Philosophical Psychology Vol. 22, No. 1, February2009, 75–79

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Philosophical Psychology Vol. 22, No. 1, February2009, 75–79

P Psychology77
Philosophical Psychology Vol. 22, No. 1, February2009,75–79 Doestopologicalperce
ptionreston amisconceptionabouttopology? RobertoCasati Inthis articleI
assessomeresultsthat purport toshowtheexistenceof atypeof s 'topologicalperception ... 

Scholar Alert: [ allintitle: "philosophical psychology" ]; Articles excluding patents Philosophical Psychology Vol. 22, No. 1, February2009, 75–79 P Psychology77 Philosophical Psychology Vol. 22, No. 1, February2009,75–79 Doestopologicalperce ptionreston amisconceptionabouttopology? RobertoCasati Inthis articleI assessomeresultsthat purport toshowtheexistenceof atypeof s 'topologicalperception ...
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Φιλοσοφική Ψυχολογία Vol. 22, αρ. 1, February2009, 75β;? 79
Doestopologicalperceptionreston amisconceptionabouttopology;RobertoCasati
Inthis articleI assessomeresultsthat φιλοδοξούν β toshowtheexistenceof atypeof s?; Topologicalperceptionβ ????, δηλαδή, perceptuallybasedclassificationoftopologicalfeatures.Strikingfindingsaboutperceptionininsectsappeartoimplythat (1) configural, παγκόσμια propertiescanbeconsideredasprimitiveperceptualfeatures, και (2) topologicalfeatures inparticularareinterestingastheyareamenabletoformal treatment.Idiscussfour interrelatedquestionsthatbearonanyinterpretationoffindingsabouttheperceptionof τοπολογική ΙΔΙΟΤΗΤΕΣ: whatexactlyaretopologicalproperties p, whatmakesthemglobal, inwhatsensethequotedfindingsmakesthemprimitive, andwhatarethehopesof ένα formaltheoryofperceptionbaseduponthem.Isuggestthatmathematicaltopologyisnot thecorrectmodelforcognitiontopologicalproperties, encethatsomeotherformalism h typeoftopologyarespelledout, theymaynotbeasglobalisticasonemayhaveexpected. Λέξεις-κλειδιά: opological erception? TP Τοπολογία? IsualPrimitives V
Manylogicallyindependent αλλά coordinatedfactorsconstrainthequest forvisual primitives.First, phenomenologytellsusthatthevisualsceneiscomplex, butatthe sametimethattherearerecurringelementsoutofwhichcomplexitymaybebuilt (Kanizsa, 1979). Δεύτερον, mathematicalmodels showhowit είναι possibletobuild complexrepresentationsoutofrepresentationsofsimplercomponents (Biederman, 1987). Τρίτον, computationalarchitecturemakesitplausiblethatthecomplexityof αμφιβληστροειδούς εισόδου (παροχή ιδιότητες informationabout μέγεθος pixel σε χώρους) να
RobertoCasatiis 22 στις 22. Correspondenceto: RobertoCasati, 22. Email: casati@ehess.frΓ? ISSN0951-5089 (εκτύπωση) / ISSN1465-394X (online) / 09/010075-5 2009Taylor & Francis DOI: 10.1080/09515080802703711
76 R. Casati
organizedatveryearlystagesintoelementaldescriptorstoreducethecomputational φορτίο subsequentstages (Palmer & Ροκ, 1994). Τέλος, συμπεριφοράς και neurophysiologicalevidence για συγκεκριμένες, downtosingle-neuronsensitivity να relativelywell delineatedfeaturesof theenvironment hasbeengatheredoverthe lastdecades, αρχίζοντας (Hubel & Wiesel, 1959). Butdothesecriteriaconverge onasinglelist των πρωτόγονων;Theydonot haveto, φυσικά? Andfindingout ότι αυτό που weexpect computationallyprimitive tobephenomenologicallyor δεν είναι sobehaviorallyorneurophysiologicallywill makeforaninterestingdiscovery. Αυτό είναι inpart theinterest του (Chen, Zhang, & Srinivasan, 2003) findingthat μικρό εγκέφαλο suchas αυτές των μελισσών εμφανιστεί μια ευαισθησία toglobal configurationalproperties, inparticulartopological propertiessuchasthepresence τρύπες ortheabsenceof IN2-ddisplays. Φαίνεται ότι αν δεν onlybees areableto distinguishbetweenconfigurationsthat differonlyintheirtopological ιδιότητες, butalsotheyareabletogeneralizetotopologicallyequivalentconfigurationsthatare μάλλον διαφορετική onmany άλλες respects.Accordingto (Pomerantz, 2003), τα ευρήματα tworeasons areinterestingfor.Thefirst reasonis ότι thetopological propertiesinquestionaregenerallyconsideredasrelativelycomplexandhardto υπολογίσουμε (Minsky και Papert, 1998) ιδιότητες butatthesametimeareverydeepandrobust του theenvironmentβ;? Theyareinvariant κάτω από τις περισσότερες μετατροπές, asopposed, ας πούμε, tometricproperties, hencesensitivitytothemwouldhaveahigh προσαρμοστική αξία. Η secondreasonis ότι η mathematicsbehindtopological featuresissufficientlywellunderstoodandformalizedaccordingly, relativelyinformalityofcharacterizationsofglobalfeatures asopposedtothe, π.