/Users/andrea/_magisterarbeit/korpus/clean/trainkorpus/30/file_240.html NN ----------------------------------------- : THE DT COMPUTER NN REVOLUTION NN IN IN PHILOSOPHY NN 1978 CD Aaron NP Sloman NP Book NP contents NNS page NN CHAPTER NN 8 CD ON IN LEARNING VVG ABOUT IN NUMBERS NNS . SENT PROBLEMS NNS AND CC SPECULATIONS NNS 8 CD . SENT 1 CD . SENT Introduction NN The DT aim NN of IN this DT chapter NN is VBZ both DT methodological JJ and CC tutorial JJ . SENT It PP should MD help VV to TO introduce VV readers NNS to TO some DT computing NN ideas NNS . SENT It PP also RB includes VVZ some DT theoretical JJ speculations NNS about IN learning VVG and CC memory NN . SENT These DT speculations NNS are VBP fairly RB complex JJ , , yet CC it PP is VBZ clear JJ that IN they PP are VBP too RB simple JJ minded VVD to TO be VB adequate JJ accounts NNS of IN how WRB children NNS perform VVP their PP$ astonishing JJ feats NNS of IN learning VVG . SENT Many JJ more JJR questions NNS will MD be VB asked VVN than IN answered VVN . SENT And CC answers NNS offered VVN will MD be VB tentative JJ and CC provisional JJ . SENT Unfortunately RB , , experienced JJ programmers NNS will MD find VV some DT of IN the DT explanations NNS below IN very RB tedious JJ and CC over RB simplified VVN . SENT I PP apologise VV to TO them PP , , and CC hope VVP that IN non JJ programmers NNS will MD not RB find VV the DT same JJ explanations NNS too RB difficult JJ . SENT Here RB is VBZ a DT typical JJ conversation NN with IN a DT child NN aged VVN between IN three CD and CC a DT half NN and CC five CD years NNS . SENT Adult NN . SENT Can MD you PP count VV up RB to TO twenty CD . SENT Child NN . SENT One CD two CD three CD four CD five CD six CD seven CD eight CD nine CD ten CD eleven NN twelve NN thirteen CD fourteen NN fifteen CD seventeen CD eighteen CD twenty CD . SENT A DT . SENT What WP comes VVZ after IN three CD . SENT C SYM . SENT One CD two CD three CD four CD four CD . SENT A DT . SENT What WP comes VVZ after IN eight CD . SENT C SYM . SENT Four NP A NP . SENT What WP comes VVZ after IN six CD . SENT C SYM . SENT Don't NP know VVP A DT . SENT What WP comes VVZ before IN two CD . SENT C SYM . SENT One CD A NP . SENT What WP comes VVZ before IN four CD . SENT C SYM . SENT Five CD A NP . SENT How WRB many JJ fingers NNS on IN my PP$ hand NN . SENT C SYM counting NN fingers NNS . SENT One CD two CD three CD four CD five CD A DT . SENT What's NNS two CD and CC three CD . SENT C SYM counting NN fingers NNS . SENT One CD two CD three CD four CD five CD . SENT Five CD . SENT Does VVZ this DT child NN grasp NN number NN concepts NNS . SENT Perhaps RB there EX is VBZ something NN wrong JJ with IN the DT question NN , , because IN number NN concepts NNS are VBP not RB simple JJ things NNS which WDT you PP have VHP either RB grasped VVN or CC not RB grasped VVN . SENT What WP are VBP number NN concepts NNS . SENT How WRB is VBZ it PP possible JJ for IN them PP to TO be VB learnt VVN . SENT How WRB is VBZ it PP possible JJ for IN them PP to TO be VB used VVN . SENT How WRB is VBZ it PP possible JJ to TO discover VV non JJ empirical JJ facts NNS about IN them PP . SENT I PP believe VVP we PP are VBP not RB yet RB able JJ to TO formulate VV adequate JJ answers NNS to TO these DT questions NNS . SENT What WP follows VVZ is VBZ offered VVN as IN a DT preliminary JJ exploration NN of IN some DT of IN the DT issues NNS . SENT The DT method NN illustrated VVD below RB is VBZ important JJ . SENT Previously RB in IN chapter NN 2 CD , , I PP argued VVD that IN a DT major JJ aim NN of IN science NN is VBZ to TO find VV out RP what WP is VBZ possible JJ and CC explain VV how WRB it PP is VBZ possible JJ . SENT We PP all RB know VVP a DT great JJ deal NN about IN what WP it PP is VBZ possible JJ for IN