80             The  Ontology  of  Mathematical  Practice             Oswaldo  Chateaubriand   Pontifícia  Universidade  Católica  do  Rio  de  Janeiro  (PUC-­‐Rio)   Conselho  Nacional  de  Desenvolvimento  Científico  e  Tecnológico  (CNPq)   ochateaubriand@gmail.com       Abstract:   As   Bernays   maintained   in   his   famous   1935   article   “On   Platonism   in   Mathematics”,   in   mathematical   practice   objects,   functions,  relations,  properties,  structures,  etc.  are  treated  as  entities   that   exist   independently   of   our   discourse   and   of   our   constructions.   Bernays   maintains   that   this   form   of   platonism   is   essentially   a   manner   of   speaking,   which   does   not   involve   a   commitment   to   a   strict   form   of   platonism.   In   this   note   I   defend   Bernays’   position   and   propose   a   more  systematic  formulation  for  this  form  of  platonism,  combining  it   with  ideas  of  Frege  and  of  Gödel.   Key-­‐words:  platonism;  Paul  Bernays;  mathematical  practice.       In   the   XIII   Colóquio   Conesul   I   gave   a   talk   defending   the   idea   that   the   ontology   of   mathematics   can   be   conceived   as   an   ontology   of   properties,   with   everything  else  characterized  in  terms  of  properties.  I  gave  a  similar  talk  at  the   IHPST  (Institut  d’Histoire  et  Philosophie  des  Sciences  et  des  Techniques)  when  I   was  in  Paris  in  2010,  which  led  to  a  big  discussion  about  mathematical  objects,   because  objects  are  seen  as  a  clear  aspect  of  the  ontology  of  mathematics  and  it   is  not  clear  how  to  deal  with  them  in  terms  of  properties.  Today  I  was  invited  to   speak  about  some  topics  related  to  mathematical  practice,  so  I  thought  I  would   speak  again  about  platonism,  but  from  a  somewhat  different  perspective.     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   81   In   his   1935   paper   “On   Platonism   in   Mathematics”,   Bernays   discusses   a   tendency   of   mathematical   practice   “which   …   consists   in   viewing   the   objects   as   cut   off   from   all   links   with   the   reflecting   subject”   (p.   275).   After   outlining   this   tendency   he   goes   on   to   say   that   he   has   “set   forth   only   a   restricted   platonism   which   does   not   claim   to   be   more   than,   so   to   speak,   an   ideal   projection   of   a   domain   of   thought”   (p.   277).   And,   he   claims,   this   in   1935,   “platonism   reigns   today  in  mathematics”  (p.  276).   Although   I   am   not   a   practicing   mathematician,   or   a   historian   of   mathematics,  I  think  we  can  characterize  present  day  mathematical  practice  as  a   form   of   platonism   uncommitted   to   a   philosophical   analysis   as   to   what   one   is   talking  about.  It  is  sometimes  said  that  mathematicians  are  more  formalist  than   platonist,  but  this  does  not  seem  to  be  true.  I  think  the  basic  intuitions  are  the   platonistic  intuitions,  although,  in  fact,  mathematicians  do  not  want  to  dwell  on   the  philosophical  questions.  They  do  not  want  to  ask  what  an  object  is,  or  what  a   property   is,   or   what   other   mathematical   entities   are   from   an   ontological   viewpoint.   They   only   want   to   ask   these   questions   with   respect   to   mathematics   itself,  not  in  terms  of  some  prior  philosophical  conception.  So,  what  I  will  try  to   do  in  my  talk  is  to  reformulate  some  of  my  ideas  in  these  terms.   The   reason   I   defend   the   notion   of   property,   independently   of   asking   questions  as  to  the  nature  of  properties,  is  that  it  has  a  very  clear  basis  both  in   practice  and  in  our  way  of  thinking.  Properties  are  expressed  by  predicates,  and   it  does  not  matter  what  they  are.  What  we  do  when  we  characterize  a  predicate,   i.e.,  when  we  give  the  semantic  content  of  a  predicate—be  it  a  singular  predicate   or  a  relational  predicate—is  to  give  conditions  of  applicability  for  that  predicate.   In  mathematics  these  characterizations  are  very  precise—as  opposed  to  ordinary   predicates,   where,   in   many   cases,   the   characterizations   are   rather   imprecise.   I   think   every   mathematician   would   agree   that   these   predicates   express   mathematical  properties.  