χ., theonefoundinthe gestaltliterature.Butwhatexactlyaretopologicalproperties, whatmakesthemglobal, inwhatsense Chenetal.β;? S (2003) findingmakesthemprimitive, andwhatarethehopesofaformal theoryofperceptionbaseduponthem;LetusfirstbrieflyreviewtheresultsChenetal.β;? Sexperiment.Honeybeeswere διαμορφώσεις trainedtochooseoneamongapairof: Άνω-shapedstimulusand ans-shapedstimulus, say.Thentheyweretestedontheirabilitytodistinguishthe O-shapedstimulusfromotherstimulithatareeithertopologicallynonequivalentto Theo-σχήμα (; suchasa Γ;-σχήμα, ora f-σχήμα) andstimuli thataretopologically ισοδύναμο: τη O-σχήμα (suchas σε ένα σχήμα), αλλά lookdifferent ως totheir μη topologicalaspect.Beessucceededinmakingthedistinctionwiththefirstsetof stimulibutfailedtomakeitwiththesecondset.Thisindicatesboththattheywere sensitivetotopologicaldifferencesandthattheycorrectlylumpedtogetheritemsthat aretopologicallyequivalent.OneshouldnoteinthefirstplacethatthedisplaysusedbyChenetal.fortesting honeybeeswere2-dpicturesrepresentingfiguresof διαφορετικές τοπολογικές shapesandvarying properties.Thereis, φυσικά, amuchgeneral problemof using2-d stimuliinordertodrawinferencesaboutavisual systemthathasadaptedtoa3-d κόσμο. Αλλά thereisalsoaspecificproblem: topologyin3-disnot αυτόματα mappedonto2-dtopology. A2-dimageliketheshapeof theletterBcanbethe projectionofa3-dletterOthathasbentoverinthemiddle.Hencefromsensitivity
Φιλοσοφικά Psychology77
to2-dtopologyonecanonlyinferwithmuchcaretoacorrespondingsensitivityto thetopologyof 3-dbodies? thisbyitself wouldquestionsanyecological-προσαρμοστική εκτιμήσεις.Theglobalityof τοπολογική tothelocalityof propertiescanbecapturedintuitivelyinopposition άλλα Aline features.Thedirectionalityof, για παράδειγμα, είναι μια intrinsicallylocalmatter.Atpointpthelinehasadirectionthatisgivenbyitstangent ATP. Littledoesitmatterhowthelinelookslikeata (επαρκή) distancefromp. Ontheotherhand, thefact ότι thelineclosesuntoitself (likeacircle) απολήξεις orhas (likeabar) cannotbemadedependonthepropertiesofasinglepoint ontheline? Manyotherpointshavetobescrutinized. Inthissensetheproperties studiedbyChenetal. areglobal, andappeartobethesubjectmatteroftopology.Themainquestionariseswhetherthetermβ;? Topologyβ;;;;isusedinthelooseand popularsenseof β?; rubbersheet geometryβ;? orbyreferencetoformal μαθηματικές έννοιες. Theissueofcontrol inexperimental designreflectsthisuncertainty. Chen etal.appropriatelypointoutthatitishardtotestfortopologicaldifferenceswithout introducingsomenon-τοπολογική differencesinthestimuli: β β thereseemtobe, κατ 'αρχήν, notwogeometricfeaturesthatdifferonlyintopologicalpropertiesβ β ????( 2003, σ. 6687);;;???. Αλλά fromtheviewpoint της mathematicaltopology, αυτό είναι ανακριβές.Anopensphereandaclosedsphereβ;? Ortheir2-dequivalents, opencircleβ aclosedcircleandan;? Havedifferent τοπολογική thesamemetricproperties propertiesbut (sameradius, astheboundaryoftheclosedcirclehasnodimension). Tobesure, thisfactmaynothaveanyconsequenceforthedesignofvisual teststimuli, asthe διαφορά ανάμεσα σε ένα κλειστό και ένα ανοικτό itemhas καμία οπτική ομόλογό (dimensionlessitems, onemayargue, areunderperceptual discriminationthreshold). Αλλά thisbringsustoanimportant σημείο. Whenwetalkabout τοπολογική differencesinvisualdisplays, wemaynotbetalkingaboutthedifferencesthatarethe subjectmatterofmathematical topology.Hencetalkingofβ;? Topologyβ;;;; requiressome otherrefinement, shortofbeingloosetalk, especiallyifitistoprovidetheβ;? επίσημη theoryβ;;;; ότι (Pomerantz, 2003) επικαλείται.Onemaysuggestthatsomethinglikeaninternalizedtopologycapturesthefeatures wehaveinmindβ;? Suchastheabilitytosortoutobjectsbasedonthenumberofholes theyhave (thedifferencebetweenthelettersBandO, havingoneandtwoholes αντίστοιχα), rtoassesstheequivalencebetweenfigures (ας πούμε, theletterSandthe o letterI). Butherewehavetoexertsomecare, becausethetheorynowrequiresanew έννοια, suchashole, andsomeaccountisneededofwhatitisforthevisualsystemto processthefeatureofbeingaholeinsuchawaythatitcontributestotheexplanation oftheperformanceofdistinguishingSorBfromO.Toseethepointmoreclearly, considerare-interpretationofChenetal.(2003). Fromtheviewpointofanintuitivetopology, thedifferencebetweenhavingandnot havingoneholeisimpliedbythepresenceorabsenceofothervisualfeatures.Now, αν weredemonstratedthat processingof thesefeaturesisavailabletothevisual σύστημα, itwouldbepossibletoreassesstheclaimthattheglobalfeaturesinvokedby Chenet al. andPomerantz (2003) areperceptual αρχέτυπα.Thefollowingis ένα proposalinthatsense.