adults NNS and CC children NNS to TO do VV with IN numbers NNS . SENT So RB , , instead RB of IN collecting VVG facts NNS by IN doing VVG experiments NNS on IN children NNS , , we PP can MD generate VV requirements NNS for IN explanatory JJ theories NNS by IN reflecting VVG on IN the DT fine JJ structure NN of IN familiar JJ human JJ abilities NNS . SENT In IN other JJ words NNS , , methods NNS of IN conceptual JJ analysis NN , , typically RB practised VVN by IN philosophers NNS and CC linguists NNS , , can MD be VB an DT important JJ source NN of IN data NNS for IN psychology NN . SENT Compare VV chapter NN 4 CD . SENT I PP am VBP not RB suggesting VVG that DT conceptual JJ analysis NN suffices VVZ to TO reveal VV everything NN we PP would MD like VV to TO know VV about RB , , for IN example NN , , ordinary JJ counting VVG abilities NNS . SENT The DT claim NN is VBZ only RB that IN it PP is VBZ foolish JJ to TO embark VV on IN expensive JJ empirical JJ investigations NNS before IN making VVG a DT serious JJ and CC systematic JJ effort NN to TO articulate VV what WP you PP already RB know VVP about IN the DT subject JJ matter NN . SENT Here RB are VBP some DT of IN the DT questions NNS for IN which WDT answers NNS are VBP lacking VVG . SENT What WP exactly RB is VBZ it PP that IN a DT child NN learns VVZ in IN learning VVG about IN numbers NNS . SENT How WRB is VBZ it PP possible JJ for IN different JJ fragments NNS of IN the DT same JJ number NN system NN to TO be VB mastered VVN by IN different JJ people NNS . SENT How WRB far RB does VVZ learning VVG about IN numbers NNS depend VVP on IN very RB general JJ learning VVG abilities NNS , , and CC how WRB far RB are VBP the DT hurdles NNS specific JJ to TO numbers NNS . SENT How WRB is VBZ it PP that IN a DT child NN who WP already RB seems VVZ to TO have VH the DT knowledge NN to TO answer VV a DT question NN or CC solve VV a DT problem NN , , is VBZ often RB unable JJ to TO use VV that DT knowledge NN . SENT What WP enables VVZ the DT knowledge NN to TO be VB accessed VVN at IN some DT later JJR time NN . SENT How WRB can MD a DT child NN learn VVP new JJ truths NNS about IN what WP she PP already RB knows VVZ , , for IN instance NN learning VVG that IN two CD of IN the DT numbers NNS she PP has VHZ learnt VVN add VV up RP to TO a DT third JJ number NN she PP knows VVZ , , or CC that IN two CD different JJ additions NNS generate VVP the DT same JJ result NN , , or CC that IN some DT procedure NN e NN . SENT g NN . SENT adding VVG one PP can MD be VB repeated VVN indefinitely RB to TO yield VV larger JJR and CC larger JJR numbers NNS . SENT I PP shall MD try VV to TO show VV how WRB thinking VVG about RB such JJ apparently RB psychological JJ questions NNS can MD lead VV towards IN new JJ answers NNS to TO old JJ philosophical JJ problems NNS about IN the DT nature NN of IN numbers NNS , , thereby RB providing VVG further JJR support NN for IN the DT claim NN that IN academic JJ barriers NNS between IN philosophy NN and CC science NN are VBP artificial JJ , , Some DT implications NNS regarding VVG information NN processing NN architectures NNS for IN intelligent JJ systems NNS will MD emerge VV as IN a DT side NN effect NN . SENT Note NN added VVD January NP 2002 CD I PP have VHP just RB discovered VVN the DT fascinating JJ book NN Wild NP Minds NNS . SENT What WP animals NNS really RB think VVP , , by IN Marc NP Hauser NP Penguin NN Books NNS 2001 CD . SENT Chapter NP 3 CD , , entitled VVN Number NP juggling NN , , discusses VVZ and CC compares VVZ the DT understanding NN of IN numbers NNS in IN very RB young JJ children NNS and CC in IN other JJ animals NNS . SENT Hauser NP comes VVZ close JJ to TO asking VVG some DT of IN the DT questions NNS asked VVD here RB , , and CC includes VVZ some DT speculations