Spinning Shadows - Arts & Sciences Pages

International Phenomenological Society

Spinning Shadows Author(s): Roy Sorensen Source: Philosophy and Phenomenological Research, Vol. 72, No. 2 (Mar., 2006), pp. 345-365 Published by: International Phenomenological Society Stable URL: http://www.jstor.org/stable/40040928 . Accessed: 27/07/2013 16:10 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

.

International Phenomenological Society is collaborating with JSTOR to digitize, preserve and extend access to Philosophy and Phenomenological Research.

http://www.jstor.org

This content downloaded from 128.252.101.102 on Sat, 27 Jul 2013 16:10:49 PM All use subject to JSTOR Terms and Conditions

Research Philosophyand Phenomenological Vol. LXXII, No. 2, March 2006

Shadows Spinning ROY SORENSEN

Dartmouth College

Ifa spinningspherecastsa shadow,does the shadowalso spin?This riddleis thepoint intothenatureof shadowmovement.A generaltheory foran investigation of departure I argue notjustphysicalobjects.Ultimately, ofmotionwillencompassall movingthings, of theobjectsthatcast them. thatroundshadowsdo indeedspin.Shadowsare followers Partsof the shadowcorrespondto partsof the leader,so motionof thecaster's parts accountsformotionsof the shadow's parts.I concludewitha discussionof how the on otherpuzzlesaboutshadows. dynamicaspectsof shadowsimposesubtleconstraints "AndIsaiah theprophetcried untotheLord; and he brought theshadowtendegreesbackward,bywhichithad gone down inthesundialofAhaz." II KingsXX: 9

On January8, 2004 theroverSpirittransmittedthe firstpictureof a Martian sundial.The childrenwho helped design the sundial can visit a web site that featurestime-lapsephotographyof theshadow rotatingaroundthegnomon. Or so says NASA. If textbooklaws of motion are exhaustive, then shadows do not move. Shadows have neithermass nor energyand so do not trigger formulaslicensingascriptionsof momentum. Philosophers are especially suspicious of rotatingshadows. We will see why withthe help of the followingquestion: A spherecasts a shadow. If the spherespins,does its shadow spin as well?

Figure 1 This riddlebears on the classic controversyabout how objects change. Endurantistsbelieve objects persist by being wholly located at different times. They take pride in their ability to distinguish between a rotating homogenoussphereand a stationaryhomogenoussphere (Zimmerman 1998). Perdurantistssay that objects persist by having different temporal parts at

SPINNINGSHADOWS 345


different so perdurantists havetroublesustimes.Noneofthesepartspersist, and the distinction between the taining spherebeingstationary itrotating. Most endurantists doubtthatshadowsendure.So theyhave troublesusbetweentheshadowbeingstationary and it rotating.I taininga distinction A cautiousperthe for both the and its shadow. distinction accept sphere willmerelyinsistthestationary-rotation distinction holdseitherfor durantist to the He will drawattention boththesphereandits shadowor forneither. the dialecticaladvantageof shadowof thehomogenoussphereto neutralize theendurantist. Observabilityand indiscerniblemovement The puzzleis notthattheshadowlooksstationary. Considertheshadowof a shadowappears chippedfrisbee.Whenthefrisbeespinsrapidly,its irregular to becomeperfectly roundand stationary. reveals High-speedphotography thattheshadowis movingandneverchangesitsshape. The realproblemis thatanyrotation bytheshadowofthesphereis absoLeibniz'sprincipleof theidentity of indiscernibles lutelyindiscernible. probetween vides a metaphysical for the distinction the premise collapsing will shadowbeingstationary andit rotating. thinkers Anti-metaphysical say theempiricalequivalencemakesthe distinction meaningless.Accordingto thelogicalpositivist'scriterion a statement of significance, is meaningful if it is or The 'If answer a testable. only analytic empirically spherespins, thenso does its shadow'failsthistestandso is condemned by the logical as meaningless. positivist IfLeibnizandthelogicalpositivistsareeven-handed, theywill go on to dismissas meaningless thehypothesis 'If a sphereis stationary, thenso is its shadow'.Forhowcouldone tell whether thestationary spherehas a stationthana spinning shadow? aryshadowrather Mostpeoplearerepelledbythissymmetry. Theybelievethattheshadow of the stationary One mightsubtlyinsist that the sphereis stationary. - neitherstationary shadowis in a neutralstate nor spinning.Perhapsthe sphere'scenterpointachievesthis neutrality by virtueof lackingproper But a is state fora thingwithpartsto be parts. dynamicalneutrality strange in giventhatitis in space (as opposedto beingan aspatialnumberor Space itself).It is also strangethatmerechange(frombeingroundto beingoval) couldentaila transition fromtheneutralstateto thestationary state. The verdictof meaninglessness has technicaldrawbacks.If the spin andthestationary aremeaningless,thentheyshould hypothesis hypothesis havea meaningless But 'If a disjunction. spherespins,theneitherits shadow or its shadow is is true rather thanmeaningless. spins stationary' The truth of thisstatement assuresus thatwe arenotcommitting a catethe shadowspins. Shadows move in gorymistakewhenaskingwhether 346

ROY SORENSEN


exactly the same sense as physical objects. A balloonist can be sure that a dark patch is moving even if he is unsure whetherthat patch is a black balloon or his own shadow. G. E. Moore was puzzled by the movingpicturesat the cinema. How can we see the actors move if we are only presentedwith a rapid succession of still pictures? If theydid move, it would be in the same sortof sense in which a shadow moves, which is very similar to thatin which a wave moves: no material thingis transferredfrom one place to the other; but one surface after another, is lighted with a degree of illuminationless than that which lightssurroundingsurfaces, and this happens continuously- just as in the case of a wave, a certain formof arrangementof masses of water occurs successively in many different places, & thatcontinuously.(1962, 139)