If  one  tries  to  prod  them,  however,  and  ask  “What  do   you  mean  by  a  property?”,  one  easily  sees  that  it  is  not  the  kind  of  question  they   want  to  get  into.  They  do  not  want  to  discuss  what  they  mean  by  “property”,  but     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   82   they   probably   will   say   that   a   property   is   what   is   expressed   by   a   predicate:   what   I   define,  what  I  formulate,  and  so  forth;  these  are  properties.   The  problem  I  saw  in  my  platonistic  talk  is  that  mathematicians  also  talk   about   objects,   structures,   spaces,   etc.,   and   one   of   the   questions   that   can   be   raised   is   what   is   meant   by   “object”.   One   wants   to   talk   about   objects   in   such   a   way   that   one   is   not   committed   to   giving   a   philosophical   analysis   of   what   an   object  is.  But  we  want  to  say  that  numbers  are  objects;  many  people  want  to  say   that  sets  are  objects;  they  want  to  say  that  spaces  are  objects,  and  so  on.  These   are   all   kinds   of   objects   one   talks   about   in   mathematics,   and   how   should   we   conceptualize  that?  Well,  there  is  a  sense  in  which  it  is  rather  clear,  and  that  is   when   we   talk   about   a   domain   of   discourse.   Even   functions   can   be   treated   as   objects  if  they  are  our  domain  of  discourse.  This  is  an  idea  Frege  elaborated  in   some   ways,   even   before   the   discussion   he   has   in   the   early   90’s.   In   The   Foundations   of   Arithmetic   (p.   77   n.   2)   he   has   a   characterization   of   objects   and   concepts  according  to  which  an  object  is  that  which  can  be  subject  of  a  singular   proposition,  and  a  concept  is  that  which  can  be  a  predicate  of  such  a  proposition.   So,   he   characterizes   objects   as   subjects   of   discourse.   Something   that   can   be   a   subject  is  what  he  is  going  to  call  an  “object”.  That  is  one  of  the  reasons  he  wants   to  say  that  numbers  are  objects,  because  we  refer  to  numbers  as  subjects.  We   say  “the  number  1  is  the  smallest  positive  integer”,  “the  number  9  is  odd”,  and   so  on.  And,  of  course,  we  are  using  those  expressions  in  subject  position.  So,  in   that   context,   at   any   rate,   Frege   treats   the   notion   of   object   as   the   notion   of   what   can   be   talked   about   as   the   subject   of   an   assertion,   or   the   subject   of   a   predication.  (Of  course,  it  could  also  be  a  relational  proposition.)   When   he   gets   to   his   later   papers   “On   Concept   and   Object”   and   “Function   and  Concept”  he  seems  to  want  to  introduce  a  more  ontological  notion  of  object   and  concept,  and  then  he  comes  up  with  things  like  “objects  are  saturated  and   functions   are   unsaturated”.   And   then,   of   course,   he   gets   into   a   puzzle   (for   which   he   was   so   thoroughly   criticized)   because   he   says   that   in   “the   concept   horse   is   easily   attainable”   one   is   not   talking   about   a   concept   but   about   an   object,     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   83   because  “the  concept  horse”  is  subject  of  the  singular  proposition.  People  have   criticized   him   for   that,   but   perhaps   his   mistake   was   to   ontologize   the   distinction.   In   fact,   Frege   has   no   ontological   analysis   of   objects—or   of   functions,   for   that   matter.   When   he   characterizes   functions   he   does   so   in   terms   of   an   expression   where  one  can  substitute  certain  names;  and,  he  says,  if  I  substitute  these  names   by   other   names,   then   I   have   the   same   function.   And   what   does   he   say   about   objects?   He   says   an   object   is   anything   that   is   not   a   function.   And   that   is   the   only   characterization   he   gives.   So,   perhaps   Frege   made   a   mistake   in   trying   to   ontologize  the  distinction  of  object  and  function.  Maybe  he  should  have  stayed   with   the   original   distinction   in   terms   of   subject   and   predicate.   He   would   have   avoided  a  lot  of  problems.  Nobody  can  object  to  “the  concept  horse”  being  the   subject   of   the   predication   “is   easily   attainable”,   but   if   one   says   that   it   is   an   object,  in  some  sense  ontological,  then  it  seems  problematic.   