NNS about IN possible JJ mechanisms NNS , , but CC does VVZ not RB seem VV to TO be VB aware JJ of IN the DT full JJ variety NN of IN architectures NNS and CC sub NN mechanisms NNS that WDT might MD explain VV the DT observed JJ evidence NN . SENT He PP repeatedly RB stresses VVZ the DT important JJ point NN that IN it PP is VBZ very RB easy JJ to TO assume VV that IN the DT observed JJ behaviours NNS of IN animals NNS often RB suggest VVP a DT unique JJ interpretation NN , , until IN we PP start VVP exploring VVG possible JJ mechanisms NNS that WDT might MD produce VV those DT behaviours NNS . SENT He PP implicitly RB acknowledges VVZ that IN such JJ mechanisms NNS can MD be VB described VVN at IN different JJ levels NNS of IN abstraction NN , , not RB only RB at IN the DT level NN of IN brain NN physiology NN . SENT The DT common JJ trap NN of IN anthropomorphism NN is VBZ often RB a DT product NN of IN a DT lack NN of IN understanding NN of IN the DT variety NN of IN possible JJ information NN processing NN architectures NNS . SENT Some DT of IN them PP are VBP explored VVN in IN these DT recent JJ online JJ presentations NNS . SENT http NN . SENT www JJ . SENT cs NP . SENT bham NN . SENT ac NNS . SENT uk NP axs NP misc NN talks NNS 8 CD . SENT 2 LS . SENT Philosophical JJ slogans NNS about IN numbers NNS Here RB are VBP some DT examples NNS of IN philosophers NNS answers NNS to TO the DT question NN What WP are VBP numbers NNS . SENT , , and CC related JJ questions NNS . SENT Numbers NNS are VBP non JJ physical JJ mind NN independent JJ entities NNS , , existing VVG in IN their PP$ own JJ realm NN which WDT is VBZ different JJ from IN the DT world NN of IN spatial JJ objects NNS . SENT Platonists NNS Numbers NP are VBP perceivable JJ properties NNS of IN groups NNS of IN objects NNS . SENT For IN example NN , , the DT number NN three CD is VBZ what WP is VBZ visibly RB common JJ to TO the DT two CD groups NNS and CC Aristotle NN . SENT Numbers NNS are VBP mental JJ objects NNS , , created VVN by IN human JJ mental JJ processes NNS . SENT Facts NNS about IN numbers NNS are VBP discovered VVN by IN performing VVG mental JJ experiments NNS . SENT Kant NP , , and CC the DT Intuitionist NN mathematicians NNS Numbers NP are VBP sets NNS of IN sets NNS , , or CC predicates NNS of IN predicates NNS , , definable JJ in IN purely RB logical JJ terms NNS . SENT An DT example NN of IN this DT view NN . SENT the DT number NN one NN is VBZ the DT set NN of IN all DT sets NNS capable JJ of IN being VBG mapped VVN bi NP uniquely RB onto IN the DT set NN containing VVG nothing NN but CC the DT empty JJ set NN . SENT Frege NP , , Russell NP , , and CC other JJ logicists NNS Numbers NP are VBP meaningless JJ symbols NNS manipulated VVN according VVG to TO arbitrary JJ rules NNS . SENT Mathematical JJ discoveries NNS are VBP merely RB discoveries NNS about IN the DT properties NNS of IN this DT game NN with IN symbols NNS . SENT Formalists NNS Numbers NP are VBP implicitly RB defined VVN by IN a DT collection NN of IN axioms NNS , , such JJ as RB , , Peano's NP axioms NNS . SENT Any DT collection NN of IN things NNS satisfying VVG these DT axioms NNS can MD be VB called VVN a DT set NN of IN numbers NNS . SENT The DT nature NN of IN the DT elements NNS of IN the DT set NN is VBZ irrelevant JJ . SENT Mathematical JJ discoveries NNS about IN numbers NNS are VBP merely RB discoveries NNS of IN logical JJ consequences NNS of IN the DT axioms NNS . SENT Many JJ mathematicians NNS There EX is VBZ no DT one CD correct JJ answer NN to TO the DT question NN what WP are VBP numbers NNS . SENT People NNS play VVP a DT motley NN of IN games NNS using VVG number NN words NNS and CC other JJ symbols NNS , , and CC a DT full JJ account NN of IN the DT nature NN of IN numbers