Moore goes on to suggest that shadows (and waves) move only in the sense that"to any normalperson they would look to be moving." (1962, 139) He concedes thatthiscannotbe theonly sense of 'move' because we can say that the earthis moving even though in normal circumstancesit does not appear to be moving (1962, 118). So Moore believes thereare two senses of 'move' and restrictsshadows to thesense thatparallels the normalobserverdefinition of color (as in "Something is yellow if and only if it looks yellow to a healthyhumanbeing in daylight"). Moore's shadow is on the earth. If his shadow persists for one second, thenit mustshare the motionof the earth.Since the earth's motion is objective, then Moore's shadow must also objectively move. We can generalize: shadows sharethe movementof the earth.Therefore,even shadall terrestrial ows thatdo not appear to be movingin normalcircumstancesare moving. A passenger waitingforhis trainto leave the stationmay misconstruethe motionof a neighboringtrain'sshadow; he may thinkhe is moving through a stationaryshadow when he is reallystationaryand thereis a shadow passing over him. We distinguishapparentmotion of shadows fromreal movement of shadows. The shadow moved' cannot be paraphrasedas The shadow appearedto move'. Shadows are in the same public space as the objects that cast them. Commensurabilityof directionis assumed in the historian's explanation of why clocks move clockwise: The hour hand of the firstmechanical clocks mimickedthe shadow of a sundial. In the NorthernHemisphere, where the mechanicalclocks werepioneered,theshadow moves clockwise. When we watcha lunareclipse, we can ask how much of the darkeningis due to the movement of the Earth's shadow and how much is due to the movementof the Moon intothatshadow. Jointeffectsimply commensurable causes. Commensurabilityof speed is assumed in the astronomer'scalculation of a solar eclipse's duration:The two variables are the diameterof the moon's SPINNING SHADOWS


347

shadow and its groundspeed. The shadow's groundspeed is affectedby the curvatureof theearth.Anotherfactoris thespeed of the earthand moon. The shadow is slowed by the factthat the Earth's rotationhas the same angular directionas the Moon's orbital motion. This effectis strongestat the midpointof the shadow's transitwherethe movementof the Moon's shadow and the movementof the Moon are closest to being parallel. The astronomer'smeasurementscompel us to say that some shadows move in surprisingways. During a solar eclipse, the ground speed of the moon's shadow often exceeds 5 kilometersper second. Unlike a jet, the shadow does not produce a sonic boom when it breaks the sound barrier. Unlike physical objects, shadows can move fasterthan the speed of light (Rothman 1960). A shadow cannot initiatemovementon its own. Nor can it be directly moved. However strongyou mightbe, you cannot lift my shadow above the ground.To move my shadow you must alterotherthings- physical objects

such as my body or the lamp thatilluminatesme. Aristotleand Descartes requiredthat a mover impact the moved. Impact witha shadow is problematic.Shadows offerno resistance.The perfectease withwhichshadows are penetratedmakes it impossible to transfer energyto them. Shadows have no mass. The air inside the long shadow volume you cast at sunset weighs more than you do. (Air is surprisinglyheavy.) Your shadow weighs nothing. Those who continueto harborconceptual reservationsabout the spinning shadow riddlemightnavigate throughtechnicaldifficultiesby declaringThe shadow spins' indeterminate (ratherthan meaningless or ambiguous). Indeterminatedisjuncts can forma determinate disjunction.For instance, if you feel thattheexpansion of a balloon is a borderlinecase of movement(because its centerof mass stays the same), then you cannot say the balloon is definitelymoving and you cannot say that it is definitelynot moving. You can say that definitelythe balloon is eithermoving or stationary.Similarly, if you deem theroundshadow to be a borderlinecase of rotation,then you cannot say it is definitelyspinningand cannot say it is definitelynot stationary. You can say it is definitelyeitherspinningor stationary. This hedgingis incompatiblewith the common sense belief that a stationaryspheredefinitelyhas a stationaryshadow. Common sense privileges the reststate.We ask why a rock moved, not why it remainsstill. Aristotleendorsescommon sense by characterizingrestas the naturalstate forterrestrial objects. If Aristotlewere willing to extendthe rest privilege to shadows, he could endorsetheanswerthatthestationarysphere has a stationaryshadow. Aristotle'sphysics mightnot offera complete solution.He says the natural motion of celestial objects is circular(to explain why heavenly bodies

348

ROY SORENSEN


still move despite their infiniteantiquity).Since the moon is a celestial object thatcasts a shadow, Aristotleis committedto thereversepreferencefor the moon's shadow (when the moon is illuminatedalong its axis of rotation). Accordingto Aristotle,the moon is theonly celestial body thatdoes not emit its own light. So the lunar shadow would be the single rotatingshadow in the universe. Isaac Newton opposed Aristotle's presumption that rest is the natural between rest and motion: state. Newton's firstlaw promotes indifference in of or continues its state rest of uniform motion in a straight Every object line unless compelled to change thatstateby impressedforces.Perhaps Newtoniansprojectthislaw on to shadows. They wince at the suggestion that the spinningshadow of a ball with a spike in it stops spinning when the spike . dropsoff.

Figure 2 How could sheer change or shape (to roundness) act as an instantaneous brake?Even Aristotelianshave the intuitionthat the shadow should gradually slow down afterthe spike drops. Spinnersexploit the principlethatthe motion of a shadow is not affected by its contact with anothershadow. Consider the separateshadows cast by a spinninggold hemisphereand a spinningoak hemisphere.

