Representation_2 copy   1     Users,  Structures,  and  Representation     Mathias  Frisch   University  of  Maryland     Abstract   This  paper  defends  a  pragmatic  and  structuralist  account  of  scientific  representation  of  the   kind  recently  proposed  by  Bas  van  Fraassen.    As  I  show,  the  account  appears  to  have  the   unacceptable  consequence  that  the  domain  of  a  theory  is  restricted  to  phenomena  for   which  we  actually  have  constructed  a  model—a  worry  arising  from  the  account’s   pragmatism,  which  is  exacerbated  by  its  structuralism.    Yet  the  account  has  the  resources   at  least  partially  to  address  the  worry.    What  remains  as  implication  is  a  strong  anti-­‐ foundationalism.       1.  Introduction   My  aim  in  this  paper  is  to  offer  a  partial  defense  of  a  pragmatic  and  structuralist  theory  of   scientific  representations.    The  central  tenet  of  the  view  I  will  explore  is  the  claim  that   representation  is  an  essentially  pragmatic  notion,  dependent  on  a  context  of  use.    This   claim  has  recently  been  forcefully  defended  by  Bas  van  Fraassen  (van  Fraassen  2008)  and   in  what  follows  I  will  take  his  account  as  my  main  point  of  departure.    I  will  try  to  make   various  aspects  of  van  Fraassen’s  pragmatic  account  of  representation  plausible,  but  my   aim  here  is  not  to  offer  a  comprehensive  defense  of  this  view.    Rather  I  am  interested  in   some  of  the  consequences  of  the  account  for  philosophical  views  of  scientific  theorizing,  for   the  question  of  scientific  foundationalism  and  for  the  issue  of  inconsistent  theories.   Van  Fraassen’s  account  of  representation  amounts  to  a  critique  of  the  view  that  a   physical  theory’s  representational  content—what  the  theory  says  about  the  world—is   given  simply  by  a  statement  of  the  theory’s  basic  equations  or  by  a  set-­‐theoretic  predicate   defining  the  theory’s  representational  structures.    The  latter  view,  which  Nancy  Cartwright   has  derisively  dubbed  “the  vending  machine  view”  of  scientific  theories  (Cartwright  1999)   and  which  van  Fraassen  himself  appears  to  have  once  held  (see,  e.g.,  van  Fraassen  1980),    is     2   inadequate  precisely  because  it  ignores  the  central  role  played  by  users  of  representations.     Furthermore,  as  I  will  argue,  a  pragmatic  account  of  representation  has  radically  anti-­‐ foundationalist  implications  of  a  kind  which  van  Fraassen  himself  may  be  reluctant  to   accept.   In  the  next  section  I  will  motivate  and  partly  defend  the  claim  that  both  the  target   and  the  content  of  a  scientific  representation  depend  on  a  context  of  use  of  the   representation.    In  section  three  I  will  argue  that  just  as  selective  resemblance  plays  an   important  role  in  successful  scientific  representation,  distortions  and  selective  non-­‐ resemblance  play  an  important  role  as  well.    There  is,  as  Paul  Teller  has  argued,  no  ‘perfect   model’.    Defending  this  claim  will  lead  to  a  worry  for  a  pragmatic  account  of  representation,   however:  the  reason  why  there  can  be  no  perfect  model,  it  seems,  is  that  our  theories  only   represent  those  phenomena  for  which  we  have  actually  constructed  a  representation.    This   worry  gets  exacerbated,  if  we  take  scientific  representation,  at  least  within  physics,  to  be   structural  representation.    For  then,  as  I  will  argue  in  section  four,  we  have  to  confront   Putnam’s  model-­‐theoretic  argument  and  responding  to  the  argument  requires  a  further   emphasis  of  the  role  of  the  user  in  the  link  between  a  representation  and  its  target.    But,  as  I   will  argue  in  section  five,  the  worry  concerning  the  overly  restricted  domains  of  our   theories  can  be  met,  if  we  distinguish  between  ‘horizontal’  and  ‘vertical’  extensions  of  the   domain  of  a  theory.    A  context  of  use  can  be  extended  horizontally  from  phenomena   modeled  to  others  of  the  same  kind  that  have  not  been  actually  modeled.    But  we  cannot   similarly  extend  the  domains  of  our  theories  vertically,  as  foundationalism  claims,  or  so  I   will  argue.     2.  ‘No  representation  without  representer’   What  is  it  for  a  scientific  representation  to  represent  a  phenomenon?    Van  Fraassen  argues,   to  my  mind  convincingly,  that  representation  is  an  essentially  pragmatic,  user-­‐dependent   notion  that  we  cannot  “define”  or  “reduce  […]  to  something  else”  (2008,  7).    To  call  a  thing  a   representation  is  to  say  something  about  its  use.    Thus,  the  “Hauptsatz”  of  van  Fraassen’s   account  of  representation  is  that  “There  is  no  representation  except  in  the  sense  that  some   things  are  used,  made,  or  taken,  to  represent  things  as  thus  and  so.”  (2008,  23,  italics  in   original)    This  implies  that  there  can  be  no  ‘natural  representations’—no  naturally     3   produced  objects  or  phenomena  that  represent  other  phenomena  without  being  used  by   someone  to  represent.  Independent  of  its  actually  being  used  as  a  representation  a  pictures   has  no  representational  content:  “To  call  an  object  a  picture  at  all  is  to  relate  it  to  its  use.”   (2008,  25,  italics  in  original)1   I  do  not  here  want  to  defend  the  view  that  we  cannot  give  a  more  substantive   account  of  representation  than  what  is  expressed  in  van  Fraassen’s  Hauptsatz,  but  I  do   wish  to  claim  that  the  Hauptsatz  will  have  to  be  part  of  any  satisfactory  account  of  scientific   representation  in  particular.    For  example,  one  cannot  define  representation  independently   of  use  exclusively  in  terms  of  likeness  or  resemblance,  for  several  obvious  reasons.    As  has   often  been  pointed  out,  at  least  since  Nelson  Goodman’s  seminal  work  (Goodman  1976),   resemblance  is  a  symmetric  relation  but  representation  is  not.    Moreover,  while  perfect   resemblance  would  be  much  too  strong  a  requirement  for  representation,  partial   resemblance  is  much  too  weak:  arguably  for  every  two  objects  there  will  be  some  respect   in  which  the  two  objects  resemble  each  other.    Thus,  partial  resemblance  cannot  be   sufficient  for  representation,  for  otherwise  we  would  be  forced  to  the  conclusion  that   everything  represents  everything  else.    But  partial  resemblance  as  necessary  condition,   without  additional  constraints,  would  be  an  empty  requirement.   It  might  seem  that  by  insisting  on  the  pragmatic  character  of  representation,  I  am   thereby  disagreeing  with  recent  structuralist  accounts  of  representation,  such  as  the  partial   structure  account  proposed  by  Steven  French  and  Otavio  Bueno  (see,  e.g.  Bueno  and  French   2011),  or  the  homomorphism  account  defended  by  Andreas  Bartels  (Bartels  2005;  2006).     But  French  and  Bueno  also  acknowledge  an  important  role  for  pragmatic  and  context-­‐ dependent  considerations.    According  to  them,  “partial  isomorphism  [that  is,  the  structural   relationship  that  they  see  at  the  core  of  successful  representation]  is  not  sufficient  and  that   other  factors  must  be  appealed  to.”  (2011,  29)    Indeed,  they  emphasize  that  just  as  a  certain   structural  relationship  is  not  enough  to  fix  a  representation’s  target,  one  also  cannot  rigidly   build  a  particular  intention  into  the  representational  mechanism  that  would  permanently   fix  the  representation’s  target.    Rather  we  must  allow  for  the  flexibility  of  “pragmatic  or                                                                                                                   1  A  similar  non-­‐reductive  account  of  representation  is  defended  in  (Suárez  2004)  and  (Suárez   2010).     4   broadly  contextual  factors  to  play  a  role  in  selecting  which  [representational]  relationships   to  focus  on.”  (Bueno  and  French  2011,  31).   Initial  appearances  to  the  contrary,  Bartels  also  agrees  with  the  claim  that  structural   relationships  are  not  sufficient  for  representation.    Bartels  argues  that  the  common   symmetry-­‐objection  to  structural  accounts  of  representation  is  unsuccessful  and  can  be   disarmed  by  taking  the  appropriate  structural  relationship  between  a  representation  and   its  target  to  be  a  non-­‐symmetric  homomorphism  instead  of  a  symmetric  isomorphism.     