cannot  be  concrete  objects.  Since  persons  are  concrete  it  follows  that  persons  aren’t  numbers.  But  this  argument   is   too   quick.   It   neglects   to   rule   out   the   possibility   that   some   numbers   (Pinitely  many  of  them)  are  concrete  objects.  

Another   line   of   argument   appeals   to   the   necessary   truth   of   a   wide   range   of  mathematical   claims.   Necessary   truths   require   necessary   existences   to   serve   as   their  immutable  subject  matter.  Since  concrete  objects  are  contingent   it   follows  that   the  objects  picked  out  by  mathematical  truths  cannot  be  concrete.  But  this  argument  is  also  too  quick.  It  assumes   that   the   sentences   that   express   necessary   truths   must   refer   to   necessary  existences.  This  assumption  may  be  questioned.  The  necessary  truth  of  a  sentence  may  be  sustained  by  virtue  of  its  constituent  terms  picking  out  different  objects  at  different  possible  worlds.   In  other  words,  the  argument  assumes  that  numerical  terms  are  rigid  designators.  However,   if  numerical   terms  are  non-­‐rigid   thenfor  all   that  has  so   far  been  establishedthey  may  at  a  given  world  pick  out  concrete  objects  (Caesar  amongst  them).

A  variation  on   this  argument  appeals   to   the   role  of  numerical   terms   in   contingent  counterfactual   claims   of   applied   arithmetic.   Consider   a   range   of   counterfactual  circumstances   in  which   the  number  of   Fs   remains   the   same  even   though   concrete  non-­‐Fs  pass   either   in   or   out   of   existence   from   one   circumstance   to   the   next.   It   follows   that   the  expression  “the  number  of  Fs”  cannot  refer  to  any  concrete  non-­‐F.  For  example,  the  number  of   moons   of   Mars   is   two.   We   can   entertain   counterfactual   circumstances   in   which   the  number   of   moons   of   Mars   remains   two   even   though   Caesar   had   never   existed.   So   the  number  of  moons  of  Mars   cannot  be  Caesar.   (Of   course,   it  may   take   further  discussion   to  show  that  the  numerical  terms  that  Pigure  in  the  statements  of  pure  arithmetic  cannot  pick  

out  concrete  items  either.)  Here   it   is   assumed   that   the   numbers   applied   to   concepts   in   counterfactual  

circumstances  actually  exist   there.  Then  since   the  concrete  objects  at   issue  do  not  exist   in  those  circumstances  the  desired  consequence  follows  that  the  numbers  in  question  are  none  of   the   concrete   things.   But   this   assumption   is   far   from   obligatory.   The   application   of  numbers   to   concepts   in   counterfactual   circumstances   may   not   rest   upon   an   ability   to  identify  the  numbers  that  exist  in  those  circumstances.  Rather  it  may  rest  upon  an  ability  to  identify  and  count  with  numbers  in  the  actual  world  and  then  use  these  numbers  to  count  the  objects  falling  under  concepts  in  counterfactual  circumstances  from  here.  It  is  therefore  left   open   whether   numbers   exist   or   not   in   any   given   counterfactual   circumstance.  Alternatively,  the  relation  expressed  by  “is  the  same  number  as”  may  express  an  equivalence  relation   (a   trans-­‐world   relation),   weaker   than   identity,   between   the   shifting   referents   of  numerical   terms.   If   so,   the   fact   that   the   number   of   Fs   remains   the   same   across  counterfactual   circumstances   fails   to   determine   that   the   number   of   Fs   is   not   concrete   in  some  world.

It  may   also   be   argued   that   concrete   objects   are   contingent  whereas   numbers   are  necessary  and  so  no  person  can  be  a  number.  These  claims  may  appear  beyond  question.  Surely  it  is  “manifest”  that  Caesar  is  no  necessary  existent  (Hale  &  Wright  [2001],  p.  366)?  Surely  it  is  beyond  doubt  that  numbers  cannot  be  contingent?  But  these  assumptions  have  been   questioned.   Field   has   argued   that   numbers   are   contingent   existences;   after   all,   he  claims,  there  is  no  logical  incoherence  in  the  suggestion  that  numbers  might  fail  at  a  given  world  to  exist.    As  part  of  a  defence  of  a  simple  form  of  quantiPied  modal   logic  (including,  crucially,   the   Barcan   formula:   xx)   Linsky   &   Zalta   have   argued   that   every   object   exists  necessarily.  However,   the   point   is   not   only   that   these   assumptions   have   been   questioned.  More   signiPicantly,   the  point   is   that  a   failure   to  appreciate   that   these  assumptions  may  be  questioned   constitutes   a   failure   to   appreciate   the   problematic   character   of   the  (metaphysical)  Caesar  problem  itself.  

A  common  Pirst  reaction  to  the  Caesar  problem  is  to  take  it  as  manifest  that  Caesar  is  no  number.  So  Sortal  Exclusion  is  simply  taken  for  granted.  But  we  are  able  to  see  our  way  past   this   initial  response  when   it   is  appreciated  that   it  no  straightforward  matter   to  settle  whether  different  kinds  of  objectsthat  are  apparently  as  unlike  as  objects  can  beare  really  distinct.  The  matter  is  difPicult  to  settle  because  (in  part)  it  is  not  explicitly  written  into  the  nature  of  persons  that  they  are  not  numbers  (or  vice  versa).  But  nor  is  it  explicitly  written  into  the  nature  of  persons  or  numbers  that  they  are  contingent  or  necessary.  So  to  assume  on  manifest  grounds  that  Caesar  cannot  be  necessary  or  that  numbers  cannot  be  contingent  is   simply   to   ignore   the   problematic   character   of   the  Sortal   Exclusion   assumption.  What   is  

wanted  here   is   just   an   instance   of  what   is  wanted   generally:   a   principled   account   of  why  objects   of   one   kind   cannot   possess   features   (necessary   existence,   abstractness)   usually  associated  with  different  kinds.    

In  response  it  may  be  claimed  that  it   is  plainly  constitutive  of  being  a  person  to  be  contingent.  After  all,  persons  come  to  be  and  pass  away.  They  begin  and  cease  to  exist.  They  might   not   have   existed.   They   can   hardly   be   necessary   existents!   Of   course,   this   train   of  thought  will  hardly   settle   that  Caesar   is  not  a  number  unless   is   also   shown   that  numbers  cannot  be  contingent.  But,  more  signiPicantly,  ask  yourself  the  question:  is  that  an  accurate  statement  of  what  we  know  to  be  true  of  persons?  Might  it  not  be  more  accurate  to  say  that  persons  take  on  and  then  throw  off  a  material  guise  and  it  is  left  open—a  matter  upon  which  speculation   may   never   cease—what,   if   anything,   happened   before,   next   or   whether   they  might  never  have  existed?  I  do  not  mean  to  suggest  that  persons  continue  to  exist  without  bodies  or  as  bare  abstract  entities.  The  point  is  rather  that  to  legitimate  the  Sortal  Exclusion  assumption   that   underwrite   the   metaphysical   version   of   the   Caesar   problem   it   must   be  demonstrated   that   no   persons   is   a   number.   And   if   the   resolution   is   to   be   theoretically  satisfying  this   fact  must  somehow  be  guaranteed  by  the  underlying  nature  of  persons  and  numbers.   But   since   there   is   no   immediate   incompatibility   between   being   a   number   and  being  a  person  the   intriguing  difPiculty  we  have  to  confront   is   that  we  have  apparently  no  idea  of  how  Sortal  Exclusion  might  be  legitimated.  