78 R. Casati
Thevisual featuresinquestionare:(1) (2) boundariesofaunit Thepresenceofcompletevisual, και Theuniformityandconnectionoftheunit (Palmer & Ροκ, 1994), alongwith itsmaximality (Casati, 2002).
Thepresenceofholesiscorrelatedwiththesesimplerfeaturesinthefollowingway.Ifamaximal όριο uniformconnectedunitpossessesjustonecompletevisual, thenit έχει nohole. Αν διαθέτει twovisual όρια, thenit έχει onehole. 1 Ingeneral, foranygivenvisual απεικόνιση:(3) Formmaximaluniformconnectedfiguresandncompletevisualboundaries, thenumberofholesis (nβ;? M).
Το περαιτέρω στοιχείο που είναι thenneededis ότι η οπτική countingthefeaturesandcomparetheircardinalities somewayof systemimplement. Givenwhatis knownaboutthelimitsoftheabilitytosubitizesmallquantities, itisexpectedthat thedifferencebetweenconfigurationswith, ας πούμε, oneandtwoholeswillbeaccessible tothesystem.Atthesametime, thedifferencebetweenconfigurationswithnineand tenholes είναι expectednot είναι προσβάσιμα system.But: τη σίγουρα αυτές οι τελευταίες ρυθμίσεις είναι τοπολογικά ξεχωριστή άποψη fromthe της μαθηματικής τοπολογίας. Hencetestingtheabilityof distinguishingbetweenconfigurations με varyingnumbersofholescandecidebetweenaholisticandalessholisticaccountof visualproperties. Επιπλέον, howfar cantopological generalizationgo; Επιστολές I andJ είναι topologicallyequivalentintheintendedsense??; Butsoare, κατά πάσα πιθανότητα, I, L, KandH (thelatterthree, forinstance, canallbeβ shrunkβ toanIwithoutβ cuttingorgluingβ ????).;;;;;;;; Θα dataconfirmasensitivitytotheseequivalences Somemayexpectinsteadthat somesortof parsingbycomponentswill predictthattheseshapesareresilientto placementinasinglecategory:? AnHhasthreecomponents, anIhasonlyone.Here againtheglobalistichypothesiscanbepittedagainstothertheoretical λογαριασμούς. Itmaybequestionedwhetherfeatures (1) και (2) arereallysimplerthantheglobal featureof havingahole. Μετά από όλα, και τα δύο (1) και (2) presupposethat theunity (σύνδεση) του boththe boundaryandthe σχήμα είναι προσβάσιμες? Andassessing connectionisanotoriouslydifficultcomputational πρόβλημα. Ωστόσο, ontheone χέρι, thisisageneral πρόβλημα, onethat affectsall theoriesthat aresupposedto characterizetheentryunitsofthevisualsystem.Ontheotherhand, inordertoshow ότι sensitivitytothefeatureof δεν possessingaholeis sensitivitytoavisual πρωτόγονη, είναι enoughtoshowthat canbeexplainedinterms theformer του sensitivitytootherfeatures, withoutanyfurthercommitmenttothehypothesisthat πρωτόγονους thesefeaturesarethemselvesvisual. Toconclude, whatistheevidencethattopologicalorglobalfeaturessuchashaving aholeareprimitivesof thesystem, accordingtoChenet al. ; (2003) Theclearly delineatedcriterioninthepaperisthesizeofthecomputationalsystem: μέλισσες havesmallbrains.Thecriterionisnovelrelativetothefourcriterialistedatthetop ofthispaper.Thecriterionpredictsthatafeatureisprimitiveifitiscomputedby
Φιλοσοφικά Psychology79
asmall συστήματος.Ωστόσο, forthereasonsgivenabove, thesysteminquestionmay simplybenotsmall enoughtoprovideacogentanswer.
Σημείωση[1] Thecriterionreflectstheonegivenforcavitiesin3-dbodiesin (Casati & Varzi, 1994).
ΑναφορέςBiederman, Ι. (1987). Η αναγνώριση-από-εξαρτήματα: Atheoryof humanimage ερμηνεία. Ψυχολογική κριτική, 94.115 β?? 148. 2β;;;;Casati, R. (2002). Topologyandcognition.Encyclopediaofcognitivescience (pp.2). 2: McMillan. Casati, R., & Varzi, AC (1994). Holesandothersuperficialities. Cambridge, MA: MITPress. . Chen, Λ., Zhang, Σ., & Srinivasan, MV (2003) Παγκόσμια εγκεφάλους perceptioninsmall: τοπολογική patternrecognitioninhoneybees.PNAS, 100 (11), 6884β;? 6889. Hubel, Δ. Χ., & Wiesel, Θ. Ν. (1959). Receptivefieldsof singleneuronsincatβ;? Sstriatecortex.JournalofPhysiology, 60.106 β?? 154. 1 2: Kanizsa, Γ. . (1979) Organizationinvision: Essaysongestaltperception. Praeger. Minsky, Μ. Λ., & Papert, S. A. (1998). Perceptrons: Anintroductiontocomputational γεωμετρία (expandededition).Ambridge, MA: MITPress. C Palmer, Σ., & Ροκ, Ι. (1994).Επανεξέταση της οπτικής οργάνωσης: Ο ρόλος της ενιαίας συνεκτικότητας. PsychonomicBulletinandReview, 29β;? 55. 1, Pomerantz, J. R. (2003). Σύνολα, holesandbasicfeaturesinvision.TrendsinCognitiveSciences, 7 (11), 471β?; 473.
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Philosophical Psychology Vol. 22, No. 1, February2009,75–79