NNS would MD simply RB be VB an DT analysis NN of IN these DT games NNS including VVG the DT activity NN of IN mathematicians NNS and CC the DT roles NNS they PP play VVP in IN our PP$ lives NNS . SENT Wittgenstein NP . SENT Remarks NNS on IN the DT Foundations NNS of IN Mathematics NNS For IN more JJR details NNS , , see VVP standard JJ texts NNS on IN philosophy NN of IN mathematics NNS . SENT I PP believe VVP that IN more JJR or CC less RBR articulate VV versions NNS of IN these DT philosophical JJ theories NNS , , play VV an DT important JJ role NN in IN many JJ psychological JJ and CC educational JJ theories NNS about IN numbers NNS . SENT I PP have VHP formed VVN this DT opinion NN over IN many JJ years NNS , , from IN wide JJ but CC unsystematic JJ reading NN and CC discussion NN , , including VVG attendance NN at IN lectures NNS and CC seminars NNS . SENT So RB I PP am VBP not RB in IN a DT position NN to TO document VV the DT claim NN . SENT I PP shall MD continue VV in IN this DT chapter NN to TO make VV remarks NNS about IN psychological JJ theories NNS if IN the DT disparaging VVG ones NNS are VBP untrue JJ I'll NN be VB delighted VVN . SENT All PDT the DT views NNS listed VVD above RB combine VV elements NNS of IN truth NN with IN distortions NNS and CC oversimplifications NNS . SENT I PP think VVP that DT Wittgenstein's NP answer NN comes VVZ closest JJS to TO encompassing VVG the DT truth NN . SENT In IN his PP$ writings NNS he PP formulates VVZ many JJ problems NNS about IN mathematics NN , , which WDT are VBP not RB answered VVN by IN other JJ theories NNS , , but CC his PP$ own JJ solutions NNS seem VVP to TO me PP to TO be VB too RB shallow JJ . SENT In IN particular JJ , , the DT anti JJ mentalism NN , , or CC anti JJ psychologism NN , , which WDT pervades VVZ much RB of IN his PP$ writing NN prevents VVZ him PP from IN discussing VVG mental JJ processes NNS in IN any DT depth NN . SENT So RB he PP writes VVZ as RB if IN thinking VVG about IN numbers NNS were VBD an DT essentially RB social JJ process NN , , consistently RB with IN his PP$ conclusion NN in IN Philosophical JJ Investigations NNS that IN all DT rule NN following NN is VBZ an DT essentially RB social JJ process NN , , dependent JJ on IN the DT existence NN of IN a DT public JJ language NN . SENT This DT conflicts NNS with IN a DT computational JJ analysis NN of IN mental JJ processes NNS , , according VVG to TO which WDT it PP is VBZ perfectly RB possible JJ for IN a DT non JJ social JJ mechanism NN to TO contain VV within IN itself PP rules VVZ which WDT it PP can MD obey VV , , for IN instance NN , , programs NNS transmitted VVD genetically RB . SENT Wittgenstein's NP position NN also RB conflicts NNS with IN any DT sensible JJ account NN of IN the DT biological JJ evolution NN of IN mental JJ processes NNS in IN precursors NNS of IN homo NN sapiens JJ . SENT I PP am VBP not RB going VVG to TO try VV to TO solve VV all PDT the DT philosophical JJ and CC psychological JJ problems NNS about IN numbers NNS in IN one CD chapter NN . SENT I PP shall MD merely RB try VV to TO show VV how WRB we PP can MD get VV important JJ new JJ insights NNS into IN the DT problems NNS , , and CC perhaps RB take VV some DT small JJ steps NNS towards IN formulating VVG possible JJ answers NNS , , if IN we PP think VVP about IN the DT mental JJ processes NNS and CC mechanisms NNS as RB if IN they PP were VBD analogous JJ to TO the DT processes NNS and CC mechanisms NNS involved VVN in IN so RB called VVN list NN processing VVG computer NN programs NNS . SENT Adequate JJ exploration NN of IN these DT issues NNS has VHZ been VBN hampered VVN by IN the DT current JJ separation NN of IN philosophy NN and CC psychology NN , , and CC the DT ignorance NN among IN most JJS philosophers NNS and CC