In  any  case  appeal  to  different  principles  of  modal  existence  is  far  too  coarse-­‐grained  a   basis   upon   which   to   ground   Sortal   Exclusion.   For   suppose   that   one   were   to   become  convinced  that  persons  exist  necessarily.  Would  one  then  feel  anymore  comfortable  with  the  suggestion  that  Caesar  is  a  number?  Or,  alternatively,  suppose  that  one  already  believed  God,  or  some  other  plausibly  non-­‐numerical  item,  exists  necessarily.  Would  it  then  be  legitimate  to  suggest  God  is  a  number?  The  intuitive  response—not  to  mention  the  theological  one—is  likely   to   be   that   it   is   not.   And   until   it   is   established   that   there   cannot   be   different   (non-­‐overlapping)  kinds  of  necessary  existent  this  response  cannot  be  rejected  out  of  hand.

So   what   gives   rise   to   this   intuitive   response?   It   does   not   appear—as   one   might  initially  have  thought—to  arise  from  any  overt   incompatibility  between  the  different  kinds  in  question.  Instead  it  appears  to  result  from  the  fact  that  we  have  no  intellectual  stomach  for   irresolvable  metaphysical   inscrutability.   For   if  Caesar   is   a  number   then   this   identity   is  simply  brute.  The  different  ranges  of  properties  associated  with  being  a  person  and  being  a  number  are  so  distinct  in  kind  that  there  is  nothing  that  might  be  said  to  render  this  identity  transparent   to   the   understanding.   It   is   entirely   opaque   how   a   single   object   could   be   the  subject   of   such   diverse   properties.   Consequently   even   an   exhaustive   investigation   (at   the  limit  of  enquiry)  of  the  personal  properties  of  Caesar  will  not  enable  us  to  decide  whether  

Caesar   is   a   number,   and   if   so,   which   one.   Similarly,   no   amount   of   investigation   of   the  numerical  properties  of  4  will  determine  whether  it  is  also  a  person.

The   issues   surrounding   the   Caesar   problem   encroach   here   upon   traditional,  metaphysical  concerns  about  the  nature  of  substance,  about  what  makes  an  object  a  uniPied,  integrated  whole.    Consider  the  following  remarks  from  Leibniz:  

“I  also  maintain  that  substances  (material  or  immaterial)  cannot  be  conceived  in  their  bare  essence,   devoid   of   activity;   that   activity   is   of   the   essence   of   subject   in   general…it  must   be  borne  in  mind  above  all  that  the  modiPications  which  can  occur  to  a  single  subject  naturally  and   without   miracles   must   arise   from   limitations   and   variations   in   a   real   genus,   i.e.   of   a  constant   and   absolute   inherent   nature….  Whenever  we   Pind   some   quality   in   a   subject,  we  ought  to  believe  that   if  we  understood  the  nature  of  both  the  subject  and  the  predicate  we  would  conceive  how  the  quality  could  arise  from  it.  So  within  the  order  of  nature  (miracles  apart)   it   is   not   at   God’s   arbitrary   discretion   to   attach   this   or   that   quality   haphazardly   to  substances.  He  will  never  give  them  any  which  are  not  natural  to  them,  that  is,  which  cannot  arise  from  their  nature  as  explicable  modiPications.  …what  is  natural  must  be  such  as  could  become   distinctly   conceivable   by   anyone   admitted   into   the   secrets   of   things.”   (Leibniz  [1982],  65-­‐6)    

Two  relevant  thoughts  may  be  distinguished  here.  First,  it  is  claimed  that  substances  cannot  be  conceived  as  bare  particulars.  Second,   it   is  stated  that   the  exhibition  of  properties  by  a  substance   must   somehow   be   rendered   intelligible   by   the   underlying   “real   kind”   of   the  substance.   Both   thoughts   plausibly   militate   against   the   identiPication   of   Caesar   with   a  number.     For   if   Caesar   is   a   number   then   the   subject   that   underlies   the   relevant   personal  properties  and  the  subject  that  underlies  the  relevant  numerical  properties  can  be  no  more  than  barely  identical.  Moreover,  if  it  is  Caesar’s  nature  to  be  human  then  he  cannot  also  be  a  number.  For  the  possession  of  numerical  properties   is  rendered  not  whit   intelligible  by  an  underlying   human   nature.   Obviously   these   rePlections   present   no   decisive   case.   It   is  arguable  that  (in  certain  limit  cases)  bare  identities  may  be  properly  admitted.  Moreover,  if  one  is  willing  to  admit  such  conjunctive  kinds  as  being  a  person  and  a  number  then  Caesar’s  underlying  nature  will  render  his  possession  of  numerical  qualities   intelligible  after  all.  Of  course,  this  raises  the  question  of  whether  conjunctive  kinds  are  themselves  intelligible.  

More   signiPicantly,   Leibniz’   views   on   substance   Plow   from   his   endorsement   of   the  principle   of   sufPicient   reason—a   principle   that   demands   the   intrinsic   intelligibility   of   the  universe.   But   since   we   have   jettisoned   the   principle   it   is   difPicult   to   see   how   it   can   be  maintained  that  the  world  ought  to  be  intrinsically  intelligible.  The  demand  that  the  world  should   conform   to   the   patterns   of   our   thoughts   about   it,   that   it   should   be   transparent   to  even   our   idealised   understanding,   appears   no   more   than   conceit.   Should   we   therefore  continue   to   maintain   Sortal   Exclusion   or   should   we   be   prepared   to   simply   leave   it   open  thatfor  all  that  we  knowCaesar  is  a  number?

2.5  The  Meaning-­‐theoretic  Caesar  problem.  This  version  of  the  Caesar  problem  questions  whether  (HP)  succeeds  in  conferring  content  on  the  natural  number  expressions  it  purports  to   introduce.  Frege   initially  sought   to   introduce  numerical   terms  (“Nx:Fx”)  contextually  by  Pixing  the  content  of  identity  sentences  in  which  they  occur.  However,  (HP)  fails  to  settle  the  content  of  all  the  identity  contexts—speciPically  contexts  of  the  form  “Nx:Fx  =  q”—in  which  the   introduced   expressions   feature.   Recall:   (HP)   simply   fails   to   say   anything   about   the  signiPicance   of   contexts   that   feature   a   singleton   occurrence   of   the   numerical   operator.  Therefore,   (HP)   fails   to   bestow   the   signiPicance   of   singular   terms   upon   the   expression   it  introduces.

In  fact,  this  version  of  the  Caesar  problem  is  multiply  ambiguous.  It  all  depends  upon  what  “content”  is  taken  to  mean.  If  “content”  means  sense  then  the  complaint  comes  down  to  this.   (HP)   fails   to   determine  whether   identity   contexts   of   the   form   “Nx:Fx   =   q”   have   any  sense.   This   calls   into   question   whether   (HP)   Pixes   a   sense   even   for   sentences   of   the  superPicially   tractable   form   “Nx:Fx   =   Nx:Gx”.  When   it   is   articulated   in   terms   of   sense   the  meaning-­‐theoretic   version   of   the   Caesar   problem   may   be   developed   in   the   following  manner.  