Doestopologicalperceptionreston amisconceptionabouttopology?
RobertoCasati

Inthis articleI assessomeresultsthat purport toshowtheexistenceof atypeof s β€˜topologicalperception’,i.e., perceptuallybasedclassificationoftopologicalfeatures. Strikingfindingsaboutperceptionininsectsappeartoimplythat(1)configural, global propertiescanbeconsideredasprimitiveperceptualfeatures,and(2)topologicalfeatures inparticularareinterestingastheyareamenabletoformal treatment.Idiscussfour interrelatedquestionsthatbearonanyinterpretationoffindingsabouttheperceptionof topological roperties: p whatexactlyaretopologicalproperties, whatmakesthemglobal, inwhatsensethequotedfindingsmakesthemprimitive, andwhatarethehopesof a formaltheoryofperceptionbaseduponthem.Isuggestthatmathematicaltopologyisnot thecorrectmodelforcognitiontopologicalproperties, encethatsomeotherformalism h oughttobeusedβ€”aformofβ€˜β€˜internalizedtopology.’’However,oncetheprinciplesofthis typeoftopologyarespelledout,theymaynotbeasglobalisticasonemayhaveexpected. Keywords: opological erception; T P Topology; isualPrimitives V

Manylogicallyindependent but coordinatedfactorsconstrainthequest forvisual primitives.First,phenomenologytellsusthatthevisualsceneiscomplex,butatthe sametimethattherearerecurringelementsoutofwhichcomplexitymaybebuilt (Kanizsa,1979). Second,mathematicalmodels showhowit is possibletobuild complexrepresentationsoutofrepresentationsofsimplercomponents(Biederman, 1987).Third, computationalarchitecturemakesitplausiblethatthecomplexityof retinal input (providing pixel size informationabout properties at places) be

RobertoCasatiis 22 at 22. Correspondenceto:RobertoCasati,22. Email: casati@ehess.fr
ΓŸ ISSN0951-5089(print)/ISSN1465-394X(online)/09/010075-5 2009Taylor&Francis DOI: 10.1080/09515080802703711

76 R. Casati

organizedatveryearlystagesintoelementaldescriptorstoreducethecomputational load of subsequentstages (Palmer &Rock, 1994). Finally, behavioral and neurophysiologicalevidence for specific, downtosingle-neuronsensitivity to relativelywell delineatedfeaturesof theenvironment hasbeengatheredoverthe lastdecades,startingfrom(Hubel &Wiesel, 1959).Butdothesecriteriaconverge onasinglelist of primitives? Theydonot haveto, of course;andfindingout that what weexpect tobephenomenologicallyor computationallyprimitive is not sobehaviorallyorneurophysiologicallywill makeforaninterestingdiscovery. This is inpart theinterest of (Chen, Zhang, &Srinivasan, 2003) findingthat small brains suchas those of the honey bees display a sensitivity toglobal configurationalproperties,inparticulartopological propertiessuchasthepresence ortheabsenceof holes in2-ddisplays. It looks as if not onlybees areableto distinguishbetweenconfigurationsthat differonlyintheirtopological properties, butalsotheyareabletogeneralizetotopologicallyequivalentconfigurationsthatare rather different onmany other respects.Accordingto(Pomerantz, 2003), the findings areinterestingfor tworeasons. Thefirst reasonis that thetopological propertiesinquestionaregenerallyconsideredasrelativelycomplexandhardto compute(Minsky&Papert, 1998)butatthesametimeareverydeepandrobust properties of theenvironment—theyareinvariant under most transformations, asopposed,say,tometricproperties, hencesensitivitytothemwouldhaveahigh adaptive value. The secondreasonis that the mathematicsbehindtopological featuresissufficientlywellunderstoodandformalizedaccordingly,asopposedtothe relativelyinformalityofcharacterizationsofglobalfeatures,e.g.,theonefoundinthe gestaltliterature. Butwhatexactlyaretopologicalproperties,whatmakesthemglobal,inwhatsense Chenetal.’s(2003)findingmakesthemprimitive,andwhatarethehopesofaformal theoryofperceptionbaseduponthem? LetusfirstbrieflyreviewtheresultsChenetal.’sexperiment. Honeybeeswere trainedtochooseoneamongapairof configurations:anO-shapedstimulusand anS-shapedstimulus,say.Thentheyweretestedontheirabilitytodistinguishthe O-shapedstimulusfromotherstimulithatareeithertopologicallynonequivalentto theO-shape(suchasa  -shape,ora f -shape)andstimuli thataretopologically equivalent tothe O-shape(suchas a -shape) but lookdifferent as totheir non-topologicalaspect.Beessucceededinmakingthedistinctionwiththefirstsetof stimulibutfailedtomakeitwiththesecondset. Thisindicatesboththattheywere sensitivetotopologicaldifferencesandthattheycorrectlylumpedtogetheritemsthat aretopologicallyequivalent. OneshouldnoteinthefirstplacethatthedisplaysusedbyChenetal.fortesting honeybeeswere2-dpicturesrepresentingfiguresof different shapesandvarying topological properties.Thereis, of course,amuchgeneral problemof using2-d stimuliinordertodrawinferencesaboutavisual systemthathasadaptedtoa3-d world. But thereisalsoaspecificproblem: topologyin3-disnot automatically mappedonto2-dtopology. A2-dimageliketheshapeof theletterBcanbethe projectionofa3-dletterOthathasbentoverinthemiddle.Hencefromsensitivity