Bartels  then  proposes  a  homomorphism  condition  either  as  a  sufficient  condition  (Bartels   2005)  or  as  a  necessary  condition(Bartels  2006)  for  representation.    While  this  might  be   taken  to  suggest  a  purely  structural  account  of  representation,  he  ultimately  distinguishes   the  notion  of  potential  representation  from  that  of  actual  representation  and  maintains  that   only  the  former  notion  satisfies  the  homomorphism  condition.    Whether  a  potential   representation  is  also  an  actual  representation  is  determined  by  pragmatic  and  context-­‐ dependent  factors.   The  difference,  then,  between  a  formal,  structural  account,  such  as  French  and   Bueno’s  or  Bartels’s,  and  van  Fraassen’s  account  of  representation  (which,  after  all,  also   takes  representation  in  physics  to  be  structural  representation)  strikes  me  as  one  of   emphasis:    whereas  French  and  Bueno  stress  the  structural  relationships  that  have  to  exist   between  a  successful  scientific  representation  and  its  target,  van  Fraassen  focuses  on  the   ineliminable  pragmatic  aspects  of  the  representation  relation  and  the  insufficiency  of   purely  structural  relations  in  establishing  a  representation  relation  between  a  structure   and  its  target.   Thus,  we  can  concede  that  resemblance  plays  an  important  role  in  representation,   but  where  it  does  play  a  role  it  does  so  as  a  function  of  the  representation’s  use.    For   example,  in  certain  contexts  we  identify  a  representation’s  target  with  the  help  of  selective   resemblances  between  representation  and  target.    Yet  which  aspects  are  important  in   assessing  the  likeness  between  representation  and  target  is  given  by  the  context  in  which   the  representation  is  used.   Scientific  representation  is  representation  as:  a  representation  has  a  certain  target   and  represents  its  target  as  being  thus  and  so.    The  content  of  a  representation  is  use-­‐ dependent  as  well  and  cannot  be  simply  read  off  the  representational  structure.    As  we  will     5   see  in  somewhat  more  detail  below,  the  use  of  a  given  representation  determines  for  each   of  the  models  properties  under  which  of  a  number  of  different  categories  a  given  property   or  feature  of  the  model  falls:  some  features  are  part  of  the  model’s  representational  content   with  the  aim  of  representing  the  model’s  target  approximately  correctly;  other  features  are   part  of  the  model’s  representational  content  but  purposefully  misrepresent  its  target;  and   there  are  features  of  a  model  that  do  not  play  any  representational  role  (in  a  given  context).     The  classical  point-­‐charge  model  for  an  electric  charge  can  illustrate  all  three  types  of   feature:    We  take  it  that  the  1/r2-­‐dependence  of  the  electric  field  posited  in  the  model   accurately  represents  the  actual  field  dependence.    The  model  represents  charges  as  point   particles,  which  is  usually  taken  to  be  an  idealization,  but  arguably  the  model  does  not   represent  electric  charges  as  having  infinite  mass  and  self-­‐energy  (for  otherwise  it  would   represent  charges  as  objects  that  cannot  be  moved  by  any  finite  external  force).    The   infinities  of  the  model  are  mathematical  inconveniences  that  arguably  play  no   representation  role  in  how  the  model  is  used.2   The  success  of  a  scientific  representation  depends  on  a  selective  likeness  between   the  representation  and  its  target:    what  a  representation  r    represents  its  target  t    as,  needs   to  be  appropriately  similar  to  the  target.    And  again  which  aspects  of  the  representation  are   relevant  to  judging  its  likeness  to  the  target  and  what  counts  as  sufficiently  similar  for   success  depends  on  the  context  in  which  the  representation  is  used  and  can  change  from   context  to  context.    Thus,  one  and  the  same  scientific  model  can  provide  an  adequate  or   successful  representation  of  an  object  in  some  contexts  but  not  in  others.   Representation,  then,  is  best  thought  of  not  as  a  two-­‐place  relation  but  rather  as  a   multi-­‐place  relation,  which  includes  a  place  for  the  user  of  the  representation  and  for  its   context,  aim,  or  purpose.    If  we  take  aims  and  purposes  to  be  implicit  in  the  context,  we  can   construe  representation  as  a  four-­‐place  relation:  a  is  a  representation  of  b,  exactly  if  there   is  some  context  c  in  which  a  user  u  uses  a  to  represent  b.3                                                                                                                   2  A  partial  structure  approach  is  tailor-­‐made  to  capture  the  various  roles  that  properties   and  relations  of  a  model  can  play.    My  claim  here  is  that  which  partial  structure  adequately   captures  the  representational  content  of  a  given  scientific  representation  is  determined  by   pragmatic  and  context-­‐dependent  factors.     3  Similarly,  Ronald  Giere  has  proposed  to  understand  scientific  representation  in  terms  of   the  following  four-­‐place  relation:  ‘S  uses  X  to  represent  W  for  purposes  P’  (Giere  2006,  60).       6     3.  No  perfect  model   Van  Fraassen  argues  that  just  as  partial  resemblance  can  figure  in  successful   representation  distortion,  or  selective  non-­‐resemblance  plays  an  important  role  as  well.    As  a   scientific  example  he  discusses  the  fact  that  classical  physics  represents  objects  as  having   sharp  boundaries.    Already  the  very  idea  of  the  true  and  exact  shape  and  the  true  and   precise  boundaries  of  a  macroscopic  physical  object  might  strike  one  as  suspect.    More   troubling,  sharp  boundaries  in  a  mathematical  model  often  result  in  discontinuities  or   singularities,  where  the  physics  used  to  represent  a  system’s  behavior  sufficiently  far  away   from  the  boundaries  breaks  down.    Thus,  representing  objects  as  having  sharp  boundaries   could  not  possibly  be  completely  accurate  and  non-­‐distorting,  at  least  if  they  involve   singularities,  nevertheless  such  representations  fulfill  an  important  role  in  our  scientific   image  of  the  world  and  may  in  certain  contexts  provide  us  with  the  only  means  to  construct   useful  representation  of  certain  phenomena.   Sometimes  there  are  techniques  to  patch  over  what  happens  at  such  boundaries,  but   often  not  in  ways  that  allow  for  a  single  unified  representation  of  the  system.    Mark   Wilson’s  book  (Wilson  2008)  is  replete  with  fascinating  examples  of  how  our   representational  practices  in  classical  physics  have  to  distort  to  be  successful  at  all,   representing  phenomena  by  partially  overlapping  yet  in  some  sense  incompatible  ‘theory   façades’,  breaking  down  at  the  fault  lines  between  the  façades.    The  situation  here  is  in   certain  respects  analogous  to  multi-­‐perspectival  paintings  by  artists  like  Picasso,  which   bring  together  on  a  single  canvas  different  perspectives  on  different  parts  of  the  human   body,  without,  however,  being  able  to  combine  these  multiple  perspectives  into  yet  another   unifying  perspective  (see  also  van  Fraassen  2008,  38).   However,  I  want  to  focus  here  on  a  different  cluster  of  reasons  for  why  distortions   play  a  central  role  in  scientific  representations.    Paul  Teller  (Teller  2001)  has  argued   forcefully  against  what  he  calls  “the  perfect  model  model”—that  is,  the  view  that  our  best   scientific  theories  provide  use  with  complete  and  perfectly  accurate  models  of  physical   phenomena,  or  at  least  that  physics  is  progressing  toward  and  aiming  at  developing  ever                                                                                                                                                                                                                                                                                                                                                                         I  take  Giere’s  proposal  to  be  implied  by  my  perhaps  slightly  broader  suggestion:  purposes   can  be  understood  to  be  given  by  contexts.     