Frege   sought   to   introduce   numerical   terms   by   Pixing   the   truth   conditions   of   the  identity  sentences  in  which  they  occur.  However,  if  this  method  of  Pixing  truth  conditions  is  to  result  in  the  introduction  of  genuine  numerical  termssingular  terms  that  purport  to  stand  for  objectsthen  the  sentences  of  the  form  “Nx:Fx  =  Nx:Gx”  whose  truth  conditions  are  Pixed  must  be  genuinely  logically  complex.  Sentences  of  this  form  must  be  understood  as  saying  of  Nx:Fx   that   it   satisPies   the   predicate   “…   =   Nx:Gx”.   Consequently,   (HP)   will   only   succeed   in  conferring   individual   signiPicance   on   the   component   numerical   terms   of   the   sentences  whose  truth  conditions   it   Pixes   if   it  also  determines  a  meaning   for  such  predicates  as  “…  =  Nx:Gx”   and   “Nx:Fx   =   …”.   To   achieve   this   (HP)   must   also   Pix   truth   conditions   for   all   the  sentences   in   which   these   predicates   occur.   So   (HP)   must   also   Pix   truth   conditions   for  sentences   of   the   form   “Nx:Fx  =  q”   (where   “q”   is   any   singular   term  whatsoever).   But   (HP)  only   Pixes   truth   conditions   for   sentences   of   the   form   “Nx:Fx  =  Nx:Gx”.   It   does  not   provide  truth  conditions  for  sentences  of  any  other  form.  Therefore,  (HP)  fails  to  provide  a  basis  for  supposing   “Nx:Fx   =   Nx:Gx”   is   logically   complex   and   that   the   expressions   it   contains   are  genuine  singular  terms.

However,   if   “content”   means   reference   then   the   complaint   is   quite   different.   The  failure  of  (HP)  to  settle  truth-­‐values  for  the  identities  “Nx:Fx  =  q”  is  interpreted  as  a  failure  to  determine  a  dePinite  referent  for  each  of  the  numerical  expressions  introduced.  It  is  then  doubted   whether   expressions   of   the   form   “Nx:Fx”   are   referential   in   the   Pirst   place.  

Benacerraf  famously  propounded  a  related  argument  (see  his  [1965]  and  Kitcher  [1975]).  In  this  particular  case  set-­‐theoretic  terms  are  taken  as  values  of  “q”.  According  to  Zermelo’s  set-­‐theoretic  dePinition  of  natural  number,  0  is  the  empty  set  and  the  successor  function  takes  x  to   the   unit   set   of   x.   Von   Neumann   dePined   the   natural   numbers   a   different  way:   0   is   the  empty   set   but   the   successor   function   takes   x   to   the   union   of   x   and   the   singleton   of   x.  Benacerraf  argued   that   the  use  of  arithmetical  vocabulary   fails   to   settle  whether  ordinary  numerals  refer  to  the  Zermelo  numbers  or  the  von  Neumann  numbers  (whether  “2  =  {{}}”  or  “2  =  {,  {}}”  is  true).  The  problem  is  that  each  of  these  set-­‐theoretic  progressions  serves  as  an  equally   effective  model   of   the   number   theory   embodied   in   ordinary   usage.   He   concluded  that   the   semantic   function   of   ordinary   arithmetical   expressions   must   be   other   than  referential.  

The   problem   that   Benacerraf   isolates   for   ordinary   numerical   terms   is   often  assimilated   to   the   Caesar   problem   itself   (see,   for   example,   Shapiro   [1997],   pp.   78-­‐81).  However,  this  would  be  a  mistake  for  several  reasons.  As  we  have  seen,  the  Caesar  problem  has   epistemological   and   metaphysical   dimensions   that   Benacerraf’s   argument   fails   to  capture.  But  even  if  attention  is  focused  solely  upon  the  meaning-­‐theoretic  Caesar  problem  there  are  other  reasons  to  resist  the  assimilation.  To  begin  with  the  Caesar  problem  Frege  confronted  concerned  the  signiPicance  of  expressions  (“Nx:xx”,  “Ny:[y  =  Nx:xx]”…)  artiPicially  introduced   by   means   of   a   dePinition   (HP)   whereas   Benacerraf’s   problem   concerns   the  signiPicance   of   ordinary   terms   (“0”,   “1”…)   that  may   or  may  not   have   been   introduced   this  way.  Putting   this   issue  aside,   the   ‘sense’   and   ‘reference’   versions  of   the  meaning-­‐theoretic  Caesar  problem  need  to  be  kept  separated.  The  latter  species  of  argument  does  not  deny  the  signiPicance   of   contexts   of   the   form   “Nx:Fx   =   q”   but  moves   from   the   existence   of   distinct  eligible  referents  for  the  same  numerical  terms  to  the  conclusion  that  the  semantic  function  of   these   expressions   (“Nx:Fx”)   cannot   be   referential.   By   contrast,   the   former   sort   of  argument   moves   from   the   failure   of   (HP)   to   address   the   status   of   “Nx:Fx   =   q”   to   doubt  whether  such  contexts  are  signiPicant  at  all.  

Despite   the   important  differences   that  obtain  between   these  divergent  versions  of   the  meaning-­‐theoretic   Caesar   problem   they   are   subject   to   a   generic   doubt.   The   problematic  character  of  the  relevant  class  of  Caesar  problems  is  revealed  in  the  high  semantic  threshold  they  each  impose  on  the  introduction  of  genuine  singular  terms:

iv) Semantic  Threshold:  in  order  to  confer  signiPicance  on  the  terms  and  predicates  purportedly  introduced  by  (HP)  their  signiPicance  should  be  everywhere  Pixed.

The   obvious   doubt   to   entertain   here   is   whether   these   differing   versions   of   the   Caesar  

problem   impose   too   high   a   threshold.   More   or   less   extreme   forms   of   this   doubt  may   be  entertained.  For  example,  it  may  be  claimed  that  no  context  of  the  form  “Nx:Fx  =  q”  requires  to  have  its  signiPicance  Pixed  in  order  to  introduce  singular  numerical  terms.  A  related  view  is   evidenced   in   Carnap’s   contention   that   such   ‘mixed’   contexts   that   feature   both  mathematical  and  non-­‐mathematical  terms  are  actually  nonsense:

“2.   “Caesar   is   a   prime   number”…(2)   is   meaningless.   “Prime   number”   is   a   predicate   of  numbers;   it   can   neither   be   afPirmed   or   denied   of   a   person….The   fact   that   the   rules   of  grammatical   syntax   are   not   violated   easily   seduces   one   at   Pirst   glance   into   the   erroneous  opinion   that   one   has   still   to   do   with   a   statement,   albeit   a   false   one.   But   “a   is   a   prime  number”  is  false  iff  a  is  divisible  by  a  natural  number  different  from  a  and  from  1;  evidently  it   is   illicit   to  put  here  “Caesar”   for  “a”.  This  example  has  been  chosen  that   the  nonsense   is  easily  detectable….”.  (Carnap  [1932],  pp.  67-­‐8)

If   such  mixed   contexts   are  nonsense   then   it   can  hardly   be   an   adequacy   constraint   on   the  introduction  of  numerical   terms  that  a  meaning   is   Pixed   for   these  contexts.  A   less  extreme  doubt  will  discriminate  between  different  sentences  of  the  form  “Nx:Fx  =  q”  where  “q”  takes  different  sorts  of  values.   It  may  be   that   there   is  no  need   for   (HP)   to   Pix   the  signiPicance  of  some  of  these  contexts  in  order  to  effect  the  introduction  of  genuine  singular  terms.  It  may  also   be   that   genuinely   signiPicant   sentences   that   take   different   values   for   “q”   generate  different  obstacles  for  the  introduction  of  numerical  singular  terms.  

