Holes and Other Superficialities (Bradford Books)

Acknowledgments

We started working on this book in the summer of 1989. In the spring of that year a paper by Roberto, no parts of which survive here, had been presented to the ResearchGroup on the Philosophy of Perception in Geneva, during whosemeetingsholes were sometimesmentioned as problematic perceptual objects. At that time, Achiile was struggling with truth value gaps and other harassingsemantic nonentitles, and was starting to develop a hyperrealistic attitude towards them. As we met in July at a lecture by Kit Fine on the logic of ordinary objects, where holes and other immaterial things were evoked as troublemakers, it somehow looked natural to join our efforts and face this neglectedside of reality once and for all. Holes proved to be a fascinating, intriguing, all -absorbing subject. It was great fun to write thesepages, but above all we enjoyed the countless discussionswith philosophers, professional or natural , originated by the question What is a hole? Our first thanks go to Kevin Mulligan and Barry Smith, who in many ways encouragedand supported us through various stagesof this work. Their commentson earlier versions of the manuscript saved us from many errors in both syntax and semantics. Many other people offered us help, remarks, and suggestionsthat proved very precious ; among them we would like to mention at least Paolo Bozzi, Valentino Braitenberg, JohannesBrandi, Erica Camajoli , Antony Cohn, Jer6me Dokic , Richard Glauser, Douglas Hofstadter, Marco Nani , Marco Panza, Jerzy Perzanowski, Martina Roepke, Daniel Schulthess, Jeoffrey Simmons, Peter Simons, Alex Simpson, Gianfranco Soldati, A vrum Stroll , Sara Tamanini , Tomek Taylor , Michael Tye, Mark Vorobej, Graham White , David Wiggins, and the participants in some talks we gave during the past couple of years(in Rome, Geneva, Schaan, Padua). To Betty and Harry Sta~ton we are particularly grateful for their warm encouragement during the final stage of our work , when we also benefited from the editorial help of Teri Mendelsohn and Paul Bethge. And we are thankful to David and Stephanie Lewis for having written their marvelous piece

Acknowledgments

" Holes." Achiile also wishesto acknowledgehis debt to the inspiring a e Tecnologica environmentof the Istituto per la Ricerca Scientific SwissNational the thank to wishes Roberto while Trento in , (IRST) with work his for Foundation(SNSF Science grants10-2436.88 ) supporting .91. and 11 31211 Marianna Basileand FriederikeOursin followed the developmentof . Theysawit grow, andweendedup writing this book with specialpassion their mostof it during holidays. To both of them, with our love, our truly wholehearted gratitude.

.

Introduction

We talk about holes, we count them, we describeand measurethem. We ' ' explain other people s or even animals beh~vior by attributing intentions and other attitudes whose content includes referenceto - or representations of- holes. Peopleand animals do certain things becausethey believe they have come across a hole, becausethey want to dig a hole, to pass through a hole, to jump over a hole, to get out of a hole, or to hide inside one. Holes are something about which we commonly reason, and- like other particulars such as tables, stones, drops of oil - they seem to be indispensable in accounting for certain causal interactions. A hole in a bucket lets the water run out; a tunnel in a mountain lets a train go through . It is the shapeof the hole in a nut that holds a bolt tight . Indeed there are objects that seemto be essentially holed: a doughnut, a piggy bank, a genuinechunk of Emmenthal cheese.A colander, a needle, a lock , or a flute would not be what they are- and could not be usedto do what they are meant for - had they no holes. Even the common-sensedistinction betweenin and out (or inside and outside, or within and without ) is to someextent linked to the notion of a hole. All this seemsto point at the acceptanceof holes- and other cognate entities, such as depressions, hollows, cavities, grooves, cracks, and fissures - into our basic onto logical inventory , alongside tables, stones, and drops of oil. Yet holes are more disturbing than theseother entities, more uncomfortable to live with . Perhapsonly a dry -minded philosopher would hazard questioning the reality of tables and stones. But just ask any person " " to tell you what holes are- real, everyday holes, not the abstract holes of geometry- and he will likely elaborate upon absences , nonentities, nothingnesses, things that are not there. Are there such things? Holes are slippery, elusive entities. They are spatio-temporally localized, like tables or liquids or explosions and unlike numbers or moral values. They seem, therefore, to qualify as material particulars. However, it is much more difficult to specifyidentity and identification criteria for holes

Chapter 1

than for rnaterial particulars. Holes are not just regions of space; holes can rnove, as happensanytirne you rnove a pieceof Ernrnenthalcheese , whereas regions of spacecannot. The identity of a hole does not depend on the ' " " identity of the stuff inside it (the hole s guest, so to speak), for you can ernpty the hole of any stuff that rnight partially or fully occupy it and yet leave the hole intact. It does not seernto depend on the identity of the hole's " host" (i.e., the object where the hole is localized) either, for you can ' irnagine changing the host s stuff or even the host itself without affecting the hole. Indeed, the fact that holes are not madeof anything seernsto be a rnajor hindrance to giving adequate identity and identification criteria for thern. And this puts thern under a cloud of philosophical suspicion. Successin rnaking senseof assertions of identity between entities of a certain sort is often considereda rninirnal prerequisite for the viability of a theory resting on the idea that there are entities of that sort. (As sorne like to say: No entity without identity .) Thesedifficulties- rnaybe at tirnes cornbined with sorneform of ho" or vacui- could lead one to favor onto logical parsirnony over naive realisrn and to clairn that onto logical cornrnitrnent of sentencesor beliefsreferring to or quantifying over holes is only seerning. Perhapswe could do without holes by talking of holed objects without losing the possibility of describing all the relevant features of the world we are used to describing by appealing directly to the concept of hole. Perhaps we could spell out this elirninative strategy fully by providing an adequateaccount of every holecornrnitting sentenceby rneans of sufficiently articulated hole-free paraphrases . " When I say there are holes in sornething, I rnean nothing rnore nor lessthan it is perforated. The synonymousshape-predicates' . . . is perforated' and ' there are holes in . . . ' - just like any other shape-predicate, ' ' say is a dodecahedron- rnay truly be predicated of pieces of cheese , without any irnplication that perforation is due to the presenceof occult, irnrnaterial entities." So says Argie, the norninalist-rnaterialist character featured by David and Stephanie Lewis in their classic dialogue " Holes" ( 1970). But even if this strategy could be coherently pursued, the question rernains: A paraphraseelirninatesan entity only if it is part of a true theory of the world . And even so, paraphrasability of sentencesabout holes does not per seelirninate holes frorn the world - just as the assertibility of such sentencesdoes not autornatically create holes. To talk about valleys is, in a sense, to talk about rnountains. But would we elirninate valleys by talk -

Introduction

ing only about mountains? Would we eliminate any commitment to valleys by exercisingparsimony in the languagewe use? Further , if holes do not exist, why are we inclined to believe they do? One could think of holes as entia representationis , as-if entities, projections of the mind - as entities whoseexistencewe take for granted because we blindly and erroneously follow directions that spontaneously suggest themselveswhen our cognitive system is exposed to what is in fact a thoroughly holelessworld . Thesebeliefsmight be the by-products of some malfunction in our cognitive system, but they are neverthelessevolution fully and to find our arily justified insofar as they enable us to act success way in the surrounding world . We shall try to resist these ways out of the hole problem and to take seriously what common sensehas to say about holes. As a matter of fact, we find it suspicious to accusethe common-senseview of the world with widespreadand massiveerror , as the irrealists would do. And if there is an ontology inherent in what we ordinarily think , believe, know , and assert about the world , then this ontology comprisesholes among other things. Besides, the only way one could ultimately evaluate the successof the irrealist 's eliminative strategy is by testing it against our preanalytical intuitions - intuitions that cannot, therefore, be left in the dark. One way or another, we need a common-sensetheory of holes- a theory of what holes are in the common-sensepicture of the world - before we can venture into dangerousdiscussionsabout onto logical commitment and other such matters. We must know what we would like to reduceor to eliminate (what the eliminans looks like ) in order even to be able to attempt the ' enterpriseof reduction. One way or another, a realist s attitude is required. ' And the realist s hope is that the acceptanceof his attitude will prove inevitable, that some statementsor patterns of reasoning about holes will turn out impossibleto paraphrase. In any event, our account could receive a purely conditional interpretation : If holes exist, they are the kind of thing we are about to describe. Other reasonscould be put forward to support this way of approaching the topic , not least its practical relevance. For instance, much recent literature in artificial intelligence has focused on so-called naive physics- the ' scienceof ordinary things, space, time and motions at the ordinary man s disposal in his everyday actions. This picture of physical reality includes entities (or laws) that are not allowed or contemplated by physics proper. Supposeyou instruct a machine- say, a mobile robot with an artificial