Figure 3

SPINNING SHADOWS


349

The hemispheres arethesamesize and arerevolvingarounda commonpoint atequalspeed.As theydrawcloserandmakecontact,theirtwo shadowslook likea singleroundshadow.But contactbetweentwo shadowsis notenough to makethemone shadow.My shadowand yourshadowdo not becomea singleshadowwhenwe shakehands. is irrelevant Spinnerscan also appealto theprinciplethatlightintensity to motion.People readilydescribetheshadowof a mesh disk as spinning. For theycan see thesub-shadows cast by the gridof wires.Underintense eventheshadowofa thickly wovenmesheddiskcan be seen as illumination, theshadow.As the movingby virtueof thetinyspecksof lightpeppering The no are the because dims, photons gettingthrough. light specksdisappear to conshadowofthemeshdiskbecomescompletely black.We arereluctant cludethattheshadowof thediskhas stoppedmoving.For we do not treat as relevant to shadowmovement. changeof lightintensity Do shadowsevenexist? Zeno's responseto theenigmaofthespinning shadowswouldbe todenythat shadowsmove.The mostextreme for extremesolutionis elimithis premise nativism aboutshadows.Ifshadowsdo notexist,thenthereis no suchthing as themovement (or rest)ofshadows. One rationale foreliminativism is ontological dependence.The dependence of shadowson thedistribution of lightshowsthattherereallyare no shadows. But whystop there?The distribution of lightdependson electro-magneticwavesthatin turndependon yet morefundamental features.Spinoza reasonedthatifa substance is required tobe independent ofotherthings,there is at mostone substance:Nature(or God, depending on how you look at it). Ifthechainofdependence thenthereareno substances.Reality is bottomless, is thenan infinite regressofepiphenomena. A moderate eliminativist maysuggestthatwe onlyshaveoffthetop layer oftheontologicalhierarchy. Butshadowsarenot thetop layer.Considerthe thatpasses through theshadowsof topplingdominoes.The parapara-wave wavemovesalongtherowofshadowsbut each shadowonlymovesits own Para-wavescannotbe wavesbecausewavesmustcarryenergy.Paralength. wavesdependon shadowsthatin turndependon materialthings.Shadows can hostother"disturbances": ...a disturbanceis definable as an object or entityfound in some otherobject- not in the sense in which a lettermay be found in an envelope, or a biscuit in a tin, but in the sense in which a knot may be in a rope, a wrinkle in a carpet, a hole in a perennial border,or a bulge in a cylinder. One way telling whetheran object X is "in" an object Y in the sense peculiar to disturbances is to enquire whetherX can migratethroughY. (Karmo 1977, 147).

For instance, thebulgein theshadowof a freshly fedsnakecan move south evenif theshadowis stationary. Big bulgescan hostlittlebulgesandlittle 350

ROY SORENSEN


bulges can host littlerbulges. Therefore,the chain of dependententitiescan rise ever higher. Principledeliminativistswould like to halt the purge at some level of reductionthat respectscommon sense or science. They wish to spare whatever the legitimateauthoritiespresuppose in theirexplanations. But this conshadows. The role of shadows in physics has been servatismre-introduces documentedpanoramicallyby Roberto Casati (2003) and in telling close-ups by historiansof astronomy. The mostpowerfulreason forbelieving thereare shadows is that they are in plain sightin daily life. A second importantreason is that shadows play a role in observationin itself.We see by means of shadows. Psychologists who reveal the role of shadows in perceptionpresuppose common sense realism about shadows. Almost all theirwork priorto 1990 focuses on static shadows. One exception was interestin the three-dimensional appearanceof rotatingsilhouettes. If you bend a coat hanger into an irregularshape and attachit to a spindle of a turntable,then the silhouetteof the hanger will look like a flat scribble. When the spindle rotates, the Indeedyou can discern the shape of the silhouettelooks three-dimensional. coat hanger(Wallach and O'Connell, 1953). Nowadays the importanceof motion in perceptionis illustratedby inviting thereaderto hold a pencil over the leftside of a blank page. Fixate on a point at the rightof the page so that the shadow of the pencil is in your peripheralvision. Afterthirtyseconds, the shadow will seem to disappear. You can make it re-appearby wigglingthepencil. The lesson is that the visual systemis sensitive to dynamiccontrasts.Stationaryshadows stop registering.This explains why we do not see the silhouettesof the blood vessels which sprawl over our retinas.To see themyou have to go into a darkroom and illuminateyour eyeballs with a penlight from the side. Remember to wiggle the penlight! In the 1990s, interestin moving shadows became more systematic.Psychologistshave long studiedtheclues thathelp us turnthe flat retinalimage intoa threedimensionalrepresentation. They were given a runningstartfrom the theoryof perspective.Shadows form one class of clues. The size and shape of an object's shadow providecomputationallycheap informationabout theircasters.Thanks to advances in computergraphics,Daniel Kersten (1991) and his colleagues could test the relative strengthof anotherclue: shadow movement.Kerstensuspected that shadow size and shadow shape would prevail over shadow movementbecause morecomplicatedcalculations are needed to exploit movement.But shadow movementtrumpedthe otherclues.