Examination  of  (an  admittedly)  provisional  schedule  of  the  different  values  “q”  may  take  reveals  the  range  of  distinct  issues  involved.  First,  “q”  may  takes  values  that  in  advance  of   a   consideration   of   the  Caesar   problem  we  might   have   to   taken   to   denote   paradigmatic  extra-­‐mathematical  objects  (“Nx:Fx  =  Caesar”).  A  distinction  may  also  be  drawn  between  the  extra-­‐mathematical  cases  that  feature  reference  to  contingent  as  opposed  to  necessary  non-­‐mathematical  entities  (“Nx:Fx  =  the  Earth’s  axis”,  “Nx:Fx  =  the  True”).  Alternatively,  “q”  may  take   values   that   denote   objects   characteristic   of   the  mathematical   domain.   Some  of   these  cases   will   feature   reference   to   elements   of   progressions   that   plausibly   might   have   been  taken   to   be   mathematical,   but   not   distinctively   numerical,   progressions   ((“Nx:Fx   =   {}”)  (Benacerraf’s  examples  may  be   located  here).  Others  will   involve  reference  to  elements  of  

numerical,   but   not   distinctively   arithmetical,   series   (“Nx:Fx   =   2real”).   Finally,   there   is   the  

special   case   where   the   identities   in   question   concern   the   relation   between   the   objects  denoted  by  the  putatively  arithmetical  terms  (HP)  introduces  and  the  objects  denoted  by  the  

numerals  of  ordinary  arithmetic  (“Nx:Fx  =  2natural”).  The  ability  to  settle  the  signiPicance  of  

one  of  these  different  forms  may  not  result  in  an  ability  to  settle  the  signiPicance  of  another.  For  example,  we  may  be  able  to  determine  that  “Nx:Fx”  refers  to  a  mathematical  rather  than  

a  non-­‐mathematical  object.  But  then  we  may  be  unable  to  determine  whether  it  refers  to  an  item  drawn  from  one  rather  than  another  mathematical  progression.

These  are  not  the  only  cases  a  consideration  of  which  may  be  expected  to  shed  light  upon   the   signiPicance   of   numerical   terms.   There   are   particular   concerns   about   reference  generated   by   the   special   character   of   series   that   exhibit   non-­‐trivial   automorphisms  (Brandom  [1996]).  For  example,  in  complex  number  theory  –1  has  two  square  roots  (i  and  –i)  But  there  is  no  way  to  settle  within  the  theory  which  square  root  our  use  of  the  signs  “i”  or  “-­‐i”   denotes.   It   seems   that  we   cannot   settle   the   reference   of   these   terms.   This   is   because  every  predicate   (not   containing   “i”  or   “-­‐i”)   that   is   true  of  one  of   them   is   true  of   the  other.  There  are  also  general  concerns  about  reference  that  apply  irrespective  of  the  character  of  the  series  in  question  (Hodes  [1984],  pp.  134-­‐5).  For  the  sake  of  argument,  suppose  that  the  series  of  natural  numbers  N  have  been  singled  out  as  the  referents  of  the  ordinary  numerals.  Then  an  alternative  progression  may  be  formed  from  N  that  serves  just  as  well  as  a  source  of  eligible   referents   for   ordinary   numerals.   To   see   this   we   need   merely   permute   a   Pinite  number  of  elements  of  N  and  make  compensating  adjustments  to  the  successor  function.  For  example,  we  might  let  “4”  designate  5  and  “5”  designate  4  and  employ  “successor”  to  stand  for  the  function  that  differs  from  the  successor  function  only  in  assigning  3  to  5,  5  to  4  and  4  to  6.  Since  there  are  indePinitely  many  ways  of  so  permuting  the  elements  of  N  there  is  no  telling  which  number  is  referred  to  by  a  given  numeral.  

It  is  sometimes  thought  that  all  these  different  concerns  are  expressive  of  the  same  problem,  the  Caesar  problem.  This  would  be  a  mistake.  The  Caesar  problem  originally  arose  as   a   result   of   the   inability   of   (HP)   to   settle   the   signiPicance  of   identity   claims  of   the   form  “Nx:Fx   =   q”.   However,   the   concerns   that   have   just   been   raised   about   the   reference   of  numerical   terms   are   not   occasioned   by   any   doubt   about   whether   some   identity   is  meaningful   or   true.   In   the   former   case   they   turn   upon   the  mathematical   character   of   the  complex  number  series.  In  the  latter  case,  the  difPiculty  raised  does  not  concern  agreement  or  disagreement  about  some  object  language  sentence.  So,  for  example,  it  is  accepted  that  “4  =  4”  is  true  whereas  “4  =  5”  is  false.  The  difPiculty  confronted  stems  from  another  source.  It  stemsPiguratively  speakingfrom  stepping  back  from  our  own  language  once  the  truth-­‐values  of   all   the   sentences   have   been   settled   and   then   considering   the  myriad   different  ways   in  which  it  may  be  reinterpreted.   It   follows  that  a  happy  solution  to  the  Caesar  problem  that  generates   truth   conditions   and   values   for   sentences   of   the   form   “Nx:Fx   =   q”   cannot   be  expected  to  help  (directly)  in  resolving  more  recherché  concerns  of  this  sort.  Correlatively,  the  inability  of  a  given  solution  of  the  Caesar  problem  to  settle  issues  that  arise  once  truth  conditions   and   values   have   been   settled   need   cast   no   doubt   upon   the   credentials   of   the  resolution  in  question  qua  provider  of  truth  conditions  and  values.

3.  Two  SolutionsA  Pirst  encounter  with  the  Caesar  problem  often  occasions  a  denial,  a  denial  that  there  is  any  signiPicant  problem  to  be  addressed.  We  are  so  convinced  that  there  is  something  amiss  with  the   identiPication   of   Caesar   and   a   number   that   it   often   takes   a   good   deal   of   theoretical  orientation  before  it  is  even  appreciated  that  (HP)  fails  to  secure  the  result  that  Caesar  is  no  number.   The   preceding   discussion   sketched   in   a   preliminary   way   some   of   the   different  issues  that  underlie  the  Caesar  problem.  It  is  to  be  hoped  that  sufPicient  structure  has  been  imposed  to  enable  us  to  question  the  character  of  our  pre-­‐theoretic  conviction  and  see  that  many  distinct  epistemological,  metaphysical  and  meaning-­‐theoretic   forces  may  be  at  work  inducing  the  belief  that  Caesar  is  no  number.  A  failure  to  appreciate  or  effectively  treat  of  its  many  different  dimensions  undermines  several  proposed  solutions  to  the  Caesar  problem.  Two  such  solutionssupervaluationism  and  neo-­‐Fregeanismwill  be  examined  here.