psychologists NNS of IN computing NN ideas NNS . SENT I PP shall MD not RB be VB talking VVG about IN events NNS or CC processes NNS or CC mechanisms NNS in IN the DT human JJ brain NN . SENT Exactly RB how WRB the DT brain NN works NNS is VBZ as RB irrelevant JJ to TO our PP$ problems NNS as IN the DT detailed JJ workings NNS of IN a DT computer NN are VBP to TO an DT explanation NN of IN a DT computer NN program NN written VVN in IN a DT high JJ level NN programming NN language NN . SENT There EX may MD be VB creatures NNS on IN other JJ planets NNS , , or CC robots NNS , , whose WP$ brains NNS are VBP totally RB unlike IN ours PP in IN their PP$ physiological JJ details NNS , , yet RB such JJ beings NNS could MD well RB learn VV about IN numbers NNS , , and CC learn VV the DT same JJ concepts NNS as IN we PP do VVP , , just RB as IN two CD computers NNS with IN quite RB different JJ physical JJ components NNS can MD execute VV the DT same JJ high JJ level NN programs NNS . SENT Incidentally RB , , this DT undermines VVZ philosophical JJ theories NNS which WDT claim VVP that IN mental JJ processes NNS are VBP identical JJ with IN brain NN processes NNS . SENT This DT is VBZ as RB inaccurate JJ as IN the DT claim NN that IN computational JJ processes NNS in IN a DT computer NN are VBP identical JJ with IN physical JJ processes NNS . SENT When WRB I PP talk VVP about IN mechanisms NNS involved VVN in IN using VVG numbers NNS , , I PP am VBP not RB talking VVG about IN physiological JJ mechanisms NNS . SENT I PP am VBP talking VVG about IN aspects NNS of IN the DT way NN information NN is VBZ organised VVN and CC represented VVN , , and CC about IN the DT kinds NNS of IN symbol NN manipulating VVG processes NNS which WDT may MD be VB necessary JJ for IN accessing VVG and CC using VVG various JJ sorts NNS of IN representations NNS . SENT In IN particular JJ , , such JJ processes NNS involve VVP the DT following NN of IN rules NNS , , instructions NNS , , or CC plans NNS , , whether IN consciously RB or CC unconsciously RB . SENT This DT illustrates VVZ how WRB the DT concept NN of IN mechanism NN is VBZ extended VVN by IN developments NNS in IN computing NN . SENT 8 CD . SENT 3 LS . SENT Some DT assumptions NNS about IN memory NN Unfortunately RB , , my PP$ speculations NNS about IN mental JJ processes NNS will MD be VB intelligible JJ only RB if IN I PP introduce VV some DT technical JJ ideas NNS and CC assumptions NNS , , already RB hinted VVD at IN in IN previous JJ chapters NNS , , especially RB chapter NN 6 CD . SENT If IN the DT assumptions NNS are VBP wrong JJ , , then RB quite RB different JJ theories NNS are VBP required VVN . SENT At IN the DT moment NN , , there EX does VVZ not RB seem VV to TO be VB any DT way NN of IN avoiding VVG these DT assumptions NNS , , if IN we PP are VBP trying VVG to TO explain VV well RB known VVN facts NNS about IN what WP people NNS can MD do VV . SENT The DT main JJ assumption NN is VBZ that IN we PP can MD speak VV of IN the DT human JJ mind NN as IN storing VVG information NN in IN a DT vast JJ collection NN of IN locations NNS . SENT They PP need MD not RB be VB spatial JJ locations NNS , , like IN shelves NNS in IN a DT library NN . SENT Positions NNS in IN any DT kind NN of IN symbolic JJ space NN with IN appropriate JJ mechanisms NNS for IN storing VVG and CC retrieving VVG information NN will MD do VV . SENT So IN the DT word NN location NN is VBZ being VBG used VVN as IN a DT technical JJ term NN . SENT For IN instance NN , , radio NN waves NNS are VBP often RB used VVN to TO transmit VV information NN , , different JJ information NN being VBG transmitted VVN at IN different JJ frequencies NNS . SENT So RB information NN could MD be VB stored VVN in IN a DT collection NN of IN continually RB reverberating