So,   I   want   to   say   that   in   the   practice   of   mathematics   people   talk   about   objects  essentially  in  that  Fregean  sense.  Whatever  is  a  subject  we  can  say  it  is  an   object.   Whatever   is   part   of   the   domain   of   discourse   we   are   talking   about   we   can   say  it  is  an  object.  Of  course,  we  can  talk  about  functions,  about  spaces,  about  all   kinds   of   things,   but   as   they   appear   in   the   subject   position,   we   can   say:   well,   I   am   treating  those  as  part  of  the  objects  I  want  to  talk  about.  This  may  seem  odd,  but   we  do  it  in  practice.   In  fact,  it  is  very  interesting  that  when  we  discuss  structures  for  first-­‐order   logic   (or   for   logic   in   general),   the   elements   of   the   structure   can   be   anything.   Although  there  is  an  old  idea  that  the  elements  of  a  structure  are  individuals,  we   can   have,   for   instance,   a   first-­‐order   theory   of   properties,   where   the   properties   are  the  elements  of  the  structure  and  are  the  objects  of  the  structure.  My  point   is   that   in   practice   being   an   object   is   just   being   a   subject   of   some   kind   of   discourse.  Of  course,  something  can  occur  both  as  subject  and  as  predicate,  even   numerical  terms.  As  Frege  himself  pointed  out,  we  assert  of  some  property  that   it  applies  to  three  things,  or  that  it  applies  to  five  things,  and  these  are  second-­‐ order   predications   about   first-­‐order   properties.   We   can   also   assert   that   a     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   84   property   applies   to   finitely   many   things,   or   to   infinitely   many   things,   and   these   are   also   numerical   predications—or   cardinality   predications—about   properties.   Thus,  numbers  can  also  appear  as  predicates  and  not  only  as  subjects.   My  idea  for  getting  out  of  having  objects  and  having  only  properties  was   essentially   this:   characterize   the   numbers   in   terms   of   properties   and   say   this   is   what  numbers  really  are.  But  I  think  that  from  the  point  of  view  of  mathematical   practice,  as  far  as  I  understand  it,  one  does  not  get  into  these  questions.   We  say  that  sets  are  objects,  but  if  we  start  asking  ourselves  why  sets  are   objects,  there  does  not  seem  to  be  a  clear  answer.  Why  are  sets  objects?  Why   aren’t   they   something   else?   Sets   are   treated   sometimes   as   characteristic   functions;  so  why  is  a  set  an  object  and  not  a  function?  Also,  a  set,  in  an  arbitrary   sense,   is   a   selection   of   things,   and   why   isn’t   that   selection   a   function   rather   than   an   object?   It   seems   that   sets   are   objects   because   in   doing   set   theory   we   treat   them   in   subject   position.   And,   of   course,   we   have   relations   among   sets,   and   operations  on  them,  and  so  on.   In  the  case  of  structures  there  is  a  tendency  of  viewing  structures  as  the   fundamental   notion   of   mathematics.   This   has   been   a   recent   trend:   all   mathematics   deals   with   structures.   I   agree   this   is   a   fundamental   notion   of   mathematics,   but   I   do   not   think   one   can   characterize   everything   in   terms   of   structures.   Structures   appear   often   as   objects,   but   they   also   appear   in   a   very   general   way.   We   may   talk   about   a   particular   structure—say,   a   specific   group—as   an  ordered  pair  (say)  of  a  set  and  an  operation,  but  we  may  also  talk  about  the   group   structure   more   generally.   In   fact,   it   seems   perfectly   acceptable   to   talk   about   the   group   structure.   But   what   does   it   mean?   What   is   the   group   structure?   The   group   structure   is,   in   a   sense,   a   group   function,   a   group   operation.   That   is   what  all  groups  have  in  common;  an  operation  that  behaves  in  a  certain  way.   It   seems,   therefore,   that   all   these   notions   appear   in   mathematical   practice  in  a  way  that  is  not  ontologized.  They  are  discussed,  they  are  used,  but   they  are  not  treated  ontologically  in  a  philosophical  sense.  