3.1  The  Supervaluationist  Solution.  In  its  most  familiar  guise,  supervaluationism  provides  a   method   for   dealing   with   the   semantic   phenomenon   of   vague   predicates   (Fine   [1975]).  There  are  (apparently)  no  sharp  boundaries  between  the  objects  to  which  vague  predicates  apply  and  those  objects  to  which  they  do  not  apply.  Vague  predicates  have  borderline  cases  where   they   neither   clearly   apply   nor   fail   to   apply.   Nevertheless,   these   predicates  may   be  ‘precisiPied’:   a   sharp   boundary   may   be   Pixed   for   their   application.   But   there   are   many  different  ways  of  precisfying  a  vague  predicate  and   it  would  be  arbitrary  to  choose  one  to  express  the  ‘real  meaning’  of  the  predicate.  So  the  supervaluationist  attempts  to  account  for  the  phenomenon  of  borderline  cases  by  taking  into  account  all  the  possible  precisiPications  of  a  vague  predicate.  According  to  the  supervaluationist  account,  a  sentence  is  true  iff   it   is  true   on   all   precisiPications   of   its   constituent   vague   expressions,   false   iff   it   is   false   on   all  precisiPications  and  neither  true  nor  false  iff  it  is  true  on  some  but  not  other  precisiPications.  The   supervaluationist   interprets   vagueness   as   a   species   of   semantic   indeterminacy.  Predicates  do  not  turn  out  to  be  vague  because  they  apply  to  vague  objects.  They  turn  out  to  be   vague   because   language   users   have   not   chosen   between   different   possible  precisiPications  of  them.  

The   inability   of   (HP)   to   settle   whether   Caesar   is   a   number   may   likewise   be  interpreted  as   a   consequence  of   semantic   indeterminacy   (c.f.   Field   [1974],  McGee   [1997],  Shapiro   [this   volume]).   (HP)   determines   that   the   numerical   terms   it   introduces   refer   to  numbers   (so  presented)  but   fails   to  determine  whether  objects  otherwise  depicted  are  so  picked  out.  This  is  because  (HP)  is  a  semantically  indeterminate  principle.  There  are  many  different  possible  precisiPications  of  it  and  (HP)  does  not  select  between  them.  According  to  

some   precisiPications   of   (HP),   Caesar   is   a   number;   according   to   others,   Caesar   is   not.  Consequently,  it  is  neither  true  nor  false  that  Caesar  is  a  number.  Of  course,  it  is  open  to  us  to  constrain   the   admissible   precisiPications   of   (HP).  We  may   choose   to   unite   (HP)   with   the  additional   principle   that   no  Nx:Fx   is   a   person.   Then   there  will   be   no   precisiPication   upon  which  Caesar  is  picked  out  by  a  numerical  term.  It  will  be  false  that  Caesar  is  a  number.  

Viewed   from   the   supervaluationist   perspective   Frege   overreacted   to   the   Caesar  problem.  Frege  interpreted  the  inability  of  (HP)  to  settle  whether  Caesar  was  a  number  to  be   a   symptom   of   an   underlying   malady,   the   failure   of   (HP)   to   introduce   referring  expressions.   But   really   what   was   signalled   by   the   inability   of   (HP)   to   settle   a   dePinite  reference   for  number  words  was   the   indeterminacy  of   the   terms   it   introduced.  To  use   the  terminology  of  the  previous  section,  Frege  simply  set  the  semantic  threshold  for  introducing  referential   expressions   too   high.   He   required   that   referring   expressions   must   refer  determinately  and  therefore  failed  to  recognise  that  the  terms  introduced  by  (HP)  referred  indeterminately.

In  order  to  appreciate  some  of  the  problems  that  attend  a  supervaluationist  account  it   is   useful   to   consider   a  more   simple   and   direct   approach   to   the   Caesar   problem.  When  rePlecting   upon   the   inability   of   (HP)   to   determine   a   truth-­‐value   for   the   sentence   “Julius  Caesar  is  the  number  of  planets”,  Dummett  once  suggested  that  the  difPiculty  could  be  swept  aside  with  ease.  He  wrote:  “it  would  be  straightforward  to  provide  by  direct  stipulation  for  the   falsity   of   such   sentences”   (see   his   [1967],   p.   111).   There   are   two   relevant   difPiculties  associated  with  this  proposal.  

First,   it   is   unclear   how   a   direct   stipulation   that   sufPiced   for   the   falsity   of  all   such  sentences   might   be   constructed.   Of   course,   it   is   true   that   no   numerical   thing   is   a   non-­‐numerical  thing.  But  this  doesn’t  need  a  stipulation  to  make  it  so.  Moreover,  it  must  not  be  forgotten   that   what   is   at   issue   is   the   reference   of   numerical   terms   and   the   application  conditions   of   numerical   predicates.   So   this   truthwhose   sentential   expression   makes   play  with  the  relevant  class  of  problematic  vocabulary  the  reference  and  application  of  which  are  in  questionprovides  no  guide  to  whether  a  given  sentence  is  true  or  false  as  a  consequence  of   it.   This   suggests   that   it   might   be   better   to   proceed   piecemeal,   providing   a   range   of  stipulations   to   distinguish  numbers   from  different   sorts   of   non-­‐numerical   objects.   But,   as  we   have   seen,   it   is   no   easy   matter   to   settle   whether   numbers   do   or   do   not   possess   a  featurefor  example,  contingency  or  concretenesscharacteristic  of  a  given  sort  (section  2.4).  

This   Pirst  difPiculty   is  a  clue  to  the  second.  Suppose  that  Caesar   leads  a  double   life.  Suppose  that   in  addition  to   leading  his  material  existence  Caesar   is  also  a  number.   In   that  case  the  stipulation  that  sentences  that  say  Caesar  is  a  number  are  all  false  cannot  succeed.  For  some  of  these  sentences  will  be  true  and  true  sentences  cannot  be  stipulated  to  be  false.  

So  Dummett’s  strategy  of  directly  stipulating   the   falsity  of   the  relevant  range  of  sentences  presupposes  that  Caesar  is  no  number.  Stipulation  cannot  sufPice  as  a  basis  for  determining  that  Caesar  is  no  number.

A   similar   difPiculty   afPlicts   the   more   sophisticated   supervaluationist   strategy.  According   to   the   supervalutionist,   the   inability   of   (HP)   to   settle   the   truth-­‐value   of   such  sentences   as   “Julius   Caesar   is   the   number   of   planets”   is   a   consequence   of   semantic  indeterminacy.  In  other  words,  (HP)  may  be  precisiPied  in  a  number  of  arbitrary  ways  and  some  of   these  precisiPications  make  such  sentences   true,  others  make   them   false.  But   this  assumes  that  it  is  legitimate  to  precisfy  the  numerical  vocabulary  introduced  in  such  a  way  as  make  it  false  that  Caesar  is  a  number.  But  suppose  again  that  Caesar  is  a  number.  Suppose  that   facts  about  his   identity  and  distinctness   from  every  other  number  are  determined  by  facts  about  1-­‐1  correspondences  between  concepts.  Then  there  are  no  precisiPications  upon  which  Caesar   is  not  a  number.  Any  attempt   to  precisify   (HP)   in   this  way  will   conPlict  with  underlying  metaphysical  facts.  Consequently,  it  will  be  inappropriate  to  apply  the  semantic  machinery  of  supervaluationism  to  explicate  the  apparent  indeterminacy  of  the  sentences  in  question.   It   will   not   be   the   case   that   there   are   some   precisiPications   upon   which   these  sentences  are  true  and  some  precisiPications  upon  which  they  are  false.  The  same  difPiculty  will  attend  the  attempt  to  precisify  (HP)  by  adjoining  further  stipulations  (for  example,  that  no  number  is  contingent).