7   more  complete  and  accurate  and  non-­‐distorting  models  of  the  world.    Teller  argues  that   this  view  of  physics  is  mistaken  and  emphasizes  in  its  stead  the  importance  of  highly   idealized  models.    Idealized  models  distort  in  that  they  represent  only  some  aspects  of  the   physical  system  modeled  while  leaving  out  other  aspects  and  may  purposefully   misrepresent  some  of  the  aspects  represented.    According  to  some  of  Teller’s  arguments,   which  I  want  to  amplify  here,  distorting  models  at  the  very  least  play  an  important   explanatory  role  and  would  not  be  rendered  explanatorily  superfluous  by  complete  and   perfect  models.    But  there  are  even  stronger  arguments  that  suggest  that  the  idea  that   physics  even  in  principle  presents  us  with  perfect  models  of  the  phenomena  is  a  myth  and   that  all  scientific  representations  are  distorting.   Teller  illustrates  the  importance  of  distorting  models  by  pointing  to  different  ways   in  which  physicists  model  different  aspects  of  the  behavior  of  water.    Continuum  models   can  correctly  represent  the  wave  behavior  of  water,  whereas  particle  models  are  used  to   represent  diffusive  behavior.    While  both  kinds  of  model  represent  certain  aspects  of  the   behavior  of  water  sufficiently  accurately,  neither  type  constitutes  a  perfect  model  of  water   that  can  successfully  represent  all  of  its  properties.    And  both  types  of  model  manage  to   capture  aspects  of  the  behavior  of  water  by  distorting,  by  representing  water  either  as  a   continuum  or  as  a  collection  of  classical  particles.    The  example,  thus,  supports  the  claim   that  given  the  way  our  world  is,  for  many  physical  systems,  at  least,  there  is  no  single   model  of  the  system  that  allows  us  successfully  to  represent  the  system  in  any  kind  of   circumstance  and  we  need  to  employ  different,  and  even  in  some  sense  incompatible   models  to  represent  successfully  different  aspects  of  the  system’s  behavior.   Now,  one  might  reply  to  Teller’s  example  by  arguing  that  in  addition  to  these  two   types  of  models  there  exists  a  third  type—quantum  mechanical  models—that  do  provide   us  with  perfect,  non-­‐distorting  representations  of  water  and  that  can  unify  the  two  classical   types  of  model  by  showing  how  both  can  be  approximately  derived  from  the  correct  and   complete  micro-­‐theory  by  taking  appropriate  limits.    This  reply  can  be  read  as  arguing   either  against  the  claim  that  the  idealized  models  are  explanatorily  ineliminable  or  against   the  claim  that  all  models  in  physics  are  distorting.    I  want  to  focus  on  the  issue  of   explanatoriness  first.     8   Let  us  provisionally  grant  for  now  that  present-­‐day  quantum  mechanics  provides  us   in  principle  with  an  immensely  complex  yet  accurate—indeed,  perfect—model  of  water.     Nevertheless,  as  Teller  argues  convincingly,  even  if  we  imagined  that  we  were  given  the   solution  to  the  Schrödinger  Equation  for  a  system  of  1025  variables,  this  would  not  provide   us  with  an  explanation  of  macroscopic  waves  in  a  body  of  water,  since  being  able  to  grasp   what  this  solution  says  about  the  behavior  of  the  body  of  water  goes  far  beyond  what  is   cognitively  possible  for  us  humans.    Explanation  and  understanding  are  inherently   pragmatic  notions—explanation  is  always  explanation  for  us—and  we  absolutely  require   the  more  idealized  classical  models  to  be  able  to  understand  the  water’s  behavior.   (Frisch  1998)    makes  what  is  essentially  the  same  point  by  contrasting  classical   Newtonian  explanations  of  the  collection  of  balls  know  as  “Newton’s  cradle”  with  a  putative   quantum  mechanical  account  of  the  system  of  five  balls.    Frisch  argues  that  there  are   additional  reasons  not  to  take  a  putative  quantum  mechanical  account  as  providing  a   satisfactory  explanation.    Scientific  explanations  answer  “what-­‐if-­‐things-­‐had-­‐been-­‐ different-­‐questions”  (see  also  Woodward  2003);  that  is,  a  scientific  explanation  embeds  the   explanandum  phenomenon  into  a  pattern  of  counterfactual  dependencies.    In  the  physical   sciences  explanations  provide  mathematical  models  of  a  phenomenon  that  embed  it  into  a   pattern  of  functional  dependencies,  which  informs  us  how  features  of  the  explanandum   vary  with  the  values  of  the  variables  used  to  represent  the  phenomenon.    In  a  sense,  this   account  puts  rather  weak  constraints  on  what  counts  as  an  explanation:  functions  are  easy   to  come  by.    What  distinguishes  putative  explanations  from  one  another  is  not  so  much  the   question  whether  a  given  account  is  an  explanation  or  not—even  Moliere’s  dormative   virtue  is  weakly  explanatory,  one  could  claim—but  rather  the  relative  strength  of  an   explanation.    The  goodness  of  an  explanation  depends  on  its  accuracy,  strength,  and   simplicity.    An  explanation  is  stronger,  if  it  includes  more  factors  that  make  a  difference  to   the  occurrence  of  the  explanandum.    The  consideration  of  simplicity  pulls  in  the  opposite   direction:    An  explanation  is  simpler  if  it  does  not  add  irrelevant  details.    Both  these   conditions,  as  well  as  what  counts  as  sufficient  accuracy,  are  context-­‐dependent.    Including   certain  variables  in  a  model  of  a  phenomenon  may  be  explanatory  important  in  one   context,  while  the  same  variables  may  be  adding  irrelevant  details  in  another.    The  best     9   explanation  is  one  that  scores  best  on  some  weighted  average  of  these  criteria,  where  again   there  is  no  context-­‐independent  algorithm  for  computing  this  average.   A  consequence  of  this  account  is  that  an  explanation  that  is  more  accurate  than  all  of   its  rivals  need  not  be  the  best  explanation  of  a  phenomenon.    In  fact,  in  most  contexts  it  will   be  case  that  a  microscopic  quantum-­‐mechanical  account  of  the  state  of  all  the  molecules   composing  the  water  will  be  taken  to  offer  much  too  much  unnecessary  details  for   successfully  explaining  the  water’s  wave-­‐like  behavior.    Thus,  in  most  explanatory  contexts   a  full  microscopic  quantum  mechanical  model  would  be  explanatorily  inferior  to  the   classical  model  even  if  per  impossible  we  were  able  to  cognitively  grasp  the  details  of  the   former  model.    What  is  crucial  for  an  understanding  of  the  behavior  of  water  waves  (in   most  contexts)  is  an  understanding  of  the  general  patterns  that  transcend  the  details  of  the   given  case—patterns  that  arguably  would  be  lost  in  the  minute  details  of  a  quantum   mechanical  model  with  on  the  order  of  1025  variables  and  that  transcend  the  details  of  a   putatively  quantum  mechanical  micro-­‐model.    Thus,  even  if  we  were  to  grant  that  quantum   mechanics  provides  us,  at  least  in  principle,  with  a  perfect  model  of  the  behavior  of  water,   this  would  not  render  the  distorting  models  superfluous:    in  many  contexts  the  distorting   models  still  provide  us  with  the  best  explanation  of  the  behavior  of  water  and  their   explanatory  success  depends  precisely  on  the  fact  that  these  models  are  highly  idealized   and  distorting.   But  in  fact  we  have  granted  much  too  much  to  the  objection,  for  we  do  not  actually   possess  a  perfectly  accurate  and  complete  quantum  mechanical  model  of  wave  or  diffusion   phenomena.    This  brings  us  to  the  second  issue—the  foundationalist  assumption  that  there   exist  perfectly  accurate,  fundamental  models,  constructed  with  the  help  of  our  most   fundamental  theories,  underlying  the  idealized  higher-­‐level  models  we  use  in  practice.    I   now  want  to  challenge  this  assumption.    First,  quantum  mechanics  has  its  own  limits  in   scope  and  accuracy—current  quantum  mechanics,  too,  distorts  and  is  not  the  final  and   correct  theory  of  the  world.    