Chapter 1

" eye- to play billiards for you. Supposeyou tell it Send the ball into the " upper right comer pocket. How will it have to interpret your command? It will have to understand where you want the ball to be sent. This can be viewed as a fairly easy recognition task if the robot 's world - the billiard table- is defined as a sort of geometric map, for in that casethe robot ' s capability of recognizing a certain object or spatial region dependsessentially on the correct assignmentof geometric coordinates to that object or region. But when you are uttering your command you are not instructing the robot that way; you say " the upper right comer pocket," not " zone xyz in coordinate systemS" or something like that. You do not even mean that. Rather, you would like the robot to execute your command as a result of recognizing the right billiard pocket, with no need to resort to artificial grids and coordinate systems. You would like your robot to recognize the holes in the table and to behaveaccordingly. Now supposeyou take your robot to a golf courseand beg it " Sendthe ball into hole 1S." Will it understand this? Will it recognizethe holes in the ground as of the same family as billiard pockets? The interpretation of your commands should be so flexible as to make this possible, though differencesin material, size, relative position, and so on may be readily apparent. The principles of naive physics required for this and similar purposes- the theory depicting our beliefsabout space, time, objects, and macroscopic interactions between physical entities- need not coincide with the laws of physics proper. It must, rather, work . Its inquiry is necessary in order for us to interact efficiently with machinery of all sorts. And this inquiry must include balls, stones, drops of oil - and holes. Indeed, the point is not merely a pragmatic one. The relevanceof our investigation to naive physics is analogous to the relevanceof metaphysical investigation to physics proper. One can gain understanding of a sector of physics, and even make discoveries, once a certain physical entity is assignedto its proper category- even if the assignmentis provisional . What then are holes- as we said, real holes? As one expandsthe range of examples and case-studies, the question disclosesa very wide array of disparate philosophical problems. A way to give form to our enquiry is to study the way we perceiveholes and the complexity of beliefswe attribute to people or animals when they reason(or seemto reason) about holes. This involves a variety of perspectives - perspectivesnot only different from but sometimescomplementary

Introduction

to one another. To reason about holes seemsto involve reasoning about the shapeof an object, but also about its dispositionsto interact with other objects; about the way in which a hole is or can be generated, modified, used, destroyed; and, finally, about the way in which it is or can be perceived , identified, re-identified. We shall track all of theseclues. A certain complex pattern - a theory of holes rather than a definition of what a hole is- will gradually emerge. Take shapesfirst. If one thinks of a hole, one often thinks of the shape or form of something. Sometimesone thinks of the form of the host (modifications in the form of which may affect the hole); more frequently, one thinks of the form of the guest: one describesthe form of a hole by giving the form of its potential " filler " (the idealized piece of stuff that could be usedto fill up the hole perfectly). At the sametime, there is a clear sensein which we can say that holes themselveshave forms- forms that we recognize , measure, compare, and change. Consider, second, how important dispositional conceptsare to our investigation . To possessthe concept of a hole is to possessan intrinsically dispositional concept. This glassis empty, but it could be full . Holes link their host to the environment counterfactually: they give rise to a seriesof relational ties betweenthe object and what possiblysurrounds it . Perhaps one would not say that holes have dispositional properties. But surely if one can think of a hole one must equally be able to think that the hole mayor may not be filled , that things would move in different ways just becausethe hole is or is not present, that a thina would be uselesswere it not for the hole in it , or even that a thing would not be the thing it is were it not for its hole. In each of these counterfactual speculations, causality plays a major role, for the explanation of these dispositional features is typically a causal explanation. Third , take the suggestion that reasoning about holes is reasoning about the way holes are or can be created, operated upon, and destroyed. People plan series of actions whose objects are holes or involve holes; holes are often the results of processes of which we are the primary (though not necessarilyvoluntary ) actors. To dig, to drill , to burrow , to punch, to enlarge, to fill up, to fall in , to jump over, to look through, to hide in - all of these, and indeed many others, are things we do with , around, inside, and through holes. And theseoperations can be stylized so as to yield ideal, abstract operations that can provide us with powerful taxonomic tools. Thus, we can imagine that a fracture in a cube can spread

Chapter 1

and split the cube into a cylinder surrounded by its complement; the cylinder that fills its complement can be removed,and a string can be threaded through the resulting hole and used to handlethe object. And so on. Lastly, in the presenceof certain visual patterns, people have the impression of perceiving a hole, and they react accordingly (evenif there is in fact no hole). How doesthis impressionarise? Which patterns are naturally taken as indicating the presenceof a hole in our environment? Again, in somecontexts people are led to think that the hole they seeor think of is the sameas the one they saw or thought of on other occasions. And they are ready to give reasonsfor that: the hole is exactly of the same size, in the same spot, or in the same object. Such reasonscannot be ignored in any attempt to provide an adequate metaphysics(and adequate identity criteria) for holes.

We thereforehavea networkof interrelatedabilitiesthat call forsystematic descriptionand explanation. Our aim is to catch the shapeof this network, to substantiatethe explanation , in order to accountfor the fact that weoftenthink in termsof holes, and that wecorrectlydo so. In the next two chapterswe shall addressthe basicissues , considering variouspossibleapproach esto the question" What is a hole?" We shall, amongother things, arguethat to say that thereare holesis not a mere fafon deparler. Wewill seethat holesarenot propertiesbut particulars(of a specialtype; we call them superficialparticulars ). They are dependent particulars: they cannotexist alone; they needa host at the surfaceof which they find a placeto be. They are not abstractionsbut individuals, . They are not partsof althoughthey are not madeof anythingbut space the materialobjectstheyare hostedin (thoughit is sometimesby removing a part of the host that a hole is created ); rather, they are immaterial bodies , locatedat the surfacesof their hostsand providedwith somevery primitiveproperties(includingdispositionalones). We shalldistinguishthreebasickindsof holes: superficialhollows, perforating tunnels, and internal cavities. Each of these(along with mixed casesderivingfrom interbreeding ) hassomepeculiartopologicalproperties that do not, however , preventa uniformtreatment.To be a holeis to bea hollow, a tunnel, or a cavity. We shallsometimes rely upontopological concepts . But we shallalsoseethat topologyaloneis not sufficientto the complexityof the conceptof a hole. We needa broader encompass interpretivecontext, which we term a morphologyand whoseunity de-

Introduction

pends on a basic operation: the operation of filling . To that purpose, the way a hole can be filled will receive the outmost attention. Part of our reasoning about holes depends upon our ability to reason about their fillers and not only about their hosts- a form of complementary or dual reasoning that can be very illuminating . We shall also seethat the mor of structure the topological phological complexity of a hole is mirrored in ' its " skin" (the part of the surfaceof the hole s host that is in contact with the hole' s filler ). On this basis, we shall investigatesomemajor properties of the resulting conceptual framework. We shall seethat the .membersof the classof holes are subject, albeit in a constrained way, to a mereological structure. They can move, fuse into one another other, split . They can be causally responsible for , or causally subject to , severaldifferent phenomena. They can be born, develop, die. We shall also seethat some morphological featuresof holes are representedin perception; we shall examine such features (e.g., the link holes have to certain texture and shadow patterns), and we shall study the perceptual conditions in which we have the impression that we see, feel, or even hear a hole. A philosophical theory originates from astonishment and is judged from its ability to silence astonishment. A theory of holes is no exception. A ' " hole is there where something isn t. There is a lot to be astonishedabout when it comesto holes, and there is no shortage of puzzlesto begin with . Perhaps in the end there will be more. But what sort of things holes are, what neglectedspeciesof the onto logical fauna- that is what is our business to find out.