What  this  reveals  is  that  both  the  direct  stipulation  strategy  and  supervaluationism  rest  upon  a  common  assumption,   the  assumption  that  (HP)   is  semantically   indeterminate.  They   are   motivated   by   the   idea   that   the   vocabulary   introduced   by   (HP)   is   semantically  neutral   in   the   sense   that   it   enjoins   no   commitments   that   might   conPlict   with   antecedent  facts.   It   is  because,   they  presume,   the   introduced  vocabulary   is   free  of   such  commitments  that   it   is  possible   to  stipulate  or  precisfy   its  use  without   there  being  any  risk  of  offending  against  any  antecedent  fact.  If,  however,  the  vocabulary  introduced  fails  to  be  neutral  in  this  regard—if   the   use   stipulated   for   it   may   conPlict   with   antecedent   facts—then   the  mechanisms  employed  to  resolve  the  indeterminacy  will  misPire.  It  follows  that  we  can  have  no  assurance   that  direct   stipulation  or   supervaluationism  succeed   in   resolving   the  Caesar  problem  in  the  absence  of  an  argument  that  (HP)  is  semantically  indeterminate.  But  neither  purported  solution  shows  this.  They  suppose  (HP)   is  semantically   indeterminate  and  then  seek  to  accommodate  that  happenstance.

The   point   deserves   emphasis.   (HP)   fails   to   explicitly   address   the   signiPicance   of  identity   contexts   of   the   form   “Nx:Fx   =   q”.   The   direct   stipulation   and   supervaluationist  strategies  construe  this  silence  to  be  a  symptom  of  the  semantic  indeterminacy  of  the  terms  (HP)   introduces.  But  the  silence  of  (HP)  may  be   interpreted  differently.   It  may  be  taken  to  

rePlect  the  epistemological  inscrutability  of  impure  identities,  the  fact  that  we  can  just  never  know  whether  Caesar  is  a  number.  Alternativelyand  these  do  not  exhaust  the  alternativesthe  silence  may  betoken  the  meaninglessness  of  impure  identities,  the  failure  of  (HP)  to  Pix  any  signiPicance   for   identities   of   this   form.   In   the   former   case,   the   terms   introduced   have   a  dePinite  (albeit  unknown)  reference.  In  the  latter  case,  the  terms  have  no  sense  and  do  not  refernot   even   indePinitely.   In   either   case   the   imposition   of   a   supervaluationist   semantics  upon  contexts  of  the  form  “Nx:Fx  =  q”  will  fail  to  remedy  the  underlying  malady.

One  of  the  most  signiPicant  tasks  facing  any  resolution  of  the  Caesar  problem  is  that  of  determining  the  character  of  the  problem  itself.  Call  this  the  circumscription  problem,  the  problem  of  circumscribing  the  character  of  the  problem  that  demands  resolution.  The  direct  stipulation  and  supervaluationist  strategies  fail  to  address  this  problem.  They  assume  rather  than  show  that  the  Caesar  problem  has  a  certain  character  (semantic  indeterminacy).  As  a  result   the   direct   stipulation   and   supervaluationist   strategies   fail   to   resolve   the   Caesar  problem.  

3.2   The   Neo-­‐Fregean   Solution.   By   contrast   to   the   supervaluationist,   the   neo-­‐Fregean  proposes   a   solution   to   the   Caesar   problem   that   relies   upon   a   distinctive   “philosophical  ontology”  (Hale  &  Wright  [2001],  pp.  385-­‐96,  [this  volume]).  The  solution  is  framed  in  the  context  of  a  theory  of  categories.  A  category  is  a  collection  of  objects  that  share  a  common  criterion   of   identity.   Different   categories   are   distinguished   by   the   different   criteria   of  identity  associated  with  them.  Now  consider  the  possibility  that  an  object  belonging  to  one  category   is   identical   to   an   object   drawn   from   another.   The   neo-­‐Fregean   claims   that   such  trans-­‐sortal  identiPications  possess  a  distinctive  epistemological  feature:  “there  is  simply  no  provision   for   or   against   such   identities”;   there   are   no   encompassing   identity   criteria  available  that  would  allow  us  to  settle  whether  objects  drawn  from  distinct  categories  are  the  same  or  different  (Hale  &  Wright  [2001],  p.  394).  

On   the   basis   of   the   claim   that   trans-­‐categorical   are   evidence   transcendent   in   this  way   the   neo-­‐Fregean   presents   a   dilemma.   Either   it   is   granted   that   there   are   true   trans-­‐categorical  identities  or  it  is  not.  If  there  are  such  identities  then  Frege  may  be  convicted  of  overestimating  the  gravity  of  the  Caesar  problem.  For  this  so-­‐called  ‘problem’  arises  from  the  inability   of   (HP)   to   settle   a   statement   concerning   the   identity   of   objects   drawn   from  different  categories  (persons  and  numbers).  But   if   it   is  a  general  truth  that  such  identities  cannot  be  settled  then  it  can  signal  no  defect  in  (HP)  that  it  fails  to  settle  a  truth-­‐value  for  impure   identity   claims.   Alternatively,   it   may   be   denied   that   there   are   any   true   trans-­‐categorical   identities.  But   then  Frege  may  be   convicted  of  underestimating   the   capacity  of  (HP)   to   solve   the   Caesar   problem.   For   the   criteria   of   identity   that   (HP)   stipulates   for  

numbers  are  distinct  from  the  characteristic  criteria  of  personal  identity  (whatever  package  of   psychological   and   bodily   conditions   that   might   be).   Therefore   persons   and   numbers  belong  to  different  categories  and  this  fact  sufPices  for  their  numerical  difference.  So  either  the  Caesar  dissolves  or   it   is   solved.  Either  way,  Frege   failed   to   show   that   (HP)  provided  a  defective  mechanism  for  introducing  numbers.

The   details   of   the   neo-­‐Fregean   view   are   clearly   open   to   question.   Consider,   for  example,  the  neo-­‐Fregean  claim  that  trans-­‐categorical  identities  are  evidence  transcendent.  This  claim  is  motivated  by  the  rePlection  that  if  it  is  legitimate  to  countenance  the  identity  of  Caesar  with  a  number  then  it  is  equally  legitimate  to  countenance  the  identity  of,  say,  Frege  with  a  Roman  statue  (an  object  drawn  from  another  distinct  category).  They  both  constitute  cases  of  bare,   imponderable   identities   (Hale  &  Wright   [2001],   p.   394).  But   if   persons  and  artefacts   turn   out   to   fall   under   a   common   categorythe   category   of   physical   objectthen   it  remains  opens  that  the  facts  of  identity  and  distinctness  amongst  persons  and  artefacts  may  be  settled  by  veriPiable  considerations  (for  example,  spatio-­‐temporal  duration  and  location).  Clearly,   the   relevant   notion   of   category   requires   greater   development   before   the   neo-­‐Fregean  proposal  can  be  properly  assessed  (see  Wright  &  Hale  [this  volume],  section  IV  for  further  developments).