Second  and  more  importantly,  even  if  we  were  to  belief  that   present  day  quantum  theory  was  exactly  correct  wherever  it  can  be  applied,  it  still  would   not  provide  us  with  models  of  the  macro-­‐phenomena  at  issue  here.    We  have  so  far   imagined  that  we  were  somehow  given  a  quantum-­‐mechanical  model  of  macroscopic   bodies  of  water,  but  that  is  of  course  only  an  impossible  fiction.    To  actually  construct  any    10   such  model,  we  would  have  to  solve  the  Schrödinger  equation  for  on  the  order  of  1025   variables—something  that  is  impossible  to  do  and  far  beyond  our  computational  capacities.   At  this  point  one  common  reply  is  to  insist  that  even  if  it  is  impossible  actually  to   solve  the  Schrödinger  equation  for  macroscopic  systems,  the  theory  nevertheless  provides   us  with  models  of  arbitrary  complexity.    The  equation  defines  a  class  of  models,  many  of   which  we  of  course  never  construct  explicitly.    Indeed,  any  physical  theory  has  many,  many   more  models  than  the  ones  scientists  have  actually  constructed  and  actually  used.     Quantum  mechanics  contains  a  model  of  the  hydrogen  atom,  of  Bose-­‐Einstein  condensates,   but  also,  one  might  argue,  of  any  arbitrary  system.  If  a  solution  of  the  Schrödinger  equation   exists  for  certain  arbitrarily  complex  initial  conditions  for  systems  consisting  of  1025   particles,  then  simply  in  virtue  of  being  in  possession  of  the  equation  we  thereby  also  are   given  a  model  of  such  systems,  even  if  we  do  not  know  how  to—or  are  practically  unable   to—explicitly  construct  this  solution.   But  if  van  Fraassen’s  account  of  scientific  representation  is  correct,  then  the  mere   fact  that  an  equation  has  solutions  in  addition  to  the  ones  actually  constructed  does  not   imply  that  whenever  we  possess  a  theory  we  thereby  also  are  in  possession  of  a  large  range   of  models  of  arbitrarily  complex  systems  governed  by  that  theory.    That  quantum   mechanics  provides  us  with  a  model  even  of  macroscopic  bodies  of  water  is  supposed  to   follow  from  the  claim  that  among  the  set  of  solutions  to  the  Schrödinger  equation  are  ones   that  are  structurally  similar  to  bodies  of  water.      Yet,  as  we  have  seen,  according  to  van   Fraassen’s  account  no  structural  relationship  between  a  model  and  a  phenomenon  can  on   its  own  suffice  to  make  the  model  a  representation  of  that  phenomena  (see  also  van   Fraassen  2008,  250).    Rather  something  is  a  representation  only  if  it  is  used  to  represent  a   thing.    But  since  we  do  not  even  actually  have  the  quantum  mechanical  initial  state  of  the   system  of  water  let  alone  a  solution  of  the  Schrödinger  Equation  for  that  system—since   there  is  no  way  for  us  to  pick  out  the  appropriate  solutions  from  the  class  of  solutions   defined  by  the  Schrödinger  equation—we  cannot  use  the  putatively  existing  solution  to   represent  anything.   But  is  it  not  correct  that  the  Schrödinger  Equation  has  many  more  models  than  the   ones  actually  constructed  by  us?    And  if  we  allow  that  the  Schrödinger  equation  has  models   for  arbitrarily  complex  initial  conditions,  does  it  not  follow  that  among  the  equation’s    11   models  there  will  be  some  with  initial  conditions  representing  the  state  of  the  body  of   water?    This  reply,  however,  trades  on  an  ambiguity  in  the  term  ‘model’.    There  are  two   quite  different  senses  of  model  between  which  we  have  to  distinguish  carefully.    On  the  one   hand,  there  is  the  notion  of  model  as  representation,  according  to  which  a  structure  is  a   model  of  a  thing  just  in  case  it  is  used  to  represent  that  thing.    That  is,  something  is  a  model   of  some  object  or  system  in  virtue  of  representing  the  object  or  system.    And  if  van  Fraassen   is  right,  then  nothing  is  a  model  in  this  sense  without  actually  and  as  a  matter  of  fact  being   used  as  a  representation—the  existence  of  a  certain  structural  relation  between  the  model   and  the  target  system  is  not  enough  for  it  to  be  a  representation  of  the  system.    On  the   other  hand,  a  structure  is  a  model  of  a  set  of  sentences,  in  the  logical  or  model-­‐theoretic  sense   of  ‘model’,    just  in  case  it  satisfies  that  set  of  sentences  or  the  set  of  sentences  are  true  in   that  structure.    A  linguistic  description  of  a  theory  serves  to  specify  the  theory’s  model-­‐ theoretic  models  in  the  sense  in  which  a  set  of  equations  specifies  the  set  of  its  own   solutions.    This  second  notion  of  model  is  not  an  intentional  notion.    All  that  is  required  for   a  structure  to  be  a  model  in  this  sense,  is  that  a  mapping  from  the  structure  to  the  set  of   sentences  exists  such  that  all  the  sentences  in  the  set  come  out  true;  it  need  not  be  the  case   that  there  is  a  user  who  takes  the  set  of  sentences  to  be  true  in  that  particular  structure.   If  we  accept  van  Fraassen’s  account  of  representation  and  “there  is  no  such  thing  as   representation  apart  from  or  independent  of  our  practice”  (2008,  258),  then  it  does  not   follow  from  the  fact  that  a  set  of  equations  has  solutions  or  models  in  the  non-­‐intentional,   model-­‐theoretic  sense  that  exist  all  along  even  without  us  using  or  being  able  to  construct   these  solutions,  that  these  solutions  are  also  models  in  the  first,  representational  and   intentional  sense.    Recall  van  Fraassen’s  “Hauptsatz”:    “There  is  no  representation  except  in   the  sense  that  some  things  are  used,  made,  or  taken,  to  represent  things  as  thus  and  so.”    Thus   solutions  to  equations  that  we  have  not  found  or  constructed  cannot  be  used  to  represent   anything,  simply  because  we  cannot  use  anything  that  we  do  not  have  or  do  not  even  know   exists.    Even  if  we  assume  that  the  Schrödinger  Equation  has  a  solution  for  a  system  of  1025   variables  of  the  kind  that  we  might  possibly  use  to  represent  a  small  macroscopic  body  of   water  if  we  were  to  possess  this  solution,  the  mere  existence  of  the  solution  does  not  imply   that  there  is  a  quantum  mechanical  representational  model  of  the  waves  in  a  body  of  water   or  of  diffusion  phenomena  in  water.    The  idea,  then,  that  our  most  fundamental  micro-­‐  12   theories  provide  us  with  representational  models  covering  all  physical  phenomena  is  a   foundationalist  myth.   Surprisingly,  van  Fraassen  himself  does  not  draw  this  conclusion  or  at  least  is   ambiguous  in  his  views  regarding  this  issue.    He  considers  the  question  in  what  sense  a   theory  or  equation  can  be  a  theory  of  phenomena  not  actually  encountered  in  practice— that  is,  of  phenomena  that  we  have  not  actually  described  and  for  which  have  not  actually   constructed  a  model  with  the  help  of  the  theory  or  equation.    Van  Fraassen  would  like  to   agree  with  what  I  described  as  the  common  view  and  “would  like  to  say  that  if  the  equation   does  have  [an  appropriate]  solution—equivalently,  if  the  theory  has  such  a  model—then   that  (equation,  theory)  does  correctly  represent  that  phenomenon.”  (2008,  250)    Later  in   the  book  he  says:   The  sense  in  which  a  theory  offers  or  presents  us  with  a  family  of  models  is   just  the  sense  in  which  a  set  of  equations  presents  us  with  the  set  of  its  own   solutions.    In  many  cases,  no  solutions  to  a  given  equation  are  historically   found  or  constructed  for  a  very  long  time  …  though  mathematically  speaking,   they  exist  all  along.    When  the  equations  formulate  a  scientific  theory,  their   solutions  are  the  models  of  the  theory.  (van  Fraassen  2008,  310,  italics  in   original).   The  latter  passage  occurs  within  the  context  of  a  discussion  of  Nancy  Cartwright’s  view  that   there  are  models  in  science  that  have  an  existence  that  is  in  some  sense  independent  of  the   theories  with  the  help  of  which  they  are  constructed.    