2

SuperficialPartieulars

A spot in the wall. Let us start with some facts from daily life. Suppose you wake up one morning and look at the white wall in front of you. It is the usual wall you seeevery morning, of course. But this time, right there in the top left comer, something new catchesyour eye: you seethat a little hole is now there that was not there yesterday. How do you describewhat you see? A spot in the wall, darker than the rest, filled with shadow, that goes deep inside (though you cannot really tell how deep). It looks unitary and complete, compact, though lessdense than the wall. A thing, perhaps, but a bit mysterious. It is not made of the shadow you see. It is not evenmade of the sorts ofstufl' ordinary things are ordinarily made of: not of the air that is inside it , nor of the plaster and bits of paint that have fallen on the floor over night . In fact, if it is something, it does not seemto be made of anything. You compare it with the bookshelf on the wall. It is also unitary , complete , compact, and different in density from the wall. Theseproperties are exemplified at the region of spacewhere the shelf is localized (and which the shelf occupies). But in the caseof the hole, no properties seemto do that job . It is uncertain whether the hole really occupiesthe place where it is localized. In fact, it seemsthat there is a hole therejust insofaras nothing occupiesthat place (or insofar as something elsecould occupy that place; as a matter of fact, you are now planning to fill that hole with new synthetic filling ). If there is a hole there, it is becausethe wall has a certain shape there (and surely if the wall were not there the hole would not be there either). Unlike the shelf, the hole depends for its existenceon the existenceof a wall, and of that particular wall. The borders of the hole mark an interruption in the surfaceof the wall. A discontinuity breaks in, and you can imagine how to restore the natural continuity of the surface so as to make the hole disappear. Thus, it is by perceiving a discontinuity in the surfaceof the wall that you seemto perceivethe hole. Surfacesare the major character here.

Chapter2

. Consider for Surfacespose"ario88 pbll~ phkal problems of their OWD instance the puzzle raised by Leonardo in a memorable page of his Notebooks: " What is it . . . that divides the atmosphere from the water? It is necessarythat there should be a common boundary which is neither air nor water but is without substance, becausea body interposed between two bodies prevents their contact, and this does not happen in water with ' air." (pp. 75- 76 of MacCurdy s edition ) How can two things- the water and the atmosphere, the ocean and the sky- be in contact and yet be " " separated? What sort of thing is that common boundary that belongsto neither of the two contiguous media it separates? Indeed it is unclear whether there are one or two surfacesat the junction betweentwo distinct things. How many surfacesdoes a fish break when it jumps out of the ocean? Does it dive into the air in the sameway in which we dive into the ocean? Or consider the problem of the origin of surfacesin a continuous manifold ' , as described in Bolzano s Paradoxes of the Infinite . If we cut some chunk of continuous matter into two halves, it would seemthat one of the two pieceswill be open (i.e., will have no point that could properly qualify as the last point of that half before the surfaceof the other half ). But which half would be so open? An analogous problem is the famous Peirceanpuzzle of the color of the line dividing a black spot from its white background on a continuous surface(figure 2. 1). We do not have any reasonto say that the line is either black or white, and we could chooseamong somealternatives. It could be meaninglessto attribute any color to the line. It could be that the line is both black and white, so that there would be true contradictions in nature.

.

1. 1 F~ ' Peirces puzzle. Which color is the line of demarcation between a black spot and a white background: black, or white?

Superficial Particulars

It could also be that the line is neither black nor white, thus allowing for indeterminaciesin nature. Or we might as well agreethat the line is white 50% of the times you make a black spot and black the other times. Again: the surfaceor surfacesof an object do not to seemto have a back and a front. Yet something could be inside or outside the spacedefined by a surface. Thus, it is not so unreasonableafter all to distinguish between two sidesof a surface, or to attribute to a surface an internal complexity that mirrors this twosidedness. A pre- analytical notion of a surface. The above problems- and perhaps many others- could seriously affect any philosophical theory resting on the notion of a surface. Nevertheless, we shall try to remain independent of these difficulties. The point is simply that we seem to perceive holes when we perceive some kind of discontinuity in the surfacesof material objects- that to think of a hole is to think of some superficial part of the object. Hence, to get a grasp of what a hole is, we need some account of the intuitive , preanalytical notion of a surface upon which we rely when we describeholes. Surfacesare for us among the boundaries (or limits ) of material objects, at least in an intuitive sense(but not too intuitive , perhaps: it could be useful for us to say that liquids also have surfaces, and gasestoo ). The surfaceis the first part of a material object to come into contact with the ' object s environment. It delineatesthe form of the object by enveloping it , as it were (though a proper envelope would not be a part of the object). The surfacedefinesthe inside and the outside of the object. Although one could intuitively regard the surface of a material object as the outermost layer of the stuff of which the object is made, it is not possible to remove it from the object in the sameway in which we can remove a parcel of the ' object s stuff. And , last but not least, the surfaceis the only part we can see of an opaque object. In his book Surfacesand related works, A vrum Stroll has argued that there are two main contrasting conceptionsof surfaces, which are irreducibly different: surfacesconceived of as some sort of abstraction, with no substanceor divisible bulk (as Leonardo seemsto have suggested ), and surfacesconceivedof as somesort of physical entities, with divisible physical bulk (the " topmost layer of atoms" studied by certain branches of contemporary physics). Our view is somewhatneutral with respectto such conceptions, or perhaps derives from a wavering interplay betweenthem:

Chapter 2

it shareswith the former a geometrical (rather than physical) perspective, but unlike the former it regards surfacesas parts of the objects of which they are boundaries. From this point of view, we shall talk of surfacesin the very intuitive way in which somebodywould talk of them were he to look at a material object or at his own skin and just describewhat he primarily seemsto see, or , in general, what he primarily gets information about, or what he pri marily acts on (one cannot touch or scratch the inside of a cube). In this sense, therefore, we simply take surfacesto be the outermostparts of ob, those parts we seeor touch. In other words, the surfaceof an object j ,ects x is the part of x that overlaps (i.e., is partly sharedby) all those parts of x that are in contact with the geometrical complement of x - where the geometrical complement of an object x is simply defined as the entity wholly occupying the region of spacethat is not occupied by x. Formalities. Our definition can also be extended to cover the caseof a (non-scattered) object having two or more superficially disconnectedsurfaces , as in the caseof a spherewith one or more internal cavities. In that casewe cannot speakof the surface, and speakingof the outermostparts of objects might be misleading(it could, for instance, be more appropriate to describecertain internal " surfaces" as innermostparts). Yet the basic characterization would still do. Let a part z of an objecty qualify as maximally connectedif and only if it is self-connected (i.e., in one piece) and closed under the property of comprising all the parts of y that are in contact with it (i.e., every part of y that is in contact with some part of z is also part of z). Then a surfaceof an object x can be defined as a part of x that overlaps all those parts of x that are in contact with some maximally connected part of the geometrical complement of x. As a result, the number of surfacesis defined by the number of maximally connected parts of the complement. Relative to this definition , we can then define a superficial part of an object x simply as a part of a surface of x. An equivalent, more direct definition would be the following : z is a superficial part of x iff (i ) z is a part of x and (ii ) for every part y of x , z is a part of y only if y is in contact with the geometrical complement of x. A surface of an object x is thus one of those maximal entities whose parts are superficial parts of the object, i.e., a maximally connectedsuperficial part of x.