Philosophical Psychology77

to2-dtopologyonecanonlyinferwithmuchcaretoacorrespondingsensitivityto thetopologyof 3-dbodies; thisbyitself wouldquestionsanyecological-adaptive considerations. Theglobalityof topological propertiescanbecapturedintuitivelyinopposition tothelocalityof other features.Thedirectionalityof aline, for instance,is an intrinsicallylocalmatter.Atpointpthelinehasadirectionthatisgivenbyitstangent atp. Littledoesitmatterhowthelinelookslikeata(sufficient)distancefromp. Ontheotherhand, thefact that thelineclosesuntoitself (likeacircle)orhas terminations(likeabar)cannotbemadedependonthepropertiesofasinglepoint ontheline; manyotherpointshavetobescrutinized. Inthissensetheproperties studiedbyChenetal. areglobal, andappeartobethesubjectmatteroftopology. Themainquestionariseswhetherthetermβ€˜topology’ isusedinthelooseand popularsenseof β€˜rubbersheet geometry’orbyreferencetoformal mathematical notions. Theissueofcontrol inexperimental designreflectsthisuncertainty. Chen etal.appropriatelypointoutthatitishardtotestfortopologicaldifferenceswithout introducingsomenon-topological differencesinthestimuli: β€˜β€˜thereseemtobe, in principle,notwogeometricfeaturesthatdifferonlyintopologicalproperties’’(2003, p. 6687). But fromtheviewpoint of mathematicaltopology, this is inaccurate. Anopensphereandaclosedsphereβ€”ortheir2-dequivalents,aclosedcircleandan opencircleβ€”havedifferent topological propertiesbut thesamemetricproperties (sameradius, astheboundaryoftheclosedcirclehasnodimension). Tobesure, thisfactmaynothaveanyconsequenceforthedesignofvisual teststimuli, asthe difference between a closed and an open itemhas no visual counterpart (dimensionlessitems,onemayargue, areunderperceptual discriminationthreshold). But thisbringsustoanimportant point. Whenwetalkabout topological differencesinvisualdisplays,wemaynotbetalkingaboutthedifferencesthatarethe subjectmatterofmathematical topology.Hencetalkingofβ€˜topology’ requiressome otherrefinement,shortofbeingloosetalk, especiallyifitistoprovidetheβ€˜formal theory’ that(Pomerantz, 2003)invokes. Onemaysuggestthatsomethinglikeaninternalizedtopologycapturesthefeatures wehaveinmindβ€”suchastheabilitytosortoutobjectsbasedonthenumberofholes theyhave(thedifferencebetweenthelettersBandO, havingoneandtwoholes respectively), rtoassesstheequivalencebetweenfigures(say, theletterSandthe o letterI).Butherewehavetoexertsomecare,becausethetheorynowrequiresanew notion,suchashole,andsomeaccountisneededofwhatitisforthevisualsystemto processthefeatureofbeingaholeinsuchawaythatitcontributestotheexplanation oftheperformanceofdistinguishingSorBfromO. Toseethepointmoreclearly, considerare-interpretationofChenetal. (2003). Fromtheviewpointofanintuitivetopology, thedifferencebetweenhavingandnot havingoneholeisimpliedbythepresenceorabsenceofothervisualfeatures.Now, if it weredemonstratedthat processingof thesefeaturesisavailabletothevisual system,itwouldbepossibletoreassesstheclaimthattheglobalfeaturesinvokedby Chenet al. andPomerantz(2003) areperceptual primitives. Thefollowingis a proposalinthatsense.

78 R. Casati

Thevisual featuresinquestionare:
(1) (2) Thepresenceofcompletevisual boundariesofaunit, and Theuniformityandconnectionoftheunit(Palmer&Rock,1994),alongwith itsmaximality(Casati, 2002).

Thepresenceofholesiscorrelatedwiththesesimplerfeaturesinthefollowingway. Ifamaximal uniformconnectedunitpossessesjustonecompletevisual boundary, thenit has nohole. If it possesses twovisual boundaries,thenit has onehole. 1 Ingeneral, foranygivenvisual display:
(3) Formmaximaluniformconnectedfiguresandncompletevisualboundaries, thenumberofholesis(n–m).

The further element that is thenneededis that the visual systemimplement somewayof countingthefeaturesandcomparetheircardinalities. Givenwhatis knownaboutthelimitsoftheabilitytosubitizesmallquantities, itisexpectedthat thedifferencebetweenconfigurationswith,say,oneandtwoholeswillbeaccessible tothesystem.Atthesametime,thedifferencebetweenconfigurationswithnineand tenholes is expectednot be accessible tothe system.But surely these latter configurations are topologically distinct fromthe viewpoint of mathematical topology. Hencetestingtheabilityof distinguishingbetweenconfigurations with varyingnumbersofholescandecidebetweenaholisticandalessholisticaccountof visualproperties. Furthermore, howfar cantopological generalizationgo? Letters I andJ are topologicallyequivalentintheintendedsense;butsoare,presumably,I,L,KandH (thelatterthree,forinstance,canallbeβ€˜shrunk’toanIwithoutβ€˜cuttingorgluing’). Will dataconfirmasensitivitytotheseequivalences?Somemayexpectinsteadthat somesortof parsingbycomponentswill predictthattheseshapesareresilientto placementinasinglecategory:anHhasthreecomponents,anIhasonlyone.Here againtheglobalistichypothesiscanbepittedagainstothertheoretical accounts. Itmaybequestionedwhetherfeatures(1)and(2)arereallysimplerthantheglobal featureof havingahole. After all, both(1) and(2) presupposethat theunity (connection) of boththe boundaryandthe figure are accessed; andassessing connectionisanotoriouslydifficultcomputational problem. However,ontheone hand, thisisageneral problem, onethat affectsall theoriesthat aresupposedto characterizetheentryunitsofthevisualsystem.Ontheotherhand,inordertoshow that sensitivitytothefeatureof possessingaholeis not sensitivitytoavisual primitive, it is enoughtoshowthat theformer canbeexplainedinterms of sensitivitytootherfeatures,withoutanyfurthercommitmenttothehypothesisthat thesefeaturesarethemselvesvisual primitives. Toconclude,whatistheevidencethattopologicalorglobalfeaturessuchashaving aholeareprimitivesof thesystem, accordingtoChenet al. (2003)?Theclearly delineatedcriterioninthepaperisthesizeofthecomputationalsystem:honeybees havesmallbrains.Thecriterionisnovelrelativetothefourcriterialistedatthetop ofthispaper. Thecriterionpredictsthatafeatureisprimitiveifitiscomputedby