Thus,  van  Fraassen  here  appears  to   be  guilty  of  not  carefully  differentiating  between  the  two  notions  of  model  that  I   distinguished  above:  the  solutions  of  the  equations  are  model-­‐theoretic  models  but  not   thereby  automatically  also  representational  models.    Even  though  Cartwright  is  clearly   interested  in  models  as  representational  structures,  van  Fraassen  invokes  the  model-­‐ theoretic  notion  of  model  in  his  response  to  her.    Yet—putting  the  point  without  using  the   term  ‘model’—the  fact  that  we  can  formally  define  a  class  of  structures  that  satisfy  a  set  of   sentence  says  nothing  about  the  representational  use  to  which  we  might  put  those   members  of  the  class  that  we  have  in  fact  explicitly  constructed.    And  van  Fraassen  himself   elsewhere  in  the  book  appears  to  stress  this  very  point:    13   There  is  no  such  thing  as  representation  apart  from  or  independent  of  our   practice.    So  how  can  we  say  something  like  “this  theory  accurately   represents  that  bacterial  growth  phenomenon”  although  the  relevant  model   was  never  constructed  and  the  bacterial  colony  was  certainly  not   encountered  in  human  practice?    The  structural  relationship  between  the   model  in  question  and  the  phenomenon,  that  we  just  described,  does  not   suffice  to  make  the  model  a  representation  of  the  phenomena.  (2008,  249)   And:   Undoubtedly  in  many  contexts  something  is  called  a  model  only  if  it  is  a   representation,  and  the  sense  in  which  any  solution  of  an  equation  is  a  model   of  the  theory  expressed  by  that  equation  certainly  does  not  have  that   meaning.  (2008,  250)   Thus,  van  Fraassen  apparently  wants  to  be  committed  to  two  ideas  that  seem  to  be   in  tension  with  each  other:  on  the  one  hand,  the  idea  that  “we  would  like  to  say”  that  if  a   theory  formally  has  an  appropriate  solution  then  it  does  represent  the  phenomenon  in   question  even  if  the  solution  has  not  been  explicitly  constructed;  and  on  the  other  hand,  the   idea  that  there  is  no  representation  independent  of  its  being  used  as  such  and  that  “there  is   nothing  useful  to  be  found  in  2-­‐place  structure-­‐phenomenon  relations  alone  when  we  try  to   understand  representation.”  (2008,  258)   His  resolution  of  the  apparent  tension  is  to  understand  the  notion  of  the  empirical   adequacy  of  a  theory  in  terms  of  counterfactual  representations:   If  the  theory  is  offered,  that  amounts  to  the  offer  of  a  range  of  structures— the  structures  we  call  models  of  the  theory—as  candidates  for  the   representation  of  the  phenomena  in  its  domain.    If  this  range  contains  a   candidate  that  would  satisfy  the  structural  constraint—if  the  phenomenon  is   embeddable  in  it  […]—then  the  theory  is  empirically  adequate.  (2008,  250)   That  is,  offering  a  theory  amounts  to  offering  a  range  of  model-­‐theoretic  models—of   mathematical  structures  that  we  could  use  to  represent  phenomena.    And  a  theory   represents  a  particular  phenomenon  within  its  domain  adequately,  just  in  case  there  is  a   structure  among  the  range  defined  by  the  theory  such  that  were  we  to  use  this  structures  as    14   representation  for  that  phenomenon,  then  the  phenomenon  could  be  embedded  into  the   model.   There  is  a  certain  irony  in  the  fact  that  van  Fraassen  appears  to  be  driven  to  appeal   to  counterfactuals  here,  given  his  well-­‐known  view  that  counterfactuals  are  inherently  and   irreducibly  context-­‐dependent.    What  are  the  truth-­‐conditions  of  claims  of  the  form  ‘if  we   were  to  use  a  structure  s  to  represent  phenomenon  p,  then  p  would  be  embeddable  in  s’?     The  problem  is  that  it  is  not  clear,  independent  of  our  actual  use  of  a  structure  to  represent   a  phenomenon,  that  there  is  a  unique  answer  as  to  how  the  structure  would  be  used  to   represent  the  phenomenon  and  what  the  appropriate  embedding  relation  would  be  were   we  to  use  the  structure  as  representation.    And  again  it  is  van  Fraassen  himself  who  has   convincingly  shown,  for  reasons  having  to  do  with  Putnam’s  model-­‐theoretic  argument,   why  there  is  no  unique  embedding  relation  independently  of  our  use  of  a  given  structure.     In  that  context,  van  Fraassen  stresses  precisely  the  point  I  wanted  to  emphasize  here:    That   we  have  to  be  careful  about  an  “illegitimate  slippage  from  ‘there  exists’  to  ‘we  have’”  (van   Fraassen  2008,  233).    While  there  may  exist  solutions  to  the  Schrödinger  equation  for   systems  of  1025  variables,  simply  writing  down  the  general  form  of  the  equation  does  not   imply  that  we  thereby  have  all  of  its  solutions  of  arbitrary  complexity.   In  the  next  subsection  I  will  summarize  some  of  the  considerations  leading  up  to   Putnam’s  argument  that  are  relevant  to  our  discussion  here  and  which  will  serve  to  further   amplify  the  claim  that  the  notion  of  representation  is  essentially  tied  to  a  representations’   use.       4.  Structures  and  Use   Physical  theories  provide  us  with  mathematical  representations  of  phenomena—that  is,   with  abstract  structural  models.    Successful  theories,  it  seems,  provide  us  with  models  that   in  some  sense  resemble  the  phenomena  they  represent.    One  issue  in  this  context  is  the  one   that  divides  scientific  realists  and  empiricists  and  concerns  the  question  whether  we  can   have  good  reasons  to  believe  that  successful  models  resemble  the  physical  systems  they   represent  in  their  entirety  or  only  with  respect  to  their  observable  substructures.    This  is   not  an  issue  I  will  be  pursuing  here.    A  conceptually  prior  question  is  what  kind  of   resemblance  can  be  possible  at  all  between  abstract  mathematical  models  and  concrete    15   goings  on  in  the  world.    The  most  plausible  answer  to  this  questions  is:  structural   resemblance.    That  is,  successful  theories  provide  us  with  models  that  are  structurally   similar  to  the  phenomena  they  represent.    This  view  has  a  long  tradition  in  the  philosophy   of  science,  dating  back  at  least  to  the  Bildtheorie  of  Heinrich  Hertz  and  Boltzmann  and  is   expressed,  for  example  in  Hertz’s  famous  dictum:    “We  form  for  ourselves  inner  apparent   images  or  symbols  of  external  objects,  and  we  do  this  in  such  a  manner  that  the  necessary   consequents  of  the  images  in  thought  are  always  the  images  of  the  necessary  consequents   in  nature  of  the  objects  pictured.”4   In  a  recent  paper,  Roman  Frigg  (Frigg  2010)  has  presented  an  argument  against  the   structuralist  view  that  scientific  representation,  at  least  in  the  physical  sciences,  is   structural  representation  and  for  the  view  that  the  model  systems  at  the  heart  of  a  physical   theory  ought  to  be  thought  of  as  hypothetical  or  imagined  concrete  physical  systems.    Frigg   points  out  that  in  order  to  make  sense  of  a  structural  resemblance  between  a  model  and  its   target  one  has  to  assume  that  the  target  also  exemplifies  a  certain  structure  but,  Frigg   argues,  “this  cannot  be  had  without  bringing  non-­‐structural  features  into  play.”  (Frigg   2010,  254)    Frigg’s  argument  for  this  claim  proceeds  in  two  steps.    First,  he  argues  that   since  structures  are  abstract,  structural  claims  about  a  physical  system  cannot  be  true   unless  some  non-­‐structural  claims  are  true  as  well.    Second,  he  argues  that  the   “descriptions  we  use  to  ground  structural  claims  are  almost  never  in  fact  true  descriptions   of  the  intended  target  system”  (Ibid.)    From  which  he  concludes  that  “the  descriptions  that   ground  structural  claims  (almost  always)  fail  to  be  descriptions  of  the  intended  target   system.  Instead,  they  describe  a  hypothetical  system  distinct  from  the  target  system.”   (Ibid.)    Thus,  Frigg  wants  to  conclude,  theoretical  models  cannot  merely  be  mathematical   structures  but  are  concrete,  albeit  merely  imagined  or  hypothetical  physical  systems.   Frigg’s  second  step  begins  by  echoing  a  point  also  made  by  van  Fraassen  and  to   which  we  will  return  below:  that  structural  resemblance  is  possible  only  between  two   structures  and  hence  that  the  subject  of  the  representation  also  has  to  be  depicted  by  us  as   structured  in  a  certain  way.    