SPINNING SHADOWS


351

Are shadowsmomentary entities? Punctualistsabout shadowsconcedethat shadowsexist- but only for an akinto that instant. Theysay thereis onlyan illusionof shadowmovement causedby the framesof a movie. Punctualismcan be motivatedby the shadow'slackofimmanent causality:Each stageof theshadowdependson a lightsourceandtheobjectblockingthelight.No stageof a shadowcauses thenextstage.Therefore, shadowscannotpersist. Punctualists wouldstill facea toughquestionfromthespinningsphere: Whatis thedifference betweena seriesof shadowswhosemembersareall in anda serieswhose membersoccupyslightlydifferent thesame orientation betweena orientations? (Casati andVarzi 1994, 121) Thereis a difference of and a ofa stationary sequence photographs sphere sequenceofphotographs of the spinningsphereshow the of a spinningsphere.The photographs - even if all the photographs look identical. orientations sphereat different also recordthe shadowsof thatsphere.Why preferthe These photographs of of the spinningsphereare photographs thatthephotographs hypothesis shadowsina uniform orientation as opposedtodiverseorientations? The orientation riddleis themoregeneralenigma.The riddleaboutspinningshadowsis just a specialcase. Mirrorreversalcan also changeorientaWhenaskedwhythemirror tionindiscernibly. onlyreversesDIED in OTTO of OTTO DIED we answerthatthemirror doesreverseOTTO. The symmetry same stance when we the its reversal indiscernible. Should makes adopt just a mirrorreversalprocess?This areputthrough shadow-casters symmetrical theshadowpersists. riddlesidestepstheissueofwhether motion.In addition is translational The mostobviouscostof punctualism denies that shadowsmove the that shadows to denying rotate, punctualist denies that shadowsgrow or fromplace to place. Indeed,the punctualist thatshadowssustaintheir must even The deny changeshape. punctualist time. or locations over shapes Punctualismimpliesthatall shadowsare invisible.Anythingthatlasts of a seconddoes not last long enoughto be seen. A less thana trillionth to whatis seen in the way punctualist mightreplythatshadowscontribute ofthemovie. to theperception thattheframeofa moviecontributes Even if thiscinematographic viable, we analogywerepsychophysically thanshadows.The shadowsthemof shadowsrather wouldbe seeingeffects selveswouldbe invisible. ifthepunctualist embracesa causal The invisibility followsindependently definition of part-hood (Slote 1979). My rightfootsupportsmy body. Its contribution to theorganization of mybodymakesit partof my body.My shadowdoes notandso is notpartofmybody.Aretherepartsof my shadow thatcontribute to my shadow'sorganization? The shadowof my rightfoot does not supporttheshadowof myrightleg. The sub-shadows arecausally 352

ROY SORENSEN


ofeachother.So an applicationof thecausal definition of parts independent no All that have shadows would be shadows and simparts. simples implies to see. are too small ples Volumesand theat-attheoryof motion Othertheorists grantthatshadowspersistovertimebutdenythemmotion. Volumetheorists sayshadowsare volumesof unilluminated space. Volumes ofspace persisteventhoughone stageofthevolumedoes notcause thenext. Volumeshavesizes andshapes.However,volumesof space cannotchange shapeor position.So the volumetheoristdeniesthatshadowscan change of shadowsis reminiscent shapeor move. The volumist'sstaticconception ofEgyptianmuralsexceptitdoes allowshadowstobe three-dimensional. willconcedethatshadowschangeposition A slightly moreliberaltheorist movement.This seems contradictory to we but denythatthis constitutes ofmotion.We say motionis nothingmorethan believersin theat-attheory almostuniversally accepttheat-attheory.It changeof position.Philosophers is simpleandsolves Zeno's arrowparadox(Russell 1937, 350). However, dissidentswho thinkthereis moreto motionthan thereare distinguished Michael of Tooley (1988, 244) imaginesa worldin which change position. different placesat random.The successivepositionsof one of thingsoccupy theseherky-jerky objectshappento forma smoothpath. Tooley suggests thattheobjectis notreallymovingbecausetheearlierpositionsof theobject wouldforbid to its laterpositions.This requirement arenotcausallyrelevant with it is the shadow even from shadow though compatible moving any different positions. occupying Allowingforchangeofshape in movingshadowsweresown by David The seedsof philosophicalinterest Lewis and StephanieLewis (1970) in their dialogue "Holes". Bargle since ambushesthe materialist counterexample: Arglewithan unexpected thereareholes anda hole is wherethereis no matter,thereare immaterial talkof things.Arglefirsttriestodenytheexistenceofholes by paraphrasing thatthisreduction holesin termsof shapes.WhenArglebecomespersuaded theexisofholestoshapescannotsucceed,he changeshis mindandaffirms holes his materialist tenceof holes. Arglepreserves scruplesby identifying Since hole-linings withhole-linings. (such as a toiletroller)can spin,Argle boldlyassertssomeholesspin. each Given thatshadowsare holes in the light,Argle would identify the unilshadowwithits shadow-lining (thelightimmediately surrounding the shadow So of whether round the issue luminated volume). spinsturnson itsshadow-lining thequestionof whether spins.

SPINNING SHADOWS


353

The Lewis's dialogueinspired RobertoCasatiandAchilleVarzito writea wholebook on holes. Theyofferan immaterialist analysisof how shadows move.CasatiandVarzisayholesareconstituted by spacejust as MichelanCasati by matter(marblein particular). gelo's StatueofDavid is constituted but andVarziregardshadowsas holesinthelight.Holes "arenotabstractions but space. Theyarenot individuals, theyarenot madeof anything although of the material are hosted in...; rather, they theyareimmaterial parts object at the of their located surfaces hosts...." bodies, (1994, 6) Holes can change location.A bubblerisingthroughwateris a hole that is constituted by of holes- or, if successiveregionsofspace."Space is, in a sense,thematter is to materialobjects."(1994, 32) youprefer, space is to holes whatmatter ittakeson theshapeof its host.Theycompare Since thehole is immaterial, thehole to thewinein a spinninggoblet.The gobletspins but the wine staysstill. upon thewineso some of Actually,themotionof theglass is impressed will is thewine rotate.Butthat onlybecausethewinehas mass. Casati and Varzi'spointis thatthegoblet'scavityis immaterial andso cannotacquire motionfromcontactwiththegoblet. CasatiandVarzistillrequirean unstated premisethatrestis theuniversal defaultstateof shadows.If some shadowsarealreadyin motion,thentheir lackof masswouldjustmakeshadowsall theharderto stop. Whathappensif a circularhole in a rotatingsphereis madeelliptical? Casati and Varzi (1994, 123) denythatthe hole suddenlystartsto move. Insteadtheholejustgentlychangesitsshape.ThusCasati andVarzi'stheory of holes impliesthatall of the apparentrotationof shadowsis actually deformation. PartialRotation the ErdingSayan (1996, 87) criticizesCasati and Varzi for exaggerating extentof deformation. He doubtsthat"gentlechangeof shape"can account fortranslational motion. Sayanbelievesthata circularshadowcannotrotateandthatanyspinning shadowwitha circularcorecannotrotateas a whole.But he thinksthatan he thinksthecircularcore rotate.Specifically, ellipticalshadowcan partially oftheellipticalshadowis stationary andtherestrotates.Sayan believesthat space is themediumof a hole. Holes are a speciesof waves. Shadows,as holesinthelight,movethrough of space. spacebythesuccessivedarkening The inspiration forSayan's analysisis PeterForrest's(1984) Achillesweed.Thisfictitious plantgrowstencentimeters perhourbut a tortoiseeats one sideof itat thesame rate.None of thepartsof theweedmove. Yet the weedwindsup ina different location.