Philosophical Psychology79

asmall system.However,forthereasonsgivenabove, thesysteminquestionmay simplybenotsmall enoughtoprovideacogentanswer.

Note
[1] Thecriterionreflectstheonegivenforcavitiesin3-dbodiesin(Casati&Varzi, 1994).

References
Biederman,I. (1987). Recognition-by-components: Atheoryof humanimage interpretation. Psychological Review,94,115–148. 2– Casati,R.(2002).Topologyandcognition.Encyclopediaofcognitivescience(pp.2). 2:McMillan. Casati,R., &Varzi, A. C. (1994).Holesandothersuperficialities. Cambridge,MA: MITPress. Chen,L., Zhang,S., &Srinivasan, M. V. (2003).Global perceptioninsmall brains:Topological patternrecognitioninhoneybees. PNAS,100(11),6884–6889. Hubel, D. H., &Wiesel, T. N. (1959). Receptivefieldsof singleneuronsincat’sstriatecortex. JournalofPhysiology, 60,106–154. 1 2: Kanizsa,G. (1979).Organizationinvision: Essaysongestaltperception. Praeger. Minsky, M. L., &Papert, S. A. (1998). Perceptrons: Anintroductiontocomputational geometry (expandededition). ambridge,MA: MITPress. C Palmer, S., &Rock, I. (1994). Rethinking perceptual organization: The role of uniform connectedness. PsychonomicBulletinandReview, 29–55. 1, Pomerantz,J. R. (2003).Wholes,holesandbasicfeaturesinvision. TrendsinCognitiveSciences, 7(11), 471–473.

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http://jeannicod.ccsd.cnrs.fr/docs/00/54/42/11/TXT/CasatiTopologicalPerception_UncorrectedProofs_.txt

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Philosophical Psychology Vol. 22, No. 1, February2009,75–79

Doestopologicalperceptionreston amisconceptionabouttopology?

RobertoCasati

Inthis articleI assessomeresultsthat purport toshowtheexistenceof atypeof s β€˜topologicalperception’,i.e., perceptuallybasedclassificationoftopologicalfeatures. Strikingfindingsaboutperceptionininsectsappeartoimplythat(1)configural, global propertiescanbeconsideredasprimitiveperceptualfeatures,and(2)topologicalfeatures inparticularareinterestingastheyareamenabletoformal treatment.Idiscussfour interrelatedquestionsthatbearonanyinterpretationoffindingsabouttheperceptionof topological roperties: p whatexactlyaretopologicalproperties, whatmakesthemglobal, inwhatsensethequotedfindingsmakesthemprimitive, andwhatarethehopesof a formaltheoryofperceptionbaseduponthem.Isuggestthatmathematicaltopologyisnot thecorrectmodelforcognitiontopologicalproperties, encethatsomeotherformalism h oughttobeusedβ€”aformofβ€˜β€˜internalizedtopology.’’However,oncetheprinciplesofthis typeoftopologyarespelledout,theymaynotbeasglobalisticasonemayhaveexpected. Keywords: opological erception; T P Topology; isualPrimitives V

Manylogicallyindependent but coordinatedfactorsconstrainthequest forvisual primitives.First,phenomenologytellsusthatthevisualsceneiscomplex,butatthe sametimethattherearerecurringelementsoutofwhichcomplexitymaybebuilt (Kanizsa,1979). Second,mathematicalmodels showhowit is possibletobuild complexrepresentationsoutofrepresentationsofsimplercomponents(Biederman, 1987).Third, computationalarchitecturemakesitplausiblethatthecomplexityof retinal input (providing pixel size informationabout properties at places) be

RobertoCasatiis 22 at 22. Correspondenceto:RobertoCasati,22. Email: casati@ehess.fr

ΓŸ ISSN0951-5089(print)/ISSN1465-394X(online)/09/010075-5 2009Taylor&Francis DOI: 10.1080/09515080802703711