The   claim   that   all   trans-­‐categorical   identities   share   an   evidential   status   may   be  questioned  for  another  reason.  Suppose  Frege  has  mass  m  and  the  Roman  statue  mass  n  m.  Then  it  is  natural  to  reason  in  the  following  way:  nothing  can  have  mass  of  both  n  and  n  m  values;  so  Frege  and  the  statue  cannot  be  identical.  If  we  are  to  countenance  the  possibility  that  an  arbitrary  object  (Frege)   is  really   identical   to  an  object   located  at  another  place  (at  the  same  time)  with  different  intrinsic  properties  then  this  line  of  reasoning  will  have  to  be  shown  to  be  somehow  at  fault.   I  have  described  this  elsewhere  are  the   ‘problem  of  spatial  intrinsics’  (by  analogy  with  the  more  familiar  problem  of  temporary  intrinsics;  see  Macbride  [1998],  pp.  223-­‐7  for  further  details).  To  solve  this  problem  we  must  give  up  some  ordinary  assumptions   about   intrinsic  property  possession.  We  will   have   to   give  up   the   assumption  that  intrinsic  properties  (mass,  shape  etc.)  are  possessed  simpliciter,  that  is,  independently  of  spatial   location.   Instead  we  will  have   to   think  something  of   the   following  sort:   intrinsic  properties  are  possessed  relative  to  spaces  (and  perhaps  times  too).  And  then  there  will  be  no   incompatibility   generated   by   Frege   possessing  mass   and   shape   relative   to   the   slice   of  space-­‐time  carved  out  by  his  life,  and,  another  mass  and  shape  relative  to  the  duration  and  location  of   the  Roman  statue.  What   this  suggests   is  a  surprising  result.  Far   less  damage   is  done   to   our   ordinary   ways   of   thinking   by   countenancing   the   possibility   of   an   identity  between  numbers  and  persons  than  by  seriously  entertaining  the  idea  that  different  sorts  of  physical  objects  might  be   identical.  By  contrast   to   the   latter,   the   identiPication  of  numbers  

and   persons   does   not   force   any   revision   or   particular   view   of   the  way   in  which   intrinsic  properties  are  possessed.  This  rePlects  once  again  a  beguiling  aspect  of  the  Caesar  problem  noted   earlier:   the   fact   that   there   is   no  overt   incompatibility  between  being  a  number   and  being  a  person.

Of  course,  these  are  considerations  of  detail  that  may  very  well  be  addressed  in  the  context  of  a   fuller  development  of   the  neo-­‐Fregean  approach.  Nevertheless,   it   is  worthy  of  note   that   there   are   such   details   to   be   negotiated.   For   the   neo-­‐Fregean   intends   their  philosophical  ontology  to  be  placed  at   the  service  of  a   logicist  philosophy  of  mathematics.  The   greater   the   metaphysical   depths   the   neo-­‐Fregean   must   fathom   to   make   good   their  claims  the  less  likely  it  appears  that  the  relevant  theory  of  categories  should  draw  on  merely  logical  concepts  and  techniques  for  its  expression.

Independently  of  such  considerations  how  well  does  the  neo-­‐Fregean  solution  fare  with   respect   to   the   critical   task   of   negotiating   the   various   different   aspects   of   the   Caesar  problem?   How   does   the   neo-­‐Fregean   solution   fare   with   respect   to   the   circumscription  problem?  Unlike  the  supervaluationist  the  neo-­‐Fregeans  do  not  assume  that  contexts  of  the  form   “Nx:Fx   =   q”   are   semantically   indeterminate.   But   they   do   make   the   contrasting  assumption   that   such   contexts   are   meaningful   and   determinate.   Having   made   that  assumption  the  neo-­‐Fregean  then  sets  about  demonstrating  that  either  the  truth  values  of  impure   contexts   are   imponderable   or   settled   by   category-­‐theoretic   consideration.   Recall,  however,   the   version   of   the   Caesar   problem   that   denied   (HP)   supplied   impure   identities  with   any   meaning   whatsoever   (section   2.5).   In   that   case,   contrary   to   the   neo-­‐Fregean  solution,  impure  identities  cannot  have  any  sort  of  truth-­‐value,  unfathomable  or  otherwise.  Unfortunately,  the  neo-­‐Fregean  fails  to  address  the  circumscription  problem.  As  a  result  the  neo-­‐Fregean  fails  to  provide  a  solution  to  the  Caesar  problem.

However,   the  neo-­‐Fregean  does   offer   two   arguments   to   undermine   the   contention  that   one   might   rest   content   with   the   situation   that   Carnap   was   willing   to   toleratethe  situation  where  impure  identities  lack  a  truth  condition  or  value  (Hale  &  Wright  [2001],  pp.  340-­‐5).   One   argument   operates   at   the   level   of   understanding   and   appeals   to   Evans’  Generality   Constraint   (Evans   [1982],   pp.   100-­‐5).   Construed   as   a   linguistic   principle   this  constraint   exercises   a   control   on   the   understanding   of   sentences.   To   understand   an  expression  is   to  understand  the  contribution  it  makes  to  the  meaning  of  all   the  signiPicant  sentential  contexts  in  which  it  occurs.  So  a  subject  may  only  grasp  a  particular  sentence  if  he  or   she   grasps   the   range   of   signiPicant   sentences   that   result   from   the   permutation   of  understood   constituents.   Consider   the   possibility   currently   at   issue:   that   a   subject   may  understand   a   range   of   pure   numerical   identities   (“Nx:Fx   =   Nx:Gx”)   and   pure   personal  identities   (“Caesar   =   Julius”)   but   fail   to   comprehend   the   signiPicance   of   mixed   identities  

(“Nx:Fx  =  Caesar”).  The  Generality  Constraint  appears  to  rule  this  possibility  out.  For  if  the  subject   “fully   understands”   the   pure   identities   then   they   must   also   understand   the  sentences   that   result   from   the   permutation   of   the   constituent   terms   “Nx:Fx”,   “Nx:Gx”,  “Caesar”,   “Julius”,   and   “…=…”.   But   impure   identities   occur   amongst   the   results   of   such   a  permutation.   So   it   cannot   be   the   case   that   (HP)   serves   to   Pix   the   signiPicance   of   pure  numerical  contexts  whilst  neglecting  entirely  the  signiPicance  of  impure  contexts.

This  argument   is  open  to  question.  First,  maintaining  the  discussion  at   the   level  of  understanding,  it  may  be  granted  that  having  a  full  understanding  of  pure  identities  requires  that   a   subject  must  be  able   to  understand  all   the   signiPicant  permutations  of   themimpure  identities   included.   But   suppose   that   (HP)   provides   only   a  partial   understanding   of   pure  numerical   identities.   In   that   case   there  need  be  no   conPlict  with   the  Generality  Constraint.  For  a  subject  whose  understanding  of  pure  numerical  identities  is  mediated  by  (HP)  and  yet  fails   to   comprehend   the   signiPicance   of   impure   identities   need   be   a   subject   with   only   a  partial  understanding  of  the  signiPicance  of  pure  identities.  Second,  the  argument  stands  in  need  of  qualiPication.  The  Generality  Constraint  does  not  require  that  a  subject  understand  the   range   of   grammatical   sentences   that   result   from   the   permutation   of   understood  constituents.   It   requires   only   that   a   subject   understand   the   resulting   range   of   signiOicant  sentences.  Suppose  impure  identities  even  though  grammatical  are  meaningless.   It   follows  that  the  Generality  Constraint  fails  to  rule  out  the  possibility  of  understanding  pure  but  not  impure  identities.

The  neo-­‐Fregean  argues  however  that  impure  identities  are  meaningful  and  so  this  possibility  does  come  into  conPlict  with  the  Generality  Constraint.  They  declare  “the  thought  dies   hard   that   identity   is   categorically   appropriate   simply   to   any   object”   and   offer   two  considerations  in  favour  of  the  contended  signiPicance  of   impure  identities  (Hale  &  Wright  [2001],   p.   344,   pp.   350-­‐1).   The   Pirst   consideration   incorporates   an   appeal   to   the  contrapositive   of   Leibniz’   Law   (‘the   diversity   of   the   dissimilar’).   Suppose   that   Caesar   and  Nx:xx   belong   to   mutually   exclusive   categories   (persons   and   numbers).   Then   Caesar   is   a  person  whereas  Nx:xx  is  not.  Since  Caesar  has  a  property  (being  a  person)  that  Nx:xx  lacks  it  follows   that   they   are   distinct.   So   the   relevant   impure   identity   (“Caesar   =   Nx:xx”)   is   false  rather   than   meaningless.   However,   this   argument   presupposes   that   claims   of   trans-­‐categorical  distinctness  are   themselves  meaningful   and  capable  of   receiving  a   truth-­‐value.  But   if   trans-­‐categorical   identity   statements   are  meaningless   then   so   are   trans-­‐categorical  distinctness   statements.   So   the   argument   from   the   diversity   of   the   dissimilar   begs   the  question  against  the  view  that  impure  identities  are  meaningless.  