354

ROY SORENSEN


Peter Forrest presentsAchilles-weedas a counterexampleto the "at-at" theoryof motion.He thinksthe weed does not really move even though it is at one place at one timeand at anotherplace at anothertime. Sayan thinksthatthe Achilles-weeddoes move. He values Achilles-weed as a model for the motion of dependententities such as holes, waves, and shadows. For thesake of clarity,Sayan cultivatesa tidierdescendentof Achilles-weed: Achilles-turfis a doormat-shaped patch of lawn grass that is comprisedof grass blades.

Figure 4 The weed grows in an easterlydirectionby continual growthof new blades and is prunedat the westerlyend by a tortoise.The tortoisemoves by changing the location of its parts. Achilles-turfmoves in the way that waves move; successive blades are activatedand deactivatedas parts of the turf.In thecase of holes, points of space play the role of the blades of grass. Shadows can be understoodas holes in the light. As a region of space loses its illumination,the shadow grows. Sayan grantsshadows translationalmotion.But he denies that roundshadows can rotate.For such motiondoes not exhibitthe succession of activation and de-activationessentialto waves. Sayan's theoryis geometrical,focusing on the intrinsicfeaturesof dark patches. Here he is backed by a mathematicaltraditioninauguratedby Jean VictorPoncelet (Gray 1993). Afterbeing takenprisonerby the Russians during Napoleon's retreatfromMoscow, Poncelet kept up his spirits by developing a branchof geometryin which the key featuresof a figureare those thatit shares with its shadows. These are the projectivepropertiesof an object- thesortartistsexploit when theydepicta threedimensional object on a two-dimensionalsurface. Another French mathematician,Louis Poinsot, developed a simplegeometricaldescriptionof how a solid body rotates.This projectivegeometrywas extendedby AugustFerdinandMobius. His barycentriccalculus focuses on the object's geometricalcenterof gravity(which is calculated by assumingthattheobject has a uniformdistributionof density).

SPINNING SHADOWS


355

Sayan providesno role for the object that blocks the light. Yet movementsof shadows do seem to trackmovementsof the light blocker. Consider the shadow of a slowly spinningchipped frisbee.

Figure 5 Its moving shadow can be imitatedby rotatinga chip-shapedbeam of of theshadow of a stationaryunchippedfrisbee. lightalong thecircumference The stationaryshadow is being alteredto look as if it is spinning. Sayan must dismiss this descriptionas naive. For him, thereis nothing more to the shadow than the darkpatch. He infersthat questions about its movementsnevercruciallydependon the means by which the darkpatch is produced. AlthoughSayan forbidstotalrotation,he permitspartialrotationand distinguishesthatfromchange of shape. He takes deformationto entail change of shape or size. Thereforethemovementof a rigidshadow fromone location to anotherqualifies as translationalmotion,and not as deformation.Consider a one-inch movementof my twelve-inchruler's shadow across my desktop. Sayan says the light devouredthe southernmostinch while the northernend grew an inch. The remainingeleven inches did not move. To move as whole, theremustbe a change in the geometricalcenterof gravityof each part. So Sayan thinksthatthe ruler's shadow only partiallymoves. I agree thattheruler's shadow would move in thisgrudgingway if it were cast by an Achilles-ruler.But the shadow of my ordinaryruler moves all twelve inches of its shadow parts when it moves an inch. Each part of the shadow is a follower of some part of the ruler. Consequently, Achilles movementof shadows is as rareas Achilles-objects. Yet Sayan's theoryimplies thatall shadow movementis Achilles movement. Sayan is drivento this narrowmodel of movementby his resolution to individuateall partsof a shadow internally(in termsof the shadow's intrinsic makes contacta sufficientcondition properties).This approach inadvertently forthe unificationof two shadows. So on Sayan's theory,when the moving shadow of a bird grazes the shadow of a mountain,the result a single, huge movingshadow. We are happy to individuateparts of physical objects by other objects. The partsof a humanhand individuatethepartsof my glove. The hooks of a hookwormare understoodrelationallyas devices to grasp the intestinesof its host.We follow thesame extrinsicpolicy of partindividuationforshadows.