This  is  related  to  what   I  was  arguing  in  the  XIII  Conesul,  but  in  that  talk  I  was  trying  to  make  a  reduction,     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   85   which  I  am  not  trying  to  do  now.  The  question  that  comes  up  now  is  about  the   kind   of   epistemology   that   goes   together   with   this   practice   of   talking   about   objects  and  properties  in  a  platonistic  but  uncommitted  sense.   When   Bernays   wrote   his   paper   and   characterized   the   restricted   form   of   platonism,   he   also   said   that   there   is   a   robust,   full-­‐fledged,   platonism   which   is   what  has  been  shown  to  be  untenable  by  the  paradoxes.  Eventually,  however,  he   goes  back  on  that  when  he  reviews  Gödel’s  paper  “Russell’s  Mathematical  Logic”   in  The  Journal  of  Symbolic  Logic.  There  he  says  (p.  75)  that  full-­‐fledged  platonism   leads   to   paradox   only   if   it   is   combined   with   an   aprioristic   epistemology.   If   the   epistemology  is  not  aprioristic,  as  Gödel’s  is  not,  then  the  situation  is  different.   Gödel   is   a   very   famous   full-­‐fledged   platonist,   but   he   also   considers   the   possibility  of  separating  a  realistic  platonism  from  a  more  restricted  form.  In  fact,   there   is   a   very   explicit   passage   in   the   appendix   he   wrote   for   the   paper   on   the   continuum  problem,  where  he  says  (p.  272):   The  question  of  the  objective  existence  of  the  objects  of  mathematics   (which,   incidentally,   is   an   exact   replica   of   the   question   of   the   objective   existence   of   the   outer   world)   is   not   decisive   for   the   problem   under   discussion   here.   The   mere   psychological   fact   of   the   existence   of   an   intuition   which   is   sufficiently   clear   to   produce   the   axioms   of   set   theory   and   an   open   series   of   extensions   of   them   suffices   to   give   meaning   to   the   question   of   the   truth   or   falsity   of   propositions  like  Cantor’s  continuum  hypothesis.   Moreover,   one   of   the   aspects   of   his   epistemology,   emphasized   by   Bernays,  is  to  suggest  that  we  can  evaluate  mathematical  propositions  by  looking   at   their   consequences—especially   consequences   in   number   theory,   which   he   compares   with   the   most   elementary   domain   of   clear   intuition.   So,   if   one   has   principles   that   have   new   consequences   in   number   theory,   this   brings   evidence   for  those  principles  even  if  they  do  not  have  intrinsic  evidence.  And  that  is  the   reason   he   compares   the   mathematical   situation   with   the   physical   situation.   He   says  explicitly  (p.  265)  that  the  reason  one  accepts  a  mathematical  principle  can   be   the   same   reason   one   accepts   a   physical   law;   because   there   are   so   many   verifiable   consequences   of   the   principle—verifiable   in   mathematics,   not   somewhere   else.   Gödel   is   thinking   of   something   like   number   theory   as     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   86   corresponding   to   the   domain   of   perception,   but   the   epistemological   claim   is   much  more  general  than  that,  and  I  think  the  example  of  the  axiom  of  choice  is   an  excellent  example.   As   long   as   one   is   talking   about   classical   mathematics,   there   is   very   little   doubt  that  people  accept  the  axiom  of  choice  today  in  their  everyday  work.  The   reason   for   this   is   that   the   axiom   of   choice   has   many   consequences   in   every   domain  of  mathematics  and  that  these  consequences  have  been  verified  in  those   domains.   Even   the   results   that   were   thought   to   be   odd,   such   as   the   Tarski-­‐ Banach   paradox,   were   diagnosed   as   such   because   the   decompositions   effected   by  the  axiom  of  choice  are  not  at  all  like  ordinary  geometric  decompositions.  The   conclusion   to   which   all   the   verifiable   consequences   and   explanations   led   was   that   there   is   clear   indirect   evidence   of   this   kind   for   accepting   the   axiom   of   choice,  and  that  is  exactly  what  happened  in  practice.   Of  course,  a  constructivist  can  have  reasonable  objections  to  the  axiom  of   choice,   but   now   we   are   getting   into   a   specific   conception   of   mathematics,   which   I  do  not  think  is  the  normal  practice  of  mathematics.   