Particulars Superficial

Holes as superficial particulars. The notion of a superficial part is important for us. Holes and other discontinuities, such as borders and edges, are located at some superficial parts of objects. Even if a hole is completely hidden inside an object (think of an internal hole in a wheel of Emmenthal chees :e), where there is the hole there is a superficial part of the object (an internal surface, to wit ); and wheneveryou think of a hole, you think of a surface. This is therefore our starting point . We summarize it by saying that holes are superficial entities; holes go hand in hand with surfaces. There are various other examples of superficial entities. A comer, a bump, a ridge, a crack, a groove, and a dent are also superficial- each is located at some superficial part of an object. What is common to these examplesis that they introduce discontinuities into an otherwise macroscopically continuous world. In fact, thesediscontinuities do not mark any qualitative discontinuity . They are in this respect different from surfaces, which as a rule limit or enclose a qualitatively homogeneousregion of space, thereby separatingit from other regions possibly of a different quality . Comers and edgesare separators in a weaker sense. They float in a region that is qualitatively homogeneous: they are in marble or ice, whereas surfacesare of marble or ice and mark the limits of an object made of thesestuffs. Now , though a surface is a boundary and marks a discontinuity, our elementary notion of a surfaceis the notion of something fairly even and continuous. One assumesthat one can move on a surfacewithout coming acrossany obstaclesor abrupt changesin curvature. If such an encounter occurs, what one comes across is a superficial discontinuity, normally classified according to some ( possibly vague) elementary patterns. For instance, grooves and ridges are dual (figure 2.2): a ridge introduces a protrusion intn the surface, while a groove definesan intrusion - and the

~

-

F1iweU Theapparentdualitybetweena ridge(left) anda groove(right).

. .

Chapter2

to one another. A related characterization could be given with the help of the notion of " a disturbance" introduced by Toomas Karmo ( 1977). A disturbance may " be thought of as an object or entity found in someother object, not in the sensein which a letter may be found in an envelope, or a biscuit in a tin , but in the sensein 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 (p. 147). The idea of regarding holes as disturbances is indeed appealing, and comesclose to the notion of a discontinuity that we have in mind. However ' , Karmo s suggestionis that a way of telling whether x is a disturbance " in " an " objecty is to enquire whether x can migrate throughy . A knot is a disturbance becauseit may slip along the rope in which it is tied, and a hole is a disturbance becauseit can be pictured as moving around the " flowerbed in which it was dug (as a vortex may move in a pool). (ibid ) Surely this criterion is too weak for us. For instance, it would have the ' consequencethat even a moving red patch on a cube s side would qualify as a disturbance. Holes are disturbancesof a more substantial, geometrically relevant sort. Surfaces , potential surfaces, and parts. We would like to talk of these disturbing entities as of individuals - as opposed to , say, properties or other abstractions. There might be some awkwardnessin considering an individual 's (proper) parts as individuals in their own right , for this would imply that every individual carries along with it an infinitv of other individuals . Some philosophers tend to consider the parts of an individual as merely potential entities- things that are not proper individuals but which could so qualify under different favorable circumstances. This applies particularly to parts that have little internal or functional connected" " nessand which seemto be obtained only by arbitrary conceptual cuts (say, the top left portion of a glass, as opposedto its stem). Presumably the samewould also hold for those parts of an object whosesurfacescomprise superficial parts of the object. However, in an important sensetheselatter parts (such as a in figure 2. 3) are more privileged and have more claims to individuality than those parts that are purely internal (such as b), for they shareat least a part of the object's surface, and that surfaceis actual. The easiestway to conceive of a potential part is to provide it with a potential surface (or surfaces). Were the surface actual, the part would

Superficial Particulars

Fil . -eU A privilegedpart (a) anda purelyinternalpart (b).

becomean object itself and the host object would split . In casethe surface is only half actual and half potential (or in casewe take a part with two or more surfaces, someactual and somepotential), the part is potential but is already on the way to acquiring a life of its own. To be an individual is, amo~g other things, to be distinguished from the environment. And in the case at issue it is as if the processof individuation were only half completed - which means that it has at least started. Of course, if we were to . complete the processby providing our part with a complete actual surface (e.g., by cutting it away from the rest of the object) it would not be a part any longer: it would be an independent object. Parts, and superficial particulars , cannot be taken out of objects without becoming full -fledged objects themselves. One of the arguments that may be offered to support the view that the parts of an object are proper individuals as opposed to , say, properties - is that they can come into existence, passaway, and be counted, localized, and measuredin a way in which properties cannot (unlessone allows for individual properties). Extended parts and superficial particulars have spatio-temporal histories, and they come into being and perish as the results of various operations. Whether we take them as actual or potential, superficial entities are spatio- temporally determined in a word, they are particulars. Discontinuities and disturbancesean move or change in size. A striking example is provided by waves. This indicates that superficial particulars are " parasitic" on their hosts and at the sametime have lives of their own. One can say that both wavesand whalesare in the sea, but the relation of " " being in is different in the two cases: you can take a whale, but not a wave, out of the sea. Nevertheless, the seastays the sameafter the passage

Chapter2

of a wave, just as it does after the passageof a whale. The relation an object bearsto a discontinuity is from somepoints of view the sameas the relation it could bear to another object: it can host it , be affectedby it , or recover from an interaction with it . Here there is an interesting analogy betweencomers and wavesas disturbances. If a cube undergoesa continuous, imperceptible replacementof matter, with no changein its form , its comers can be conceivedof as stiff or frozen waves. Also, if a wave gets gradually slower as the liquid in which it propagatesbecomesmore and more viscous(think of a stream of lava), at the end of the processthe wave will be a comer. Vice versa, a comer in a melting object can turn into a wave (think of a large pieceof a wax on an inclined plane, and imagine heating it up very rapidly). Holes are alwaysin (or through) something. This is in part what we mean when we say that holes are parasitic on their hosts. But what do we mean exactly? And exactly what kind of things are they in (or through)? Take a plasticine brick. Make a hole. Is the hole in the brick (the object), or in the plasticine (the matter the brick is made of )? There seemto be points speakingin favor of both views. A simple argument to the effect that the hole is in the brick is the following : If you change the matter gradually (e.g., by replacing small fragments of plasticine with matching piecesof cement), you can say that the hole survivesbecauseof the survival of the brick - hence, the hole is in the brick , not in the matter. Another argument: Supposeyou lend some plasticine to a friend, who then models it in the shape of a statue. Your friend might well get upset at somebody who made a hole in it (thus ruining the statue), whereasyou would probably not care. Now consider the alternative view: that the hole is in the plasticine (i.e., the matter). What reasonscan be given to support this view? Well, suppose you fill up the hole with cement and then melt the brick ; then the brick is no longer there, but it wouldn ' t be so unwarranted to maintain that the survival of the cement makes the hole survive. In that case the hole would be in the plasticine (the matter), not in the brick (the object). Alternatively , consider again the statue modeled by your friend- say, a nice copy of Michelangelo' s David. Surely it sounds reasonableto regard the nostrils as holesnot in the statue but in the matter. (On the other hand, if the statue turned into a real person, in what would his nostrils be holes?)