356

ROY SORENSEN


Theessentialconnection betweena shadowandits blockeris obscuredby to focuson thecast shadow(thepartof thethreedimensional our tendency shadowthatis intercepted shadow by a surface).This leads us to interpret movement as an exampleofactionat a distance.As dustyairreveals,thecast shadowis just an edge of a threedimensionalshadowthatadheresto its blocker.The blockeris in contactwiththeshadowbody,so movementsof echomovements oftheblocker.Sincethespeedof theshadowbodynaturally lightis finite,thedistantportionsof some largeshadows(suchas the 150 millionkilometer shadowofCometHale-Bopp)significantly lag behindtheir the is blockers.Forterrestrial however, fidelity virtually purposes, perfect. we needto go outsidethe darkpatchto individuate To recapitulate, the thepartsoftheblocker.For shadows partsofa shadow.We needto introduce Therecan be no shadowswithout arederivative (likeechoesandreflections). The to the shadowignoresits causal structure blockers. geometrical approach - andthereby, andtherefore obscuresitsanatomy misconstrues itsbehavior. Full Rotation in calculatingrotation(Parsons Psychologistsconfirmour felt difficulty how rotation has bedeviledtheoriesof science chronicle Historians of 1995). This makes it to since Aristotle. motion tempting inferthat rotationis shadowmovement. essentialto theproblemofindiscernible translational motionis enoughto frametheissue. Thinkof Fortunately, theshadowofa conveyorbelt.Whenthebelt is turned on, does theshadow startmoving?If thereareno chinksbetweenthe segmentsof the belt, the If you suspectthatthereis some illicit belt'sshadowwilllook undisturbed. in theconveyorbelt,consideran unconventional rotation conveyorbeltthat the left end then at and worksbyhavingsegments grow disappearat theright end. The shadowof theconveyorbelt is movingbecausepartsof theblocker shadowpartsby blockerparts,we areready aremoving.Once we individuate The shadowcast by a spinningdiskis madeup shadows. to acceptspinning of shadowpartsthatfollowtheircastingparts.This is moresalientif the diskis a roundjigsawpuzzle.The shadowhas as manypartsas therearejigso is theshadow.Now spinthedisk. saw pieces.Whenthediskis stationary, Yet the shadow mustspin becausetheshadows The shadowlooksstationary. of thejigsaw piecesmove. movements. Some of One advantageof thissolutionis thatit coordinates is betweentheshadowandtheblocker.Recall theball thedesiredmatching fromit. Let thespikerotateawayfromthelight thathas a spikeprotruding source.

SPINNING SHADOWS


357

Figure6 The spike'sshadowshrinks fromtheleftside of theball. The anddisappears shadowofthespikesoonreappearson therightsideoftheball. The desireto thattheshadowof the nurtures theinference coordinate shadowsandblockers ball the motion of the ball. shares spiked spiked of theshadow,then Of course,ifwe relysolelyon theintrinsic properties thattheroundpartoftheshadowis stationwe cannotexcludethepossibility a shadow element and onlyto reappear growsanddisappears, ary spike-shaped on theoppositesideofthecircle. But whyconfineour attention to the shadow?The prisonersin Plato's allegoryofthecave arechainedso thattheycan only view shadowscast on thewall.We arefreetoexaminetheobjectsthatcasttheshadows.Like other animals,we attendto shadowschieflybecause theyprovideclues to the objectsthatcastthem. Rules that coordinatethe movementsof shadowswith theirblockers wouldbe moreinformative thanrulesthattreatedshadowsautonomously. Thus a purelygeometricalapproachto individuating shadowmovements shadwouldbe less likelyto developthana causal approachthatcoordinated ows withtheirblockers. betweenshadows.We Our appetiteforcoordination extendsto matching arekeento keepdouble-shadows a wobbling movingtogether. Illuminating cone fromits apex yieldsa roundshadowwithno discernible movement. a second source to illuminate the of the Adding light wobblingcone profile a We thatonly shadow. resist the conclusion produces swingingtriangular one of theshadowsis moving.Our reluctance whenwe maneuver intensifies thesecondlightsourceso thatthe two shadowsoverlap.It is arbitrary to describeone halfofthefusedshadowas movingwhiletheotheris stationary. Whatgoes fordouble-shadows extendsto thewholescene.We wantthe shadowsto move in a waythatmakesoverallsense.This is especiallytrue whentheshadowsexhibitbiologicalmotion. Scrolling Coordination betweenthemovement of blockersandtheirshadowsis comthe fact that shadows have fewerpartsthantheirblockers.Only promised by a minority of an object'spartsblocksthelight.Considertheshadowcast by Saturn'sring.Does itsshadowrotate? No. Instead,bands of shadowsmigrateacross Saturn until they are destroyed (by virtueof thatringsegmentno longerblockinglight). The shadowsthatmeettheirendatSaturn'snightfall arereplacedbyshadowbands 358

ROY SORENSEN


emerging along Saturn's daybreak. Let us call this analogue of rotation "scrolling". As in rotation,scrolling permitsan object withouttranslational motionto have all of its parts simultaneouslymove. This permission is not a requirement.Justas a rotatingobject can simultaneouslymove translationally, a scrollingobject can simultaneouslyundergotranslationalmovement. A cylindercan simultaneouslycast a scrolling shadow and a rotating shadow:

Figure 7 The roundshadow rotateswhile the rectangularshadow scrolls. All the parts of the rectangularshadow uniformlymove towardtheir destructionbut are steadilybeing replaced at theoppositeend by new shadow parts. A landingsurfaceis not neededfor a shadow. The earth's shadow extends into empty space. Aristotlerealizedthat a lunar eclipse is the moon's interceptionof thisshadow. He also realizedthat the earth's shadow is the night. Aristotledid not realize thatthe nightscrolls on the surfaceof the moon. When sunlightstrikesUranus at its southernaxis of rotation,the Uranian nightrotates.Uranus is the only planet thatspins like a barrel.All the other planets spin uprightlike tops. (Astronomerssuspect Uranus was knockedon its side.) If Earth rotatedperfectlyupright,then its shadow would perfectly scroll. The shadow of each portion of the earth would be destroyedbefore coming aroundagain. However, theearthis tilted23.5 degrees.This complicates the motion of our night. Happily geographershave terminologythat can guide my descriptionof how theearth's nightmoves. A Marsden square is a 10-degreelongitude-latitude square. These are subdivisionsof thegrid you see on a globe. There are 936 Marsden squares. Each has its own numberon a Marsden Square Chart.The top row of squares (ringingthe NorthPole) number from901 to 936. Since the earthis slightly tilted,the shadows of these