Frigg  then  argues  that  such  a  depiction  cannot  be  merely   structural  but  has  to  be  concretely  ‘fitted  out.’    (For  example,  that  three  iron  rods  can  be                                                                                                                   4  For  an  investigation  of  H.  A.  Lorentz’s  adoption  of  Hertz’s  Bildtheorie  see  (Frisch  2005).      16   taken  to  exhibit  a  certain  (abstract)  ordering  relation  is  true  only  in  virtue,  say,  of  the  rods   having  different  lengths.)    So  far  so  good.    But  what  Frigg  wants  to  show  is  not  that  a   structured  depiction  of  the  phenomena  is  accompanied  by  a  more  concrete  description,  but   that  the  theoretical  models  we  employ  are  hypothetical  concrete  physical  systems.    And  the   fact  that  any  structure  attributed  to  the  phenomena  needs  to  be  embedded  into  a  concrete   description  of  the  phenomena  does  not  imply  that  theoretical  models,  too,  need  to  be   concretely  fitted  out.    The  missing  step  in  the  argument  is  meant  to  be  provided  by  the   observation  that  the  concrete  descriptions  of  the  phenomena  are  almost  never  true   descriptions,  which  is  supposed  to  make  the  introduction  of  hypothetical  systems   necessary.   I  have  two  worries  about  this  step  in  the  argument,  however.    First,  it  is  not  clear   why  the  fact  that  the  descriptions  are  false  implies  that  they  “fail  to  be  descriptions  of  the   target  system.”    More  plausibly,  it  seems  to  me,  one  might  hold  that  even  a  false  description   of  a  system  is  a  descriptions  of  that  system—it  just  may  be  a  descriptions  that  we  do  not   take  to  be  completely  true  but  that  nevertheless  plays  a  useful  role  in  our  understanding  of   the  system.    Consider  as  an  analogy  a  caricature  that  depicts  Barack  Obama  as  having  huge   ears.    It  does  not  follow  from  the  fact  that  the  caricature  misrepresents  Obama  that  there  is   some  hypothetical  person,  which  the  caricature  represents  completely  truthfully.    Similarly,   it  does  not  follow  from  the  fact,  say,  that  a  prepared  description  represents  a  certain   wooden  beam  in  an  idealized  manner  as  perfectly  rigid,  that  the  description  describes  a   hypothetical  rigid  object.    Rather  the  description  represents  the  actual  wooden  beam  as   perfectly  rigid.    That  is,  what  the  description  “literally  describes,”  as  Frigg  puts  it,  is  the   actual  wooden  beam,  even  though  what  the  description  says  about  the  beam  is  strictly   speaking  false.    Frigg  worries  that  this  proposal  leaves  the  notion  of  idealized  description   unanalyzed  and  that  any  plausible  attempt  to  analyze  it  would  need  to  introduce  the  notion   of  a  hypothetical  system  after  all.    But  one  plausible  way  to  cash  out  the  idea  that  a   description  is  idealized  is  by  reference  to  some  other  description  or  representation  of  that   very  system.    For  example,  we  might  say  that  a  description  D  of  a  system  S  is  idealized   relative  to  some  other  description  D*  of  S  just  in  case  D  ignores  certain  features  attributed   to  S  by  D*  or  simplifies  certain  features  attributed  by  D*.    Both  description  literally  describe   the  target  system,  but  D  is  idealized  relative  to  D*.    17   My  second  worry  is  that  even  if  we  were  to  grant  that  an  idealized  description  of  a   target  system  required  that  we  introduce  a  hypothetical  or  fictional  system  that  the   idealized  description  truthfully  describes,  it  does  not  follow  that  we  also  need  to  think  of   theoretical  models  as  hypothetical  concrete  physical  systems.    Frigg  proposes  that  we   simply  identify  the  hypothetical  systems  which  satisfy  the  idealized  prepared  description   of  a  physical  system  with  the  “model  systems”  of  our  theories.    But  this  presupposes  that   our  theories  imply  or  at  least  are  logically  strictly  compatible  with  the  prepared   descriptions  of  the  phenomena,  for  otherwise  our  theories  could  not  be  true  of  the  putative   hypothetical  system.    By  contrast,  if  rather  more  plausibly  we  require  only  that  our   theoretical  models  approximately  resemble  our  representations  of  the  target  system,  then   the  hypothetical  physical  systems  that  would  concretely  realize  the  structures  of  a  class  of   theoretical  models  cannot  be  identical  to  the  hypothetical  systems  that  satisfy  an  idealized   depiction  of  the  phenomena.    But  if  we  need  to  distinguish  between  a  prepared  description   of  the  phenomena  and  the  theoretical  models  that  are  meant  to  approximately  resemble   the  former,  we  still  need  an  argument  for  why  we  need  to  think  of  both  kinds  of  models  and   not  merely  of  the  prepared  descriptions  as  hypothetical  concrete  physical  systems.   Frigg  does  offer  a  second  argument  in  support  of  the  claim  that  model  systems   cannot  be  purely  structural  appealing  to  the  fact  that  “scientists  often  talk  about  model   systems  as  if  they  were  physical  things”  (2010,  253).    This  is  surely  right,  but  it  is  unclear   what  lesson  we  should  draw  from  this  observation.    One  option  might  be  to  argue  that  this   is  merely  points  to  a  surface  feature  of  scientific  practice  and  ought  not  to  be  understood   literally.    Another  possibility  is  that  what  scientists  understand  by  a  model  system  or  a   theoretical  representation  differs  widely  from  discipline  to  discipline,  from  context  to   context  and  even  from  scientist  to  scientist.  In  some  contexts,  especially  in  the  more   fundamental  theories  of  physics,  theoretical  models  might  consist  of  purely  structural,   mathematical  representations  of  the  phenomena,  in  other  contexts  the  models  might  be   concrete  yet  imagined  natural  systems.    One  might  even  grant  that  the  structural  models  in   physics  may  sometimes  be  concretely  ‘fitted  out’  for  didactic  reasons  or  because  physicists   might  find  it  fruitful  to  think  of  a  concrete  analogy  of  the  system  represented,  even  though   their  commitment  is  only  to  a  structural  resemblance  between  the  theoretical  models  and   the  systems  modeled.    Perhaps  the  planetary  model  of  the  atom  is  an  example  of  this  kind,    18   which  may  be  best  thought  of  as  involving  two  kinds  of  model—a  purely  mathematical   model  that  is  taken  to  structurally  resemble  actual  atoms,  and  a  hypothetical  physical   system  that  is  proposed  as  a  useful  concrete  analogy  of  the  system  modeled.    The  way  in   which  atoms  are  proposed  to  be  like  planetary  systems  is  exhausted,  however,  by  the   structural  resemblance  postulated  in  the  mathematical  equations  governing  the  Bohr  atom.   Thus,  contrary  to  the  view  that  all  scientific  modeling  involves  hypothetical  concrete   physical  systems  it  seems  to  be  more  plausible,  as  Godfrey-­‐Smith  has  argued  (Godfrey-­‐ Smith  2009,  104),  that  scientific  modeling  involves  a  “gradient  of  abstraction”.    On  the  one   end  of  the  spectrum  we  find  approximate  or  idealized  concrete  descriptions  of  a  target   system,  without  the  introduction  of  fictional  entities  representing  the  target,  as  I  suggested   above.    On  the  other  end  of  the  spectrum  are  the  mathematical  models  of  theoretical   physics,  which  sometimes  are  investigated  as  purely  mathematical  structures  in  their  own   right,  to  some  extent  independently  of  possible  empirical  applications.    Somewhere  in  the   middle  are  fictional  concreta—hypothetical  physical  systems—which  can  play  a  variety  of   different  roles,  including  that  of  direct  representations  of  the  phenomena,  as  Frigg  argues,   or  that  of  offering  concrete  structural  analogies  to  systems  represented  mathematically.     Pace  Frigg,  then,  I  agree  with  van  Fraassen  that  scientific  representations,  at  least  in   the  mathematical  sciences,  are  in  the  first  instance  abstract  mathematical  structures  that   are  intended  to  structurally  resemble  their  target.    