Superficial Particulars

It seemsthat the attempt to accommodatesuch competing intuitions is doomed to fail. The question " What are holes in (or through)?" is undoubtedly a crucial question for any theory of holes. Yet it does not seem to admit of a single, unambiguous answer, and a raw battle of intuitions can hardly prove decisive. Relative to the simple example we started with , we would like to say that the hole is in the brick in virtue of its being in the matter- though this is in need of further specification. There is a rather different sensein which we can say, for instance, that we are in Italy in virtue of our being in Venice. As an alternative, we could simply say that holes are in the object, not in the matter; but we would also like not to ' overlook the importance of the object s matter. For the time being we shall therefore content ourselveswith a preliminary answer. We shall say that one always makesa hole in (or through) an ' object, but that one does so by acting upon the object s matter e.g., by even it or by expanding removing a portion of it , or by compressing , perhaps it or adding some more matter. (We shall come back to all these " " holemaking techniques in chapter 10.) Of course, not every hole is an" " artificial hole that someagent has made. The relation whereby a natural hole is in an object is somewhat hidden under the surface, and it will be our aim to make it emerge. When the hole takes over. A related problem is the following : Can there be an object that is completely holed? That is, can there be a hole in an object that occupiesall of the object? The answerseemsto be No : One can only make a hole in a part of the object, by acting on a portion of its matter. In fact, we shall maintain that every hole host has a proper part that entirely hosts the hole. Here too , however, some cautionary remarks are in order. First , one can easily think of more complicated caseslying near the borderline. Consider, for instance, taking a cubic block made of homogeneous matter and perforating it with three squareparallelepipedalholes of ' width equal to 1/ 3 of the block s side. If each hole is co-centeredwith the block and orthogonal to its sides, this will leave you with a block perforated by a hole shapedlike a triadic cross, with a cubic spike strung along each of six directions (figure 2.4, left). Now repeat the processand cut out 20 more triadic -cross holes shapedas before but reduced in the ratio 1/ 3. If you keep repeating this processad infinitum, you will be left with what has come to be known as a Sierpinski- Menger sponge: the volume of the

Chapter2

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cubic block vanishes, while the surfaceof the hole lining is infinite. In this case, it is perhapsnot so improper to speakof the block as of a completely holed object: the hole has- so to speak- taken over. Moreover, the question whether there can be completely holed objects must be kept separatefrom the question whether there is a maximumhole that can be hosted by a given object. One could be tempted to answer in the positive, though the exact answer would likely depend on the type of hole. For instance, one could suggestthat an object could host one and ' only one maximum internal cavity, leaving only the object s surface. (In the case of a perforating tunnel there would be an infinite number of maximum candidates, one for each way of enlarging an initial discontinuity .) Such a suggestion, however, is not compatible with our notion of a surface. For us, surfacesare parts of objects, but they are not bulky . Hence, one cannot be left with just a surface. A surfaceexists only if some arbitrary thicknessof internal matter exists. The ontological dependenceof holes. The fact that holes are always in something- and can be created only by action upon something- makes them onto logically dependententities: a hole cannot be removed from its host. As we already mentioned, this is not to deny that holes have lives of their own. They do; they have birthdays and careers(they have a curriculum vitae), and they can changeshape, size, depth, and width. But a hole cannot do such things- and such things cannot happen to it - without the support of some object " hosting" it . The foundation of a hole' s being lies in its host- or , better, in the surface of its host: a hole needs some " " object at the surfaceof which to grow and find a place to be.

Particulars Superficial

This type of onto logical dependence- which we label superficial dependence , to stress the fact that holes go hand in hand with surfaces- is a matter of de re necessity. It is to be distinguished from the relation of dependence that is implicitly expressedby a sentencesuch as " Every mother has a child." The latter can be classifiedas a caseof notional (or de dicto) ' . It is true, on the usual understanding of the words 'mother dependence ' ' and child , that there cannot exist a mother without a child ; but it is not true of any mother that she- that woman- could not exist had sheborne no child. By contrast, when we say that every hole has a host we mean exactly that: no hole can exist alone, without the object in which it is a hole. One should also take care, however, in distinguishing this sort of dependence from other types of de re onto logical dependence , which can even be stronger. Consider, for instance, the dependenceof a smile on its face(with the possible exception of the grin on the Cheshire Cat). This relation is onto logically coercive, " for not only can no smile exist unless some face exists, but every smile is rigidly dependent on its face; it could not have " existed except as a configuration of just the face it is in (Peter Simons, Parts, p. 300). By contrast, a hole is not so related to the host: it needsa host, but any host is just fine, as it were. At the beginning of this chapter we describeda hole in a wall as dependingfor its existenceon that particular wall. In fact this is what we mean by de re onto logical dependence , . As will gradually become but without the strength of a rigid dependence ' clearer, one can generally changethe host s matter, its shape, or even the entire host without affecting the hole. Concavities. Let us go back to our hole in the wall. We argued that it is ' the presenceof a discontinuity in the wall s surfacethat makesyou seethe hole- the dark , shadowy spot that goesdeepinside. It is the discontinuity that marks the hole and gives it the individual integrity that caught your attention - unlike other superficial parts of the wall that you never noticed and perhapsnever will . And it is this particular type of discontinuity that makes this a hole as opposed to , say, a bump or a protuberance: the dark spot is a hole becausethe discontinuity involves a concavity. We have said that the first basic feature of holes is that they are superficial particulars, that is, superficially dependent particulars. The second basic feature is that wheneverthere is a hole there is a superficial discontinuity ' that involves a concavity: you may fall into a hole, but you can t

Chapter2

, you can bumpinto one; you can hi ,deinside a hole, not behind a or a . into and hide behind comer , , bump protrusion) In trying to spell out our intuitions here, we are taking a common-sense ; i.e., we seethem from the outside. We perspectiveon material substances ~~ " generally do not think of ourselvesas burled inside a material object such as a table or a stone, observing its surfacefrom the inside. We typically stay outside. That is, we can be inside an object, as when we enter a ' telephone booth, but we cannot be inside the object s stuff (with some exceptions: for instance, we can dive into a swimming pool full of water). This general point is not as trivial as it might seem. There is indeed an important interplay betweenthe implications of our earlier remark that a hole is perceivedwhere the continuity of the surfacecan ideally be restored and the fact that the concept of a hole involves the concept of a concavity. Why, for instance, are we not inclined to take figure 2. 5 as a caseof a hole in a sphere? The answerseemsto be that although there are both a discontinuity and the possibility of eliminating it by ideally reconstructing a sphere, there is no concavity here. That is, there is no hole to hide in. On the other hand, imagine that the sphere is much bigger- the size of the Earth. In that case, someonecould think it appropriate to speakof a hole. In fact, imagine cutting the Earth as in figure 2. 5 and pouring water on the flat plane. The center of the plane is closer to the center of the Earth than are the plane's borders; thus, evenif the section is flat , the borders are ~~up" relative to the center, which is " down," and gravity will organize water so as to produce a sort of liquid dome whose surface will be the natural continuation of the Earth' s surface. There is no hole here, but inhabitants of the planet, coming to the shore and perceiving the sea as flat , would

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Superficial Particulars

regardit as hostedby a sort of big hole. Their impression is deceiving, will correcttheir naivemistake. though; geographers Conatores and supenleial gestalts. The extent to which concavity is a crucial concept for the purpose of telling whether or not a hole is there (and for that of distinguishing holesfrom other types of superficial particulars , such as comers or ridges) will becomemore apparent as we look at the many surprising ways in which patterns of concaveand convex superficial parts are organized in what may be called superfICialgestalts. In fact, it is not the presenceof a curvature in itself but the presenceof a certain pattern of curvature that matters. An object x is convex if the straight line segmentconnecting any two points of x doesnot intersect the surfaceof x. More generally, this criterion holds if we take x to be any part of a given object: If x satisfies the above condition, then it is convex; otherwise, x has at least one concavity. Now , holes require concavities; i.e., the existenceof a hole entails that a concavity be present. But the converse does not hold; i.e., from the presenceof a concavity one should not always infer the existenceof a hole. This is the case for the object depicted in figure 2.6- a sphere with a protruding horn. This object is not convex (although it has many convex parts), for the straight line segmentconnecting points a and b intersectsthe surface. Yet again we would not say that the object has holes in it . A pattern of curvature is the way in which some connected superficial parts are arranged in space. Under a different description, it is a distribution of curvature over a region of space. This pattern is a curvaturegestalt.