SPINNING SHADOWS


359

the rotationof the squaresrotate(castinga shadowthroughout uppermost a earth)buttheshadowsofmostsquaresscroll(casting shadowforonlypart oftheearth). oftherotation VolcanicChain illustrates how a physicalobject The Hawaiian-Emperor can scroll. The islandscomposingthe chaindevelopas the Pacific Plate volcano. movesnorthovera fixedhotspot.The hotspotcreatesa submarine Pillowlava fromthevolcanocreatesan island.The volcanobecomesdormant afteritridespastthehotspot.Itsislandbeginsto erodeaway intothePacific Ocean.Meanwhilethehotspotbeginsto forma newisland. The movementoflightsources Can theshadowof a stationary objectrotate?We areslow to ask thisquestionbecausehumanbeingsoperateunderthedefaultassumptionthatobjects arebeingilluminated fromaboveby a singlelightsource.We also tendto the lightsourceas a point.Finally,we simplifythe lightingby treating is stationary. operateundera defaultassumptionthatthelocal environment of a shadowto movement is to our instinct trace first any Consequently, movement of itsblocker. Whena lightsourcemovestothesideofa disk,theshadowchangesfrom thatobjects roundtooval. Sincewe also operateunderthedefaultassumption default between our the conflict the same assumptionsstimulates stay shape, thethought thatthelightsourceis moving. If thediskis also spinning,thenthechangeof shadowshape illustrates how an oval shadowcan indiscernibly rotate.The changeto side lighting to whichpartsof the much how shadow corresponds merelyre-apportions will indiscernibly oval the shadow disk.If thelightsourcecircles disk,the theblocker rotate.Thisshowshow a shadowcan rotateeventhoughneither northelightsourceis rotating. About70% of thoseattending my lecturesdenythatthe roundshadow spins. My explanationis thatpeople combinea defaultassumptionthat withan intrinsic approachto shadows.Since the only thingsarestationary relevantdatumis thedarkpatchas it presently appears,thereis nothingto override thestationary defaultassumption. An implicationfortheYale shadow puzzle As one grantsmoremovementto shadows,the rangeof viable theories ofshadows.Recall the shrinks. Formovement becomesan extradifferentiator Yale shadowpuzzle(TodesandDaniels,1975).(See Figure8.) Is theshadow castby High or Low? Since one objectis enoughto block thelight,either theappearance of the cast shadow. objectcan be removedwithoutaffecting Some respondto theoverdetermination thattheshadowis cast by asserting on spinning fusionof High andLow. Our meditations by themereological 360

ROY SORENSEN


shadows suggest an objection. What if High and Low are disks spinning in opposite directions?In whichdirectiondoes theshadow spin? rotation.Let East be a high The objection cannotbe avoided by restricting disk travelingeast and West be a low spheretravelingwest. What happens when East meets West? Is the shadow below them moving east or west? Translationalmotionis enough to raise the objection to the fusionview.

Figure 8 Accordingto theblocking theory,a shadow is an absence of light caused and so on). Only diffraction, by blockage (as opposed to absorption,filtering, the shadow is idle. Low the blocks Therefore, belongs to causally light. High how from must be movement The shadow's High blocks computed High. light. case that serves as a preceThe blocking theorysuggests an intermediate dent for acceptingindiscerniblemovementsof shadows. First, let the spinningspherebe perfectlyblack. It will look stationarybut is spinning. Now considerthe silhouetteof the spinningsphere (which can be viewed by looking fromthe opposite side, intothe light). The silhouettealso looks stationary. But thereis still no reason to conclude it is not spinning. The silhouette is the layer of the sphere that blocks the light. Unlike the shadow of the sphere,the silhouetteis part of the sphere itself (Sorensen 1999). You can touch it. To change the silhouette,you have to change the object- and the silhouettewill carryalong thatmark,unlike a shadow. Just as the frontsurface of a black sphere is visible because of the light it fails to reflect,the silhouetteis visible by virtueof the lightit blocks. The silhouettehas mass.

SPINNING SHADOWS


361

a forcecan be impressed Consequently, uponit.This layerspins in thesame that the reflective surface of an way objectspins. Theshadow'sboundary doesnotincludethesilhouette.Since noneof the partsofthespherearepartsof theshadow,partsof thespherearenotso inrelatedtotheshadowas to silhouette.But thereis a causal connectimately tion.Partsofthecasterhavepartsoftheshadowsas effects. This encourages theinference thattheshadowinherits themotionofitscaster. Superpositionof shadows JohnLocke said thattwothingsof thesame kindcannotbe in same place at thesametime.Gottfried Leibnizrepliesthat This is a reasonable assumption; but experience itselfshows thatwe are not bound to it when it comes to distinguishingthings.For instance, we find that two shadows or two rays of light interpenetrate,and we could devise an imaginary world where bodies did the same. Yet we can still distinguishone ray fromthe otherjust by the direction of their paths, even when they intersect.(Leibniz 1981, II-xxvii-1)

to Locke's Contemporary physicistswouldsay waves are counterexamples When an wave with a one-meter meets a westprinciple. east-moving trough wave with one-meter thentheybriefly coincidewitha trough moving trough, of twometers.

Figure9 The twowavesthenresumetheirone-meter depths.The two wavesjust pass each other. through ofshadowmovement vindicates Leibniz'sclaim thatshadows Acceptance andthattheirpathscan distinguish them.Considertwo flashinterpenetrate lightseach withhalfof a coin gluedto the bottomsof theirlenses. Each castsa shadowof its coin half.The starting flashlight positionof theflashis Then lights straight theypivottowardseach othermakingtheirshadup. ows intersect.