VVG radio NN waves NNS , , with IN different JJ symbols NNS stored VVN at IN different JJ frequencies NNS . SENT Each DT possible JJ frequency NN would MD then RB be VB a DT location NN in IN the DT sense NN required VVN here RB . SENT Similarly RB , , possible JJ structures NNS of IN a DT certain JJ class NN of IN molecules NNS could MD define VV addresses NNS in IN a DT space NN . SENT Storing VVG information NN at IN a DT certain JJ address NN would MD mean VV attaching VVG that DT information NN to TO molecules NNS with IN the DT structure NN represented VVD by IN the DT address NN . SENT This DT could MD be VB done VVN more RBR or CC less RBR simultaneously RB in IN many JJ different JJ physical JJ places NNS . SENT But CC the DT information NN would MD still RB be VB stored VVN in IN one CD symbolic JJ place NN , , just RB as IN a DT name NN occurs VVZ at IN only RB one CD symbolic JJ location NN in IN a DT telephone NN directory NN , , even RB though IN there EX may MD be VB millions NNS of IN physically RB distinct JJ copies NNS of IN the DT directory NN containing VVG the DT name NN . SENT So RB from IN now RB on IN , , when WRB I PP talk VVP about IN locations NNS , , this DT is VBZ neutral JJ as RB to TO what WP sorts NNS of IN locations NNS they PP are VBP . SENT I PP shall MD assume VV then RB that IN a DT mechanism NN is VBZ available JJ which WDT can MD store VV symbols NNS in IN some DT space NN of IN locations NNS . SENT Further RBR , , I PP assume VVP that IN it PP is VBZ possible JJ for IN some DT of IN the DT symbols NNS to TO represent VV locations NNS in IN this DT space NN . SENT For IN instance NN , , a DT directory NN , , or CC catalogue NN , , can MD contain VV entries NNS which WDT refer VVP to TO the DT location NN of IN other JJ entries NNS , , by IN page NN or CC section NN number NN . SENT Thus RB the DT space NN can MD contain VV information NN about IN itself PP . SENT A DT symbol NN representing VVG a DT location NN can MD be VB called VVN a DT pointer NN , , or CC an DT address NN . SENT So IN the DT storage NN mechanism NN can MD be VB given VVN an DT address NN and CC asked VVD to TO produce VV the DT symbol NN located VVN there RB . SENT In IN other JJ words NNS , , when WRB given VVN a DT pointer NN , , it PP can MD determine VV what WP symbol NN is VBZ pointed VVN at IN . SENT What WP is VBZ pointed VVN at IN may MD be VB a DT complex JJ structure NN containing VVG a DT symbol NN which WDT is VBZ itself PP an DT address NN of IN some DT other JJ location NN , , that WDT is VBZ a DT pointer NN to TO another DT symbol NN . SENT See VV Figure NP 1 CD . SENT So IN the DT space NN may MD contain VV chains NNS of IN pointers NNS . SENT In IN more JJR elaborate JJ systems NNS , , the DT addressing VVG may MD be VB relative JJ to TO a DT context NN or CC mode NN of IN operation NN . SENT That DT is VBZ , , which WDT location NN is VBZ represented VVN by IN a DT given VVN symbol NN may MD depend VV on IN the DT current JJ state NN of IN the DT accessing VVG sub NN mechanism NN . SENT Some DT of IN the DT flexibility NN of IN behaviour NN of IN the DT system NN may MD depend VV on IN such JJ systematic JJ changes NNS in IN the DT meaning NN of IN symbols NNS . SENT The DT concept NN of IN a DT symbolic JJ structure NN containing VVG pointers NNS into IN itself PP , , and CC the DT investigation NN of IN processes NNS in IN which WDT such JJ things NNS are VBP manipulated VVN and CC used VVN for IN solving VVG problems NNS , , are VBP among IN the DT important JJ contributions NNS of IN computing NN science NN . SENT I PP shall MD try VV to TO show VV how WRB these DT ideas NNS help VVP us PP to TO think VV about IN a DT child's JJ ability NN to TO count VV , , an DT ability NN which WDT provides VVZ the DT substratum NN fo NP