There  is,  however,  a  well-­‐known   problem  for  the  view  that  scientific  representation  is  purely  structural  representation:   structural  resemblance,  it  seems,  is  much  too  easily  to  be  had.    This  point  is  expressed,  for   example,  in  Putnam’s  model-­‐theoretic  argument,  which  argues  the  following:    as  long  as  the   physical  system  that  we  want  to  model  is  composed  of—or  is  taken  to  be  composed  of—a   sufficiently  large  number  of  parts,  there  will  always  be  a  mapping    from  the  variables  of  the   model  onto  parts  of  the  system  such  that  the  system  and  the  model  exhibit  similar   structures.    Van  Fraassen  illustrates  this  point  with  the  example  of  a  blank  sheet  of  paper   that  can  be  taken  to  provide  an  accurate  map  of  the  streets  of  Paris.    As  long  as  the  sheet  of   paper  has  enough  distinguishable  elements  (due  to  small  unevennesses  of  the  surface  of   the  sheet),  then  there  will  be  some  function  that  maps  the  landmarks  and  streets  of  Paris   onto  the  sheet  of  paper  such  that  the  sheet  of  paper  thus  structured  is  isomorphic  to  the   structure  given  by  the  network  of  Parisian  streets    The  problem  is  how  to  single  out  from    19   the  myriad  possible  mappings  from  a  model  onto  a  phenomenon  one  particular  mapping  as   the  intended  one,  with  the  possibility  that  our  model  turns  out  not  to  resemble  the   phenomenon  under  the  intended  mapping,  as  we  would  like  to  conclude  in  the  case  of  the   sheet  of  paper.    If  there  is  no  additional  constraint  that  allows  us  to  distinguish  permissible   from  impermissible  mappings,  then  the  claim  that  there  exists  an  appropriate  structural   resemblance  between  a  model  and  some  physical  system  turns  out  to  be  nothing  more  than   a  claim  about  the  number  of  elements  of  the  model  and  the  system.   David  Lewis  replied  to  Putnam’s  argument  by  arguing  that  there  is  an  additional   constraint  on  the  mappings  given  by  preferred  or  natural  divisions  and  relations  among   objects  in  the  world.    A  representation  is  successful,  according  to  Lewis’  proposal,  not   merely  if  the  physical  system  represented  can  be  structured  in  some  way  that  is  isomorphic   to  the  representation,  but  only  if  the  representation  is  approximately  isomorphic  to  the   structure  given  by  the  natural  kinds  out  of  which  the  system  is  composed.    That  is,   according  to  Lewis,  nature  itself  has  a  preferred  or  natural  relational  structure  and  a   theory’s  models  are  intended  to  represent  this  structure  as  accurately  as  possible.    But   aside  from  worries  about  the  metaphysical  commitments  implied  by  Lewis’s  reply,  it  is  not   clear  how  general  his  strategy  can  be  applied,  since  the  kinds  invoked  by  all  but  our  most   fundamental  scientific  theories  are  not  good  candidates  for  being  natural  kinds.   Putnam’s  own  solution  to  his  puzzle  is  to  advocate  a  deflationary  view  of  reference   combined  with  the  view  that  our  use  of  our  representations  fixes  their  meaning  (see  also   Frisch  1999).    Van  Fraassen  (2008)  agrees  at  least  with  the  second  part  of  this  and  further   amplifies  it  as  follows.    Rejecting  Lewis’s  anti-­‐nominalism  appears  to  leave  us  with  a   problem:    how  can  an  abstract  mathematical  structure  resemble  something  in  nature  that  is   not  abstract?    We  said  that  the  appropriate  resemblance  relation  between  the   mathematical  models  of  our  theories  and  the  phenomena  they  are  intended  to  represent  is   one  of  structural  resemblance.    But  if  we  reject  the  idea  that  the  world  itself  exhibits  a   preferred  relational  structure  given  by  the  natural  kinds,  then  what  are  the  structures  in   nature  that  our  scientific  models  are  intended  to  resemble?    Van  Fraassen’s  answer  to  this   problem  once  more  emphasizes  the  role  of  the  user  in  representation.    Theoretical  models,   he  maintains,  are  intended  to  resemble  data  models  of  the  phenomena,  which  are   constructed  through  the  “selective  relevant  depiction  of  the  phenomena  by  the  user  of  the    20   theory  required  for  the  possibility  of  representation  of  the  phenomenon.”  (2008,  253).     That  is,  the  phenomena,  which  our  theoretical  models  are  meant  to  represent  are   structured  by  us  relative  to  our  interests:  “the  phenomenon,  what  it  is  like  taken  by  itself,   does  not  determine  which  structures  are  data  models  for  it—that  depends  on  our  selective   attention  to  the  phenomenon.”  (2008,  254)   The  overall  picture,  then,  that  emerges  consists  of  two  stages.    We  represent  a   phenomenon  by  what  van  Fraassen  calls  a  ‘data  model’  of  the  phenomenon  and  which   provides  a  structured  depiction  of  the  phenomenon;  a  successful  theory  provides  us  with   theoretical  models  into  which  data  models  of  the  phenomena  within  the  theory’s  domain   can  be  approximately  embedded,  where  it  is  our  use  of  the  theoretical  models  that  singles   out  the  intended  embedding.    Thus,  the  user  enters  the  account  of  scientific  representation   at  two  places:  first,  in  the  depiction  of  a  phenomenon  as  structured  in  a  certain  way;  and   second,  in  taking  a  model  to  represent  the  phenomenon,  depicted  as  thus  structured.   One  might  object  that  in  requiring  of  our  theories’  models  only  that  data  models  can   be  approximately  embedded  into  them  we  are  committing  ourselves  to  van  Fraassen’s   empiricism.    Yet  the  account  of  representation  is  independent  of  constructive  empiricism.     Someone  who  takes  theories  not  merely  to  represent  observational  substructures  of  a   phenomenon  could  replace  van  Fraassen’s  notion  of  a  data  model  with  a  notion  of   phenomenological  model  that  includes  in  its  depiction  of  a  phenomenon  also  an   unobservable  substructure  not  represented  in  the  data  model.    One  might  also  worry  with   Cartwright  that  the  first  step  in  depicting  a  physical  system  is  not  yet  a  mathematically   precise  data  model,  which  often  already  involves  a  significant  amount  of  theoretical   analysis,  but  rather  a  still  somewhat  informal  prepared  description.    The  resulting  picture   is  somewhat  more  complicated  than  the  one  suggested  by  van  Fraassen  and  includes  a   prepared  description,  data  models,  phenomenological  models,  and  theoretical  models.    We   test  our  theories  by  examining  whether  a  data  model  can  be  approximately  embedded  into   a  theoretical  model;  if  the  answer  is  ‘yes’,  then  this  provides  us  with  reasons  to  believe  that   the  theoretical  model  structurally  resembles  the  phenomenon  as  formally  depicted  in  the   phenomenological  model,  and  that  our  initial  prepared  description  has  proven  itself.   Another  worry  is  that  on  this  account  it  may  seem  that  we  lose  the  ultimate  link  of   our  theories  to  reality,  since  we  never  appear  to  get  beyond  the  phenomena  as  described    21   by  us  (either  in  a  data  model  or  in  a  phenomenological  model).    Van  Fraassen  reply  to  this   objection  is  to  argue  that  the  worry  disappears  once  we  appreciate  that  for  us  there  is  no   difference  between  the  question  whether  a  theory  fits  a  phenomenon  and  the  question   whether  a  theory  fits  that  phenomenon  as  represented  by  us.    That  the  two  questions  are   equivalent  for  us,  he  maintains,  is  a  pragmatic  tautology.    Thus,  the  gap  that  the  objection   tries  to  exploit  cannot  be  coherently  expressed  by  us.    Of  course  we  can  ask  if  someone  else’s   structured  depiction  of  a  phenomenon  is  adequate  or  appropriate.    But  we  cannot  do  this   by  contrasting  the  depiction  by  the  phenomenon  with  the  phenomenon  itself,  but  only  by   comparing  the  depiction  with  the  phenomenon  as  represented  by  us.     5.  Anti-­‐fundamentalism   For  our  present  purposes  the  crucial  point  is  that  considerations  from  Putnam’s  model-­‐ theoretic  argument  further  amplify  the  pragmatic  dimension  in  the  notion  of   representation  and  provide  additional  support  for  van  Fraassen’s  Hauptsatz  according  to   which  a  structure  is  a  representation  only  if  it  is  used  as  a  representation.    