Figure10

362

ROY SORENSEN


one shadowpartis movingto the rightwhile the At the intersection, conothershadowpartis movingto theleft.In otherwords,theintersection shadowparts,nota singleshadowpartsharedby two shadtainstwodistinct ows. Iftherewerea singlesharedpart,itcouldnotbe movingbothrightand to say it moves in one direction ratherthanthe left.It wouldbe arbitrary to therightnorto left,thenwhataboutthe other.Andifitis movingneither shadowpartsthat lie beyondthe intersection? They have a directionof cannotbe tracedthrough theimmobileintersecbutthemovement movement areavoidedby allowingshadowsto pass through tion.These interruptions eachother. of shadowsbut not theinterpeneto acceptinterpenetration Ourtendency illusion.Whentwo dots tration ofobjectsmayexplainthe"motion-bounce" we them as in an X see cross movingthrougheach other. pattern, silently we see thedotsas bouncnoise at the a collision But ifthereis intersection, is thatthenoise promptsa moduleof the ingoffeachother.My conjecture thanshadows. thedotsas objectsrather visualsystemto interpret

Figure11 we can arrangea case in which Once we resignourselvesto superposition, twoshadowstotallycoincidewitheachother.Gluea roundpennyto thecenter of each of two flashlights.Aim themdirectlyat each other.Their Thereare two shadows,not one sharedshadow. shadowsoverlapperfectly. Now thereare two spinning in oppositedirections. Rotatethe flashlights shadowsinexactlythesameplaceat thesametime. A completetheoryofmotion his G. E. Moore(1939) provedtheexistenceofmaterial thingsby presenting handsas specimens.BeneathMoore's handswereshadows.Those arespecimensofabsences.Thisis Moore'sshadowproofofabsences. to theshadowproof,he is committed Moore didnotpropound Although contains an the external world of to his it. The analyproof preface accepting in space but cannotbe met arepresented sis ofexternalthings.After-images inspace.Shadowscan be metin space. Moore(1939, 275) deniesthatshadows arematerial things. should covereverythingthatmoves,notjust themovementof Physics focusedon themovement ofshadowsbecauseshadows I have material things.

SPINNING SHADOWS


363

areaccessibleandeasy to manipulate. A completetheoryof motion,howto must on and else thatis outthere. ever, holes,cracks, everything go Acknowledgements This paperbeganits lifeas lecturesat theUniversity Bristol of Edinburgh, ofLeeds,University ofBritishColumbia,andBilkent University, University I think Walter andErdin£Sayan forespecially Sinnott-Armstrong University. comments. illuminating References

Casati, RobertoandVarzi,Achille(1994) Holes and OtherSuperficialities MIT Press). (Cambridge: Casati, Roberto(2003) The ShadowClub trans.AbigailAsher,New York: Knopf. Peter(1984) "Is MotionChangeofLocation"Analysis84: 177-178. Forrest, mechanics"in Mobius and his (1993) "Mobius's geometrical Gray,Jeremy bandOxford:OxfordUniversity Press:79-103. G. (1973) "Visualperception Johansson ofbiologicalmotionanda modelfor 14: 201-211. itsanalysis".Perceptionand Psychophysics Karmo,Toomas(1977) "Disturbances" Analysis37: 147-148. Kersten,D. J., Mamassian,P., & Knill, D. (1991). "Movingcast shadows and generateillusoryobject trajectories" InvestigativeOphthalmology Visual Science,32, 1179 Kersten,D., Mamassian,P. & Knill,D. Moving(1994) "Cast Shadowsand thePerception of RelativeDepth"Max-Planck-lnstitut fuer biologische TechnicalReportNo.6. ftp://ftp.mpik-tueb.mpg.de/pub/mpiKybernetik memos/TR-006.ps.Z trans.Peter Leibniz,Gottfried (1981) NewEssayson HumanUnderstanding, Remnant andJonathan Press. Bennett, Cambridge: Cambridge University Lewis, David andStephanieR. Lewis (1970) "Holes"AustralasianJournal 48: 206-212. ofPhilosophy Moore,G. E. (1939): "A Proofof theExternalWorld,"Proceedingsofthe BritishAcademy25: 273-300. Moore,G. E. (1962) The CommonplaceBook 1919-1953London:Allen & Unwin. Parsons,L. M. (1995). "Inabilityto Reason Aboutan Object's Orientation Using andAxis andAngleof Rotation",Journalof Experimental PsyHuman and 1259-1277. chology: 21(6): Perception Performance, MiltonA. (1960) "ThingsThatGo FasterThanLight"Scientific Rothman, American203/1(July):142-152. 2ndedition,London: Russell,Bertrand (1937) ThePrinciplesofMathematics Allen& Unwin. 364

ROY SORENSEN


Sayan,Erdin$(1996) "A MereologicalLook at Motion"PhilosophicalStudies 84(1): 75-89. SekulerR, SekulerA.B., R Lau (1997) "Soundaltersvisual motionperception"Nature385: 308. Slote, Michael(1979) "Causalityand the Conceptof a Thing'" Midwest 4: 387-400. Studiesin Philosophy Sorensen,Roy (1999) "SeeingIntersecting Eclipses"Journalof Philosophy 96: 25-49. Todes,SamuelandCharlesDaniels (1975) "BeyondtheDoubt of a Shadow: and LinguisticAnalysisof Shadows"in D. Hide A Phenomenological and ExistenandR. M. Zaner,eds, Selected Studiesin Phenomenology tialPhilosophy (The Hague:Nijhoff)86-93. Tooley,Michael(1988) "In Defenseof theExistenceof States of Motion" PhilosophicalTopics 16(1): 225-253. WallachH, O'Connell,D.N. (1953) "The kineticdeptheffect".Journalof 45: 205-217. Psychology Experimental Causation:The Dean Zimmerman, (1998) "TemporalPartsandSupervenient Australasian Journalof PhiofTwo HumeanDoctrines" Incompatibility losophy76: 265-88.

SPINNING SHADOWS


365