Gödel  also  thinks  that  the  axiom  of  choice  has  an  intrinsic  justification.  As   far   as   I   know,   Gödel   was   the   first   person   to   make   the   distinction   between   the   logical   conception   of   set,   which   he   attributed   to   Frege,   and   the   mathematical   conception   of   set,   which   he   attributed   to   Cantor.   The   logical   conception   is   the   conception   of   dividing   the   totality   of   things   into   all   those   that   have   a   certain   property  and  those  that  do  not.  This  is  Frege’s  conception  of  extensions,  which   led   to   paradox.   The   other   conception,   which   Gödel   conceptualizes   in   terms   of   the  operation  “set  of”,  consists  in  starting  with  a  well-­‐defined  totality  of  things   and  “generating”  (in  a  manner  of  speaking)  sets  of  these  things,  and  sets  of  the   things   thus   generated,   and   so   on.   This   mathematical   conception,   in   terms   of   selections,   does   not   give   rise   to   any   paradox,   unless   one   makes   the   mistake   of   supposing  that  the  process  has  to  come  to  an  end  in  a  set  of  all  sets.  Moreover,   Gödel  sees  the  axiom  of  choice  as  being  part  of  the  very  idea  of  these  selections.     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   87   He  says,  in  fact,  that  the  axiom  of  choice  and  the  axiom  of  definable  subsets  are   the  axioms  that  characterize  the  iterative  conception.   I   think   that   in   practice   this   epistemological   conception   is   the   main   conception   used   by   mathematicians,   though   I   do   not   think   it   is   conceptualized   in   exactly  the  same  way  it  is  conceptualized  by  Gödel.   One   thing   I   have   found   out   by   talking   to   my   colleagues   in   the   mathematics   department   is   that   the   idea   many   logicians   have   that   mathematicians   work   in   terms   of   axiomatic   systems   is   just   not   true.   Mathematicians  do  not  work  axiomatically.  In  fact,  in  most  cases  where  one  has   axioms,  the  axioms  are  really  definitions:  groups,  rings,  fields,  topological  spaces,   etc.   And   in   other   cases   mathematicians   do   not   work   from   axioms;   in   fact,   they   often   do   not   even   know   what   the   axioms   are—I   suspect   that   a   lot   of   mathematicians   do   not   know   the   Peano   axioms,   for   instance.   They   work   from   what?   They   work   from   practice,   from   a   common   tradition   of   working   in   analysis,   or  in  number  theory,  or  mixing  things  up,  or  doing  many  different  things.     So,  just  as  there  is  not  an  ontology  separate  from  mathematical  practice,  I   think   there   is   not   an   epistemology   separate   from   mathematical   practice.   And   again,   if   one   tries   to   talk   epistemology   to   mathematicians,   they   tend   to   blank   out—although  they  may  be  polite  about  it.  Talking  about  epistemological  issues   or  ontological  issues  is  not  the  kind  of  question  they  want  to  discuss;  and  they  do   not  think  it  is  relevant  to  discuss  it  for  their  work.   My   impression,   then,   is   that   Bernays’   idea   that   his   restricted   form   of   Platonism  reigns  in  mathematics  is  still  true  today.  Following  a  suggestion  of  José   Ferreirós  at  the  end  of  my  talk  we  may  say  that  what  reigns  in  mathematics  is  a   methodological  platonism.               Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.   88   References     BENACERRAF,   P.   and   PUTNAM,   H.   (Eds.)   (1964)   Philosophy   of   Mathematics:   Selected  Readings.  Prentice-­‐Hall,  New  Jersey.   BERNAYS,   P.   (1935)   “On   Platonism   in   Mathematics”.   In   Benacerraf   &   Putnam,   274-­‐286.   _____.  (1946)  Review  of  K.  Gödel  “Russell’s  Mathematical  Logic”.  The  Journal  of   Symbolic  Logic,  v.  11,  75-­‐79.   FREGE,  G.  (1884)  The  Foundations  of  Arithmetic:  A  Logico-­‐Mathematical  Enquiry   into  the  Concept  of  Number.  Blackwell,  Oxford,  1950.   GÖDEL,  K.  (1944)  “Russell’s  Mathematical  Logic”.  In  Benacerraf  &  Putnam,  211-­‐ 232.   _____.   (1947/1964)   “What   is   Cantor’s   Continuum   Problem?”   In   Benacerraf   &   Putnam,  258-­‐273.     Notae  Philosophicae  Scientiae  Formalis,   vol.  1,  n.  1,  p.  80  -­‐  88,  abril  2012.