76 R. Casati

organizedatveryearlystagesintoelementaldescriptorstoreducethecomputational load of subsequentstages (Palmer &Rock, 1994). Finally, behavioral and neurophysiologicalevidence for specific, downtosingle-neuronsensitivity to relativelywell delineatedfeaturesof theenvironment hasbeengatheredoverthe lastdecades,startingfrom(Hubel &Wiesel, 1959).Butdothesecriteriaconverge onasinglelist of primitives? Theydonot haveto, of course;andfindingout that what weexpect tobephenomenologicallyor computationallyprimitive is not sobehaviorallyorneurophysiologicallywill makeforaninterestingdiscovery. This is inpart theinterest of (Chen, Zhang, &Srinivasan, 2003) findingthat small brains suchas those of the honey bees display a sensitivity toglobal configurationalproperties,inparticulartopological propertiessuchasthepresence ortheabsenceof holes in2-ddisplays. It looks as if not onlybees areableto distinguishbetweenconfigurationsthat differonlyintheirtopological properties, butalsotheyareabletogeneralizetotopologicallyequivalentconfigurationsthatare rather different onmany other respects.Accordingto(Pomerantz, 2003), the findings areinterestingfor tworeasons. Thefirst reasonis that thetopological propertiesinquestionaregenerallyconsideredasrelativelycomplexandhardto compute(Minsky&Papert, 1998)butatthesametimeareverydeepandrobust properties of theenvironment—theyareinvariant under most transformations, asopposed,say,tometricproperties, hencesensitivitytothemwouldhaveahigh adaptive value. The secondreasonis that the mathematicsbehindtopological featuresissufficientlywellunderstoodandformalizedaccordingly,asopposedtothe relativelyinformalityofcharacterizationsofglobalfeatures,e.g.,theonefoundinthe gestaltliterature. Butwhatexactlyaretopologicalproperties,whatmakesthemglobal,inwhatsense Chenetal.’s(2003)findingmakesthemprimitive,andwhatarethehopesofaformal theoryofperceptionbaseduponthem? LetusfirstbrieflyreviewtheresultsChenetal.’sexperiment. Honeybeeswere trainedtochooseoneamongapairof configurations:anO-shapedstimulusand anS-shapedstimulus,say.Thentheyweretestedontheirabilitytodistinguishthe O-shapedstimulusfromotherstimulithatareeithertopologicallynonequivalentto theO-shape(suchasa  -shape,ora f -shape)andstimuli thataretopologically equivalent tothe O-shape(suchas a -shape) but lookdifferent as totheir non-topologicalaspect.Beessucceededinmakingthedistinctionwiththefirstsetof stimulibutfailedtomakeitwiththesecondset. Thisindicatesboththattheywere sensitivetotopologicaldifferencesandthattheycorrectlylumpedtogetheritemsthat aretopologicallyequivalent. OneshouldnoteinthefirstplacethatthedisplaysusedbyChenetal.fortesting honeybeeswere2-dpicturesrepresentingfiguresof different shapesandvarying topological properties.Thereis, of course,amuchgeneral problemof using2-d stimuliinordertodrawinferencesaboutavisual systemthathasadaptedtoa3-d world. But thereisalsoaspecificproblem: topologyin3-disnot automatically mappedonto2-dtopology. A2-dimageliketheshapeof theletterBcanbethe projectionofa3-dletterOthathasbentoverinthemiddle.Hencefromsensitivity

Philosophical Psychology77

to2-dtopologyonecanonlyinferwithmuchcaretoacorrespondingsensitivityto thetopologyof 3-dbodies; thisbyitself wouldquestionsanyecological-adaptive considerations. Theglobalityof topological propertiescanbecapturedintuitivelyinopposition tothelocalityof other features.Thedirectionalityof aline, for instance,is an intrinsicallylocalmatter.Atpointpthelinehasadirectionthatisgivenbyitstangent atp. Littledoesitmatterhowthelinelookslikeata(sufficient)distancefromp. Ontheotherhand, thefact that thelineclosesuntoitself (likeacircle)orhas terminations(likeabar)cannotbemadedependonthepropertiesofasinglepoint ontheline; manyotherpointshavetobescrutinized. Inthissensetheproperties studiedbyChenetal. areglobal, andappeartobethesubjectmatteroftopology. Themainquestionariseswhetherthetermβ€˜topology’ isusedinthelooseand popularsenseof β€˜rubbersheet geometry’orbyreferencetoformal mathematical notions. Theissueofcontrol inexperimental designreflectsthisuncertainty. Chen etal.appropriatelypointoutthatitishardtotestfortopologicaldifferenceswithout introducingsomenon-topological differencesinthestimuli: β€˜β€˜thereseemtobe, in principle,notwogeometricfeaturesthatdifferonlyintopologicalproperties’’(2003, p. 6687). But fromtheviewpoint of mathematicaltopology, this is inaccurate. Anopensphereandaclosedsphereβ€”ortheir2-dequivalents,aclosedcircleandan opencircleβ€”havedifferent topological propertiesbut thesamemetricproperties (sameradius, astheboundaryoftheclosedcirclehasnodimension). Tobesure, thisfactmaynothaveanyconsequenceforthedesignofvisual teststimuli, asthe difference between a closed and an open itemhas no visual counterpart (dimensionlessitems,onemayargue, areunderperceptual discriminationthreshold). But thisbringsustoanimportant point. Whenwetalkabout topological differencesinvisualdisplays,wemaynotbetalkingaboutthedifferencesthatarethe subjectmatterofmathematical topology.Hencetalkingofβ€˜topology’ requiressome otherrefinement,shortofbeingloosetalk, especiallyifitistoprovidetheβ€˜formal theory’ that(Pomerantz, 2003)invokes. Onemaysuggestthatsomethinglikeaninternalizedtopologycapturesthefeatures wehaveinmindβ€”suchastheabilitytosortoutobjectsbasedonthenumberofholes theyhave(thedifferencebetweenthelettersBandO, havingoneandtwoholes respectively), rtoassesstheequivalencebetweenfigures(say, theletterSandthe o letterI).Butherewehavetoexertsomecare,becausethetheorynowrequiresanew notion,suchashole,andsomeaccountisneededofwhatitisforthevisualsystemto processthefeatureofbeingaholeinsuchawaythatitcontributestotheexplanation oftheperformanceofdistinguishingSorBfromO. Toseethepointmoreclearly, considerare-interpretationofChenetal. (2003). Fromtheviewpointofanintuitivetopology, thedifferencebetweenhavingandnot havingoneholeisimpliedbythepresenceorabsenceofothervisualfeatures.Now, if it weredemonstratedthat processingof thesefeaturesisavailabletothevisual system,itwouldbepossibletoreassesstheclaimthattheglobalfeaturesinvokedby Chenet al. andPomerantz(2003) areperceptual primitives. Thefollowingis a proposalinthatsense.