But  if  all  this  is   correct,  then  it  seems  to  follow  that  the  threat  of  Putnam’s  argument  can  only  be  avoided   for  structures  that  we  are  actually  using  as  models  and  this  conclusion  seems  highly   counterintuitive.    In  accepting  Newtonian  physics,  say,  are  we  not  committed  to  the  claim   that  the  theory  successfully  applies  to  planetary  systems  yet  to  be  discovered  and  systems   of  billiard  balls  never  explicitly  modeled?    Any  theory’s  domain,  it  seems,  extends  well   beyond  the  class  of  phenomena  for  which  we  have  actually  constructed  models.    How,  then,   can  we  combine  this  seemingly  obvious  point  with  the  lessons  of  Putnam’s  argument?   It  seems  to  me  that  we  need  to  distinguish  carefully  between  the  kind  of  example   van  Fraassen  himself  mentions  when  he  discusses  this  issue—that  of  a  colony  of  bacteria   located  somewhere  in  Antarctica  long  before  the  first  humans  appeared  on  Earth—and  the   example  we  discussed  above:  a  putative  quantum  mechanical  micro-­‐model  for  the  macro-­‐ behavior  of  water.5    Van  Fraassen  asks  whether  we  can  say  that  a  theory  of  exponential                                                                                                                   5  Van  Fraassen  himself  (2008,  25-­‐6)  discusses  the  putative  worry  as  to  how  our  models  might  be   able  “to  represent  something  that  has  not  yet  entered  our  acquaintance”.    I  agree  with  van  Fraassen   that  this  worry  does  not  genuinely  arise.    My  worry  here  is,  as  it  were,  the  mirror-­‐image  of  this:     How  can  we  represent  anything  with  models  that  have  not  yet  entered  our  acquaintance?    22   growth  adequately  represents  the  evolution  of  this  colony,  even  though  by  hypothesis  no   model  for  this  particular  phenomenon  was  ever  offered.    His  reply,  as  we  have  seen  above,   is  to  appeal  to  a  counterfactual  account  of  empirical  adequacy:  the  theory  is  adequate  if   among  the  solutions  to  its  equations  is  one  defining  a  structure  that  would  satisfy  the   relevant  constraints  on  adequacy  if  it  were  used  to  represent  the  colony’s  evolution.    The   worry  raised  by  Putnam’s  argument  is  whether  this  counterfactual  has  reasonably  well-­‐ defined  truth  conditions.    I  want  to  suggest  that  the  answer  is  ‘yes’  in  the  present  case,  since   scientists  actually  and  as  a  matter  of  fact  use  models  of  bacterial  colonies  to  represent  their   growth  and  arguably  this  practice  sufficiently  constrains  how  we  would  represent  the   Antarctic  colony  were  we  to  do  so.    That  is,  scientists  actually  depict  bacterial  colonies   through  data  models  that  are  appropriate  for  a  representation  of  the  colonies’  evolution  in   terms  of  exponential  growth  models;  and  scientists  actually  use  the  latter  models  to   represent  bacterial  colonies.    Arguably,  this  practice  sufficiently  constrains  what  it  would   be  to  provide  a  data  model  of  the  Antarctic  colony—that  is,  what  it  would  be  for  us  to   selectively  structure  the  phenomenon  in  a  way  that  is  relevant  to  exponential  growth   theory.    As  van  Fraassen  correctly  emphasizes,  however,  the  notion  of  relevance  here  is   relative  to  a  user  and  a  specific  context  of  use:   There  is  nothing  in  an  abstract  structure  itself  that  can  determine  that  it  is  the   relevant  data  model,  to  be  matched  by  the  theory.    A  particular  data  model  is   relevant  because  it  was  constructed  on  the  basis  of  results  gathered  in  a   certain  way,  selected  by  specific  criteria  of  relevance,  on  certain  occasions,  in   a  practical  experimental  or  observational  setting,  designed  for  that  purpose.   (2008,  253,  italics  and  emphases  in  original)     There  is  a  well-­‐defined  answer  to  what  it  would  be  to  depict  the  Antarctic  colony  and   embed  its  data  model  into  a  model  of  exponential  growth  only  because  the  situation  so   closely  resembles  phenomena  we  have  actually  modeled  and  which  can  therefore  provide   appropriate  criteria  of  relevance.     The  situation  is  dramatically  different  in  the  case  of  the  putative  micro-­‐model  of   water—for  the  solution  to  the  Schrödinger  equation  for  1025  variables  which  we  have   posited  exists,  but  which  do  not  actually  possess  and  cannot  as  a  matter  of  fact  use  to   represent  macroscopic  bodies  of  water.    We  do  not  actually  have  the  relevant  prepared    23   descriptions  for  macroscopic  bodies  of  water  to  be  matched  by  a  microscopic  quantum   mechanical  model  and  we  do  not  have  actual  examples  of  how  a  data  model  might  be   embedded  into  a  putative  quantum-­‐mechanical  micro-­‐model.    Thus,  there  is  neither  a  well-­‐ defined  answer  what  the  relevant  data  model  would  look  like,  completely  independently  of   any  actual  modeling  practices,  nor  a  well-­‐defined  answer  what  the  appropriate  embedding   relation  would  be.    It  is  utterly  unclear,  then,  what  it  would  be  for  the  range  of  structures   defined  by  the  Schrödinger  equation  to  satisfy  van  Fraassen’s  condition  of  empirical   adequacy—that  is,  for  the  range  to  “contain  a  candidate  that  would  satisfy  the  structural   constraint”  that  the  phenomenon  would  be  embeddable  in  it  (2008,  250).    The  lesson  of   Putnam’s  argument  is  that  there  will  be  some  mapping  from  the  1025  variables  onto  bits  of   the  body  of  water  such  that  the  theory  comes  out  true,  no  matter  what  its  details  are.    And   in  this  case  there  is  no  additional  constraint—no  practice  of  actually  modeling  the  wave  or   diffusion  behavior  of  macroscopic  bodies  of  water  microscopically—that  can  serve  to   single  out  an  intended  mapping.   In  response  one  might  try  to  appeal  to  our  actual  practices  of  modeling  simple   microscopic  systems  quantum  mechanically  as  providing  the  relevant  constraints,  but   implicit  in  such  an  appeal  would  be  a  commitment  to  natural  kinds—a  commitment  that   independently  of  our  actual  practices  of  modeling  macroscopic  bodies  of  water  in  certain   practical  and  experimental  settings  there  is  a  preferred  and  privileged  microscopic   relational  structure  that  would  be  the  correct  phenomenological  model  for  an  application   of  quantum  mechanics  to  the  system.    Contrary  to  the  nominalism  defended  by  van   Fraassen  one  would  have  to  be  committed  to  the  idea  that  for  each  phenomenon  taken  by   itself  there  is  a  determinate  answer  which  structures  are  appropriate  data-­‐  or   phenomenological  models  for  it,  independently  of  “our  decisions  in  attending  to  certain   aspects,  to  represent  them  in  certain  ways  and  to  a  certain  extent”.  (2008,  254).     6.  Conclusion   If  the  pragmatic  and  structuralist  account  of  representation  outlined  and  defended  here  is   correct,  then  this  has  far-­‐reaching  implications  for  how  we  think  about  scientific  theorizing.     First,  the  view  undermines  what  Cartwright  has  called  “the  vending  machine  view”  of   theories,  according  to  which  the  representational  content  of  a  theory  is  given  simply  by    24   stating  a  set  of  sentences  or  by  defining  a  model-­‐theoretic  class  of  models,  independently  of   a  theory’s  users.    According  to  van  Fraassen’s  account  presented  here  theories  do  not  have   any  representational  content  independently  of  our  actually  using  them  to  construct  models   of  the  phenomena.   Second,  the  view  has  radically  anti-­‐reductionist  and  anti-­‐foundationalist   implications  of  a  kind  van  Fraassen  himself  appears  to  be  reluctant  to  accept.    At  first  sight   it  might  seem  that  the  view  severely  and  dramatically  restricts  the  domain  of  a  theory  to   those  phenomena  for  which  we  actually  have  constructed  a  model.    This,  I  take  it,  would   amount  to  a  reductio  of  the  account  of  representation.    The  consequence  can  be  avoided  by     distinguishing  between  ‘horizontal’  and  ‘vertical’  counterfactual 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