The   neo-­‐Fregean   therefore   offers   a   second   consideration.   They   argue   that   it   is  “utterly  unclear”  how  a  case  may  be  made  for  the  claim  that  impure  identities  lack  a  sense.  

An  appeal  to  intuitions  of  signiPicance  is  likely  to  be  unsatisfactory  because  our  intuitions  do  not   speak   in   unison.   An   appeal   to   stipulation   is   also   out   of   order.  He   concludes:   “what   is  needed  is  a  principled  reason  for  denying  that  this  conPiguration  of  individually  signiPicant  words   adds   up   to   an   expression   which,   taken   as   a   whole,   expresses   something   true   or  false”  (Hale  &  Wright  [2001],  p.  351).  But  this  argument  also  appears  to  beg  the  question.  The  meaning-­‐theoretic  version  of  the  Caesar  problem  arises  from  the  fact  that  (HP)  fails  to  settle  any  signiPicance   for   impure   identitiesit   leaves  a   semantic  gap   there.  This  provides  a  principled   reasoning   for   doubting   whether   impure   identities   do   express   something.   Of  course,   if   the   default   assumption   is   made   that   identity   contexts   exhibit   a   free   wheeling  compositionalityso   every   grammatical   permutation   of   them   is   meaningfulthen   the  difPiculties  encountered  in  settling  the  signiPicance  of  impure  identities  provides  no  grounds  for  doubting  that  they  have  a  meaning.  They  will  have  a  meaning  regardless.  But  since  the  Caesar  problem  raises  the  question  of  whether  every  grammatical  identity  has  a  sense  the  assumption  of  free  wheeling  compositionality  can  hardly  be  relied  upon  to  bolster  a  solution  to  the  problem  itself.    

The   neo-­‐Fregean   also   supplies   a   metaphysical   argument   to   undermine   the  contention   that   impure   identities   are   meaningless.   Appeal   is   made   to   Frege’s   avowed  “Platonism”:   the   contention   that   numbers   belong   to   an   inclusive   domain   of   objects.   They  reason   that   if   numbers   are   to   belong   to   such   a   domain   then   there  must   be   a   fact   of   the  matter  about  which  objects  the  numbers  are.   In  other  words,  there  must  be  a  determinate  truth   about   whether   a   number   is   identical   or   distinct   to   any   other   object   (however  presented).  The  neo-­‐Fregean  concludes  that  if  Platonism  is  to  be  a  legitimate  position  then  it  cannot  be   the  case   the   impure   identities  are  meaningless.  But   if   there   is  any   legitimacy  to  the  concern  thatfor  all  (HP)  settlesimpure  contexts  lack  a  sense  then  this  argument  simply  places  a  question  mark  over  whether  Platonism  is  a  legitimate  position.  This  in  turn  raises  a  doubt  concerning  the  effectiveness  of  any  purported  solution  to  the  Caesar  problemthe  neo-­‐Fregean  solution  includedthat  presupposes  Platonism.

4.  ConclusionLet   us   return   to   Frege’s   original   formulation   of   the   Caesar   problem   (section   2.1).   Frege  linked  the  notions  of  object  and  identity  and  then  claimed  that  if  a  symbol  a  is  to  be  used  to  denote   an   object   then   we   must   have   available   a   criterion   that   determines   in   every   case  whether  b  is  the  same  as  a.  We  have  seen  that  considerable  difPiculties  confront  any  attempt  to   supply   such  a   criterion.  This   suggests   that   it  may  be   time   to   reconsider  whether  Frege  was  right  to  tie  together  the  notions  of  object  and  identity  in  the  manner  he  proposed.  

It   is   true   that   objects  may   be   identiPied   and   re-­‐identiPied   and   seen   from   different  

perspectives.  No  doubt  it  is  the  capacity  of  objects  to  be  identiPied  and  re-­‐identiPied  in  this  way   that   is   responsible   for   our   treating   discourse   about   objects   in   a   realist   fashion.   But  there   does   not   appear   to   be   anything   in   our   ordinary   interaction   with   objects   that  determines  objects  must  be  capable  of  being  identiPied  and  re-­‐identiPied  from  every  point  of  view.     Rather   it   appears   that  we   ‘track’   objects   across  a   range   of   relevant   situations   and  perspectives.  It  would  be  an  imposition  to  suppose  that  the  ‘tracking  conditions’  with  which  we  habitually  operate  are  identity  criteria  that  tacitly  determine  the  presence  or  absence  of  an   object   across   all   situations.   One   may   therefore   wonder   whether   Frege   made   any  legitimate   demand   when   he   required   identity   criteria   for   the   objects   he   planned   to  introduce.  

The   doctrine   that   the   notions   of   object   and   identity   cannot   be   separated   has,  however,   become   deeply   entrenched.   It   is   therefore   unlikely   that   this   suggestion   will   be  readily   received.   But   there   is   something   perplexing   about   this   persistent   adherence   to  Fregean   doctrine.   For,   despite   the   conviction   that   every   object   has   identity   criteria,  proponents  of  the  view  have  been  beggared  to  provide  any.  This  goes  for  all  kinds  of  objects,  all  the  way  up  from  quantum  particles  to  persons.  Even  setsoften  presented  as  the  paradigm  of  objects  with  clear  and  distinct   identity  criteriafail   to  meet   the  prevailing  standards.  For  the  Axiom  of  Extensionality  that  purportedly  provides  criteria  of  identity  for  sets  neglects  to  make   any   mention   of   times   or   possible   worlds.   As   a   consequence   Extensionality   fails   to  determine  whether   sets   are   identical   or   distinct   at   different   times   or   different  worlds.   So  even   someone   who   submits   that   there   are   well-­‐dePined   identity   criteria   for   sets   must  overcome   a   version   of   the   Caesar   problem.   Serious   engagement  with   the   Caesar   problem  and   its   presuppositions   may   well   assist   in   identifying   and   assessing   the   conPlicting  intellectual  forces  that  give  rise  to  this  perplexing  situation.

AcknowledgmentsThanks   to   audiences   at   the   Universities   of   Bristol,   Dusseldorf   and   St.   Andrews   for   their  helpful   comments.   I   would   also   like   to   thank   Peter   Clark,   Bill   Demopoulos,   Katherine  Hawley,   Alex   Oliver,   Michael   Potter,   Graham   Priest,   Stephanie   Schlitt,   Stewart   Shapiro,  Crispin   Wright   and   an   anonymous   reader   for   Oxford   University   Press.   I   gratefully  acknowledge   the   support   of   the   Leverhulme   Trust   whose   award   of   a   Philip   Lervehulme  Prize  made  possible  the  writing  of  this  paper  .  

Department  of  Logic  and  MetaphysicsUniversity  of  St.  Andrews,  Fife,  KY16  9AL

Olpm@st-­‐andrews.ac.uk

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