78 R. Casati

Thevisual featuresinquestionare:

(1) (2) Thepresenceofcompletevisual boundariesofaunit, and Theuniformityandconnectionoftheunit(Palmer&Rock,1994),alongwith itsmaximality(Casati, 2002).

Thepresenceofholesiscorrelatedwiththesesimplerfeaturesinthefollowingway. Ifamaximal uniformconnectedunitpossessesjustonecompletevisual boundary, thenit has nohole. If it possesses twovisual boundaries,thenit has onehole. 1 Ingeneral, foranygivenvisual display:

(3) Formmaximaluniformconnectedfiguresandncompletevisualboundaries, thenumberofholesis(n–m).

The further element that is thenneededis that the visual systemimplement somewayof countingthefeaturesandcomparetheircardinalities. Givenwhatis knownaboutthelimitsoftheabilitytosubitizesmallquantities, itisexpectedthat thedifferencebetweenconfigurationswith,say,oneandtwoholeswillbeaccessible tothesystem.Atthesametime,thedifferencebetweenconfigurationswithnineand tenholes is expectednot be accessible tothe system.But surely these latter configurations are topologically distinct fromthe viewpoint of mathematical topology. Hencetestingtheabilityof distinguishingbetweenconfigurations with varyingnumbersofholescandecidebetweenaholisticandalessholisticaccountof visualproperties. Furthermore, howfar cantopological generalizationgo? Letters I andJ are topologicallyequivalentintheintendedsense;butsoare,presumably,I,L,KandH (thelatterthree,forinstance,canallbeβ€˜shrunk’toanIwithoutβ€˜cuttingorgluing’). Will dataconfirmasensitivitytotheseequivalences?Somemayexpectinsteadthat somesortof parsingbycomponentswill predictthattheseshapesareresilientto placementinasinglecategory:anHhasthreecomponents,anIhasonlyone.Here againtheglobalistichypothesiscanbepittedagainstothertheoretical accounts. Itmaybequestionedwhetherfeatures(1)and(2)arereallysimplerthantheglobal featureof havingahole. After all, both(1) and(2) presupposethat theunity (connection) of boththe boundaryandthe figure are accessed; andassessing connectionisanotoriouslydifficultcomputational problem. However,ontheone hand, thisisageneral problem, onethat affectsall theoriesthat aresupposedto characterizetheentryunitsofthevisualsystem.Ontheotherhand,inordertoshow that sensitivitytothefeatureof possessingaholeis not sensitivitytoavisual primitive, it is enoughtoshowthat theformer canbeexplainedinterms of sensitivitytootherfeatures,withoutanyfurthercommitmenttothehypothesisthat thesefeaturesarethemselvesvisual primitives. Toconclude,whatistheevidencethattopologicalorglobalfeaturessuchashaving aholeareprimitivesof thesystem, accordingtoChenet al. (2003)?Theclearly delineatedcriterioninthepaperisthesizeofthecomputationalsystem:honeybees havesmallbrains.Thecriterionisnovelrelativetothefourcriterialistedatthetop ofthispaper. Thecriterionpredictsthatafeatureisprimitiveifitiscomputedby

Philosophical Psychology79

asmall system.However,forthereasonsgivenabove, thesysteminquestionmay simplybenotsmall enoughtoprovideacogentanswer.

Note

[1] Thecriterionreflectstheonegivenforcavitiesin3-dbodiesin(Casati&Varzi, 1994).

References

Biederman,I. (1987). Recognition-by-components: Atheoryof humanimage interpretation. Psychological Review,94,115–148. 2– Casati,R.(2002).Topologyandcognition.Encyclopediaofcognitivescience(pp.2). 2:McMillan. Casati,R., &Varzi, A. C. (1994).Holesandothersuperficialities. Cambridge,MA: MITPress. Chen,L., Zhang,S., &Srinivasan, M. V. (2003).Global perceptioninsmall brains:Topological patternrecognitioninhoneybees. PNAS,100(11),6884–6889. Hubel, D. H., &Wiesel, T. N. (1959). Receptivefieldsof singleneuronsincat’sstriatecortex. JournalofPhysiology, 60,106–154. 1 2: Kanizsa,G. (1979).Organizationinvision: Essaysongestaltperception. Praeger. Minsky, M. L., &Papert, S. A. (1998). Perceptrons: Anintroductiontocomputational geometry (expandededition). ambridge,MA: MITPress. C Palmer, S., &Rock, I. (1994). Rethinking perceptual organization: The role of uniform connectedness. PsychonomicBulletinandReview, 29–55. 1, Pomerantz,J. R. (2003).Wholes,holesandbasicfeaturesinvision. TrendsinCognitiveSciences, 7(11), 471–473.

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