Fiaure2.6 A spherewith a horn is not convexbut hasno holes.

Chapter2

It is remarkable that certain singularities on curved surfacesassumean important role in the building of these gestalts. Singularities are, for instance , points of a curvature opposite in sign to the curvature of all surrounding points, or lines constituted by points of this kind . Some such singularities can be usedto separate, in a natural way, two or more superficial parts (figure 2.7). However, sometimesthere is no such separability, as is the casein the presenceof a punctual singularity that does not separate anything from anything. U nsurprisingly, singularities act as conceptual attractors or conceptual organizers. Though a superficial part could be almost completely convex, the presencein it ofa concavesingularity prompts us to consider the whole part as a concavity. We do organize surfacesand their properties by starting from singularities, and we sometimesdo not seemto care about other

Figure1.7 A concavityline separatingtwo convexsuperficialparts(left) and a concavitypoint sur. roundedby a convexsuperficialpart (right).

Figure2.8 An internalcavitywith a convexsingularityS(left), anda heart-shapedcavitywith opposite " ' singularitiesS andS (right).

Superficial Particulars

non-singular properties or parts. This is not to deny that the superficial parts of an object may have theselatter properties, or that we can organize our segmentation(both conceptually and perceptually) by starting from these. Simply, we would say that in the first (schematic) account of how the ' object s surfaceis presentedto us we can overlook the actual overall curvature and focus on local singularities. Thus, for example, the pattern representedon the right in figure 2. 7 would mark a concavity, although every superficial part of the object that does not include the singular concavity point is convex. The purpose of this discussionis to stressthat we often usethe presence . of a concave singularity as a criterion for saying that there is a concavity somewhere. Sometimes, however, what really matters is the overall concavity of the surfaceor of somesuperficial parts. Consider figure 2. 8. In the object depicted on the left, even if there is a convex singularity S, the overall concavity of the surfacemakes it possible to speak of a cavity. All that seemsto matter is what we can do with thesegestalts. And this has a geometrical foundation , but it cannot simply be reduced to geometrical facts. That something is a concavity or a convexity dependscrucially on, but is not reducible to , the presenceof concave or convex points and singularities on its surface. Likewise, in the object depicted on the right in ' figure 2.8, the presenceof two singularities of opposite sign (S convex and " S concave) does not affect the overall concavity of the pattern, which makesit possible to speak of a cavity. Surfacesand their geometrical properties matter a lot . But we have not even begun to entrap holes. In the next chapter we will start to approach issuesin a more direct way by discussing some theories concerning the nature of holes.

( ImmaterialBodies

" " Ludovieian holes. In their 1970 paper Holes, David and Stephanie Lewis distinguished clearly- and tried to solve- the major difficulties surrounding a nominalist cummaterialist account of holes. Their reasoning goesas follows: We easily talk of, refer to , or quantify over holes; however, holesdo not seemto be made of matter, and we seemunable satisfactorily to paraphrasetalk about holesinto talk about perforated material objects. The suggestion, then, is that holesare material things. They are not, as one " " might think , the fillers (the airy plugs that fill up every hole); rather, they ' are the hole-linings (figure 3.1). " The matter isn t inside the hole. It would be absurd to say it was: nobody wants to say that holes are inside themselves . The matter surrounds the hole. The lining of a hole, you agree, is a material object. For every hole there is a hole-lining; for every hole-lining " there is a hole. I say the hole-lining is the hole. (p. 5 i.nthe 1983reprint) Several bizarre consequencesof this account are discussed by the ' Lewises. Here are some of them, together with the Lewises own replies. ( 1) We accept that holes cannot be hole-fillers, for they cannot be inside themselves. Now , if holes are hole-linings, and if hole-linings surround ' ' holes, then holes surround themselves. Reply: Surrounds has two different meanings: when said of ordinary things, it has its ordinary meaning; ' ' when said of a hole, it means is identical with . " , but hole-linings can. Reply: We do (2) Holes cannot be made of cheese say that cavesare holes in the ground and that someof them are made out " of limestone. (3) If you were to take a paper-towel roller , spin it clockwise, put atoilet paper roller inside it , and spin it counterclockwise, then something would spin in one way when a part of the samething spins in the opposite way. Reply: The little hole inside the big hole is not a part of it . -lining . (4) The volume of a hole could be less than the volume of a hole ' . You have ' ' ' ' ' ' ' " read bottle also Reply: For hole read bottle ; for hole lining

Chapter3

Fia8e 3. 1 A hole(left); oneof its hole-linings(right ). i.e.. a hole tout court according to the Lewises: a materialbody.

the sameparadox. . . . Contextual cluespermit us to keep track of which we mean." ' (5) The hole s volume is arbitrary , depending on which hole-lining we choose. Reply: " What we call a single hole is really many hole-linings." " (6) How can a single hole be identical with many hole-linings that are not identical with one another?" Reply: a and b are the samehole " when " they have a common part that is itself a hole. (7) Then a hole is really two , at least: the left half and the right half of a paper-towel roller are not the samehole, as they have no part in common. If you were to cut the two halvesapart, you would say that they are holes, since they are shaped like holes. Thus there are two holes here, and not one. Reply: These are parts of a hole, but they are not holes. " I can't say ' ' why they aren t. 1 know which things are holes, but 1 can t give you a " definition. Dangerousambiguities. We have indulged in reporting theseobjections and replies becausethey are indicative of the difficulties involved in the proposed account. The Ludovician identification of holes with holelinings does provide an appealing way of treating holes as (parts of ) material objects, yet it gives rise to numerous puzzleswhose deged solutions seemin somecasesto be hardly tenable. Note in particular that the given identification implies a radical change in the meaning of certain predicates. For instance, ' surrounds' becomes'is identical with ' (or , better, it becomesa reflexive relation; as Jerzy Perzanowski pointed out to us, the Lewises' account here is stronger than it need be). This and similar changesare required; insofar as a hole-lining

Bodies Immaterial

Fipre3 .1 ' ' ' ' If hnle.cArehole-lininP :-", insideand outsidecan be ambiguous.

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Fiaare3. 3 A fIShyhole.

surrounds a hole, if the hole is a hole-lining it will surround itself. A revisionary proposal is not in itself unacceptable. Yet some predicates seemto require specialcare- their translations becomevery problematic. If holes are hole-linings, ' inside' turns out to be ambiguous when its second ' ' argument is a hole. It means both what inside ordinarily means ' ' (the internal part of the hole-lining, which is material) and also outside (the empty part that, on the common-senseunderstanding, correspondsto the hole). But there are two outsides here. In figure 3.2, let h be what we would ordinarily take as a hole, let I be one of its hole-linings, and let 0 be whatever is not inside I (in the previous sense) and does not overlap h. In ' ' the revisionary account, ' inside means' outside . But now, does ' outside r " "' " refer to 0 or to h? Look , one could say, outside' means the smaller outside. h is smaller than o. Therefore 'outside' meansh." Surely this is not an acceptableanswer. Or , imagine cutting the profile of a fish out of a sheet of paper, as in figure 3.3. Look at the sheet. If somebody asks you what profile you see, you will naturally speakof a fish outline, though that is actually the shape of the hole. Now , if somebody asks you to put something outside the hole' s profile, where will you put it? We believe you will not put it inside

Chapter3

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the hole (in A). Yet this is preciselywhat would be required by the identification of holes with hole-linings. A much more problematic predicate is ' to enlarge'. To enlarge a hole is not necessarilyto enlargeits hole-linings; sometimesit is exactly the opposite ! In figure 3.4, for instance, the hole on the right is bigger than the one on the left, but it definitely has a smaller hole-lining . In this case, not only cannot the predicate preserve its ordinary meaning, but it receivesdifferent interpretations with no apparent uniform translation algorithm . Sometimesto enlarge a hole is to enlarge the hole-lining, but not always. The Lewises would insist here on the importance of contextual clues. Nevertheless, one could doubt whether context is always sufficient to disambiguate . Supposeyou are asked to enlarge the hole in the left object of figure 3.4. The context is clear. But what does the question amount to? If holes are hole-linings, the question is really a requestto enlarge the holelining . But does that determine an enlargementof what we take to be the hole (as in casea of figure 3.5), or an enlargementthat does not affect our hole (as in caseb)? The context is clear, but there is no way to resolve the ambiguity without somehow relying on our pre- Ludovician notion of a hole. And , finally , one would prefer some systematicity, even if one does not seeany objection to constant implicit referenceto the context- apreference whose plausibility is especially dramatic if we keep in mind that we are dealing here with some basic spatial and material predicates which provide the stuff for our more metaphorical (and surely more contextual) thoughts. From this point of view, the account we are pursuing can claim to be not only more intuitive but also more systematic, and therefore preferable.

Immaterial

Bodies

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Figure3.5 hole- buthow? a Ludovician Enlarging Other worries could be added to the list. It sounds bizarre, for instance, ' ' to affirm that explosions produce hole-linings. (Don t say now that pro duce' really means 'enlarge'.) And in ordinary discourse it makes little " ' " senseto speak of a hole s being removed (unless this is taken to mean that a hole is destroyed or eliminated); by contrast, one can certainly remove a hole-lining (an operation which, however, would likely produce a bigger hole than the original , not a smaller one). Further , one must addressthe basic question of how one can identify holes with their linings, since it seemsthat a hole may have indefinitely many linings. Here the Ludovician answer is that one need not decide: " Really there are many different holes, and each is identical with adifferent " hole-lining . But all thesedifferent holes are the same hole. (p. 6) As was mentioned above, the intended solution here is to draw a distinction '' between identity and the relation of " being the same hole as, the latter " " amounting to having a common part that is itself a hole. But this can be quite odd. For instance, with referenceto figure 3.6 we may consistently hold that (a) every part of / 1that primafacie qualifies as a lining ofh1 is the samehole as 11, (b) every part of 12that prima facie qualifies as a lining of h2 is the samehole as 12, and (c) 11is not the same hole as 12, for they do not have any common parts. On the other hand, 13has many parts in common with 11which are themselvesholes, hence it must be the same hole as 11; and it has many parts in common with 12which are themselves holes, hence it must be the same hole as 12. In view of (c) above, this

~hIh2 ~l.,hI h2 Chapter3

hI

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3. 6 ~ ffbeing the samehole is having a common part that is a hole, then ' 1 is not the samehole asi , though they are both the samehole as 13"

meansthat the relation is-the-same-hole-as is not transitive, at least on the usual understanding of part -whole relations. But then the basic question remains: How many holes are there in the cube? Two or three? The Ludovician account seemsto yield the secondanswer- and that is hardly tenable. There is, finally , some onto logical question left open by the very notion of a hole-lining when this is different from the entire host. A hole-lining is in such a casea proper part of the host, and a potential part only . Therefore , if we do not have complete clarity about the onto logical status of potential parts, we should avoid onto logical commitment to them, and the whole project of explaining away holes by referenceto hole-linings becomes at least suspicious. Moreover, if holes are anything, they are actual - one cannot detect potential holes in an object. A radieal alternative. Such difficulties also arise with other attempts to explain away holes in terms of linings. For instance, Frank Jackson has argued that there is another responseavailable, not considered by the Lewises: " Holes are not hole-surrounds [ i.e., hole-linings] , for they are ' ' nothing at all; nor can statements about holes in things be translated in terms of unstructured one-place predicateslike ' is perforated' or ' has four holes'; but what can be done is to translate statementsputatively about holes in terms of statementsabout hole-surrounds. "There are many holes ' in that piece of cheese just says that it contains many hole-surrounds; ' Thereare the samenumber of holes in A as in B' just saysthat A and B have the samenumber of hole-surrounds; and so on and soforth ." (Perception , p. 132; our italics)

Immaterial Bodies

One might well agree that " to otTerthesetranslations is not to identify holes and hole-surrounds anymore than to translate statementsabout the averagefamily in the usual way is to identify the averagefamily with the families that are there" (ibid .). But how should one fully spell out the details of the clause" and so on and so forth"? Surely this is not a matter of trivial details needing only to be routinely worked out. The systematic feasibility of such a treatment is far from obvious, and some of the abovementioned difficulties prove just as threatening in this approach as in the materialist account of the Lewises. Although hole-linings are no longer being assertedto coincide with holes, some etTectiveway of relating our putative holes with their linings is still wanted. We need a definite notion of what a hole-lining is, and it is not clear how one should define that. In fact it is not evenclear whether sucha notion can be adequatelycharacter. ized, as our argument referring to figure 3.6 suggests Two theories, and more. We have seenthat there are at least two different ways to construe holes. First there is the possibility of relying heavily on ' the suggestion that holes are superficial parts- parts of an object s surface . The properties of a hole are all the properties of that part of an ' object s surface(theory 1; we may call this the Superficial Theory). Then there is the Ludovician idea that holes are parts of material objects- they are hole-linings (theory 2). These two theories do not , of course, exhaust the matter. For example, one can combine them. According to the resulting variant (which we label theory 2 ), holes are superficial hole-linings. They are the surface of any ' hole in the senseof theory 2; i.e., they are any part of the object s surface that comprises what, in theory 1, is a hole. Moreover, there exist some interesting relationships betweenthesethree theories. For instance, holes in the senseof theory 1 are minimal holes both in the senseof theory 2 and in the senseof theory 2. One could also consider the radical view according to which the hole coincides with the entire host. Such a viewcall it theory 2+- can be kept separatefrom the philosophical position of reism, which would say that holes do not exist at all insofar as only material + objects exist. Rather, according to theory 2 , holes exist: they are maximal hole-linings and coincide spatially with the entire holed object. All these theories (see figure 3.7) rest on a very strong relation tying holes to their hosts, or at least to material parts thereof (a relation that in theories 2 and 2+ is so strong as to becomeidentity). We shall now intro -

Chapter 3

(1)

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Fia8e 3.7 Four different ways of construing holes.

duce an entirely different account that conceptually freesholes from their hosts. As we shall see, such a theory bears some interesting relations to any of the theories above. But let us addressthe matter indirectly . What are holes made of ? Consider the following hypothesis: Holes are madeof space. They are not identical with some region of space, but they are constituted by space. Spaceis, in a sense, the matter of holes- or , if you prefer, spaceis to holes what matter is to material objects. The analogy is far-reaching. Just as we have flux of matter for material objects, we have flux of spacefor holes. And just as portions of matter can coincide with objects, portions of spacecan coincide with holes. However, the analogy neednot be complete, and we neednot acceptall the implications . Some questions then arise immediately. Are material objects made of spacetoo? If so, they are made of two different matters: ordinary matter and space. Is that plausible at all? And if objects are made of space, how do they differ from holes? The answer to the last question is not difficult : Both holes and material objects are made of space; however, unlike material objects, holes are made of spaceonly. This raises the further question of the difference between holes and regions of space, for one can say that regions of space, too , are made of space. But again the solution is not difficult : Holes are things that can both (i ) be made of spaceonly and (ii ) be not identical with someregion of space. (Rememberthat holes can move, whereasregions of spacecannot.) In fact, one could wonder whether holes are the only things satisfying conditions i and ii , in which caseone could even consider taking this as a sort of definitory axiom for holes. One could be tempted here to think of a counterexample. Imagine somequalitatively filled region of space(say, a

Immaterial Bodies

volume of pinkish expanse), and imagine that a shadow- a colorlessfloating spot- finds its way into it . There is no quality exemplified at that region; thus, it is made of spaceonly . That is, condition i is satisfied. And the region is not identical with a region of space, for it can move. Hence, condition ii is also satisfied. Yet what we have here is a shadow, not a hole. The counterexample neverthelesssuggestsa way out. Floating expanses are indeed metaphysically floating; they enjoy independence , whereas holes would be lost if they did not have a sustaining material object. There is no object the floating expanseis parasitic upon, whereasholes are always closely tied to something they are holes in. . In regard to the first question above (are material Matter and space made of space too?), common sensedoes not forbid us to treat objects space as a kind of stuff. And according to some theorists it would be reasonableto consider matter as qualified space. For instance, Jonathan Bennett has suggestedthat " we can be helped to understand the notion of a thing in spaceif we analyze it in terms of qualitative variation of space. The basic idea is that for there to be an atom in a given region of spaceis " for that region to be thus rather than so. (Eventsand Their Names, p. 117) ' Indeed, the idea can be traced back to Plato s account of the Receptacleor empty spacein the Timaeus, a notion that has sometimesbeendefendedas " a basis for the explanation of the ultimate grounds of individuation of " spatio-temporal (concrete) individuals and of their objectivity (Richard Gale, Negation and Non-Being, p. 63). Even without being pushedthat far, this way of looking at things would provide a coherent solution to the puzzle. We would avoid having two kinds of stuff (spaceand ordinary matter) constituting a material object; spaceonly - qualified space- constitutes material objects. Space, on this view, is not another sort of matter alongside ordinary matter (and with somespecialproperties: it could be penetratedby ordinary matter); it is the only stuff objects are made of. If one acceptsthis pieceof metaphysics, one may properly regard holes as being made of unqualified matter (that is, " bare" matter - which is what ) space, according to this view, really is. To fill a hole would then be to qualify or requalify the matter (space) inside it ; to empty a hole would be to requalify or dequalify that matter. Holes as immaterial bodies. This, then, is how the new account goes: " " Holes are spacious; they are made of space; they consist of bare, unquali-

Chapter3

Fiaure3.8 " Holescan be construedas " immaterialbodies complementaryto their material hosts. 3.7. (Comparefigure )

fied rnatter. They are- we shall say- immaterial bodies, " growing" parasitically, like negativernushroorns, at the surfacesof rnaterial bodies. This way of looking at things, which we will label the Irnrnaterial Theory , or sirnply theory 3, shareswith theories I and 2 the idea that holes are superficial particulars (they are located at the surfaces of rnaterial objects). However, it also provides a way of rnodeling our preanalytical intuition that holes are sornehow complementaryto rnaterial objects (see figure 3.8). It does not detach holes frorn their hosts (the onto logical link rernains), but it provides a rneansof conceptuallyunchaining thern frorn the things they are in. It draws our attention not to the hole's actual host but to its actual or possible guests- to what is or could be hosted by the hole. Of course this new theory needsto be further scrutinized. Our putative " irnrnaterial bodies" threaten to turn out to be a sort of philosophical another to corne closer to their real nature- ether. Are they phlogiston, so ethereal as to vanish altogether? Let us consider sornernore objections and basic puzzles. " " " Actuality vsopotentiality. Holes, one could argue, are actual. But the immaterial bodies they are identical with are not always completely delimited by actual surfaces, and are therefore not actual. Thus holes are both actual and not actual." The answerto this objection is that if we acceptregions of spacewe also accept immaterial bodies, be they actual or not. But are we saying that immaterial bodies are individuals proper in other worlds only? We are not saying that immaterial bodies are material bodies in some other world though it is obviously true that they are penetrated by material bodies in other worlds. Holes are fillable ; therefore they are filled in other worlds.

Immaterial Bodies

There is, however, an important difference here. Consider the case of potential parts of material bodies. One can say that it is not, for them, the same as for regions of space. The latter are all actual. They do not need any real surface to get individuated (though we could use surfaces to indicate them). The surface of the hole' s potential filler , including those parts of it that are not actual, can be regardedas the actual surfaceof the hole. " Guests, hosts, and immaterial ghosts. Someonecould say: Holes arejust their own potential fillers- the imaginary plugs that could be used to fill " them up completely. After all , holes and fillers coincide spatially. The answer to this objection is, of course, that the identity between holes and fillers is unwarranted, insofar as holes can exist unfilled. Even if all holescould be filled in our world , there will be possibleworlds in which " " they are not. The relation between a hole and its possible guests is indeed a crucial one, and we shall try to investigate it more closely later on . It is not , however , a relation of identity . Likewise , consider the objection that holes are necessarily dependent on having an actual host - that immaterial bodies can exist even outside and in the absence of material objects , but holes cannot . The reply is simply that to regard holes as immaterial bodies does not mean that the converse should also hold : not every immaterial body is identical with some hole . (Think , for instance , of the metaphysical shadows discussed above .) Holes " " are only a subclass of immaterial bodies , those that grow inside material objects or at their surfaces. And this subclass of bodies cannot exist out -

side or in the absenceof material objects. They are, as we saw in the previous chapter, superficially dependententities. More objections could be raised at this point , for it seemsas if we are going to be left with lots of immaterial bodies of different sorts. We know that penetration is exclusive: an immaterial body can be penetrated by a material body (as when you fill up a hole). But how can such a body be limited by a host? Immaterial bodies do not interact. It is not as if you pushedan immaterial body inside a material one, thereby creating a hole. Perhaps, in a world nomologically different from ours, immaterial objects could be responsiblefor annihilation of stuff? Well, then, in such a world holes would be unfillable, or elseour conceptsof filling and impenetrability would have to be changed(we shall come back to this point in chapter " 9, where we discussthe strangecaseof " cartoon-holes ).

.

Chapter3

Be it as it may, an account is available which claims that when two holes (and, in general, two immaterial bodies) penetrate each other they share space- up to the point where they are made of exactly the same space. A minimal theory of immaterial bodies- aad aU problems solv. . In a minimal theory there are just as many immaterial bodies as material ones, plus (maybe) the complement of all the latter taken together. In figure 3.9, for instance, there are three immaterial bodies: a, b, and complement c. Bodiesa and b, but not body c, are occupied by material bodies. Note that the complement of a body a and the complement of a body b are not objectscontemplated by this theory, as opposedto their joint complement c; the minimal theory does not allow for partial penetration. Moreover, according to this theory, only cavities among holes (along with occupied immaterial bodies) are immaterial bodies proper (they are internal complements ). Other types of holes, such as superficial hollows and perforations , would be parts of immaterial bodies ( parts of the object's complement). In figure 3.10, hollow h is part of body c. Moreover, body a

Fia8e3.9 Th~ immaterialbodies . two or which (a andb) arepenetrated by materialbodi5.

Fia8e 3.10 Two immaterialbodies c, anda hole, II. whichis part of c. , a andits complement

Immaterial Bodies

can be moved in its complement. Body a will stay unchanged, and body c will continuously change its shape as body a moves. In the minimal theory, immaterial bodies do not occupy, though they could be occupied by, material bodies. Thus, the correct expressionof theory 3 seemsto be as follows: Holes are either immaterial bodies proper (cavities) or parts of immaterial bodies (hollows and perforating tunnels). In the latter case, when a hole is perfectly filled, it gets separatedfrom the immaterial body it was part of; it becomesan immaterial body proper. Holes can be made of space only , without having to be identical to any region of space, and they live an intrinsically parasitic life. Summary. Our inquiry into holes led us to a discussionof different theories of holes, different ways of construing them: as superficial parts (theory 1); as material parts, viz. hole-linings (theory 2); or as immaterial bodies located at the surfacesof material bodies (theory 3). (Look again at figures 3.7 and 3.8.) We suggestedthat the first theory is too reductive, and we argued against the secondtheory (including various possiblevariations sur Ie theme) on account of its counterintuitiveness. By contrast, theory 3 (the Immaterial Theory) provides what seemsto be a natural way of modeling our pre-analytical intuition that holes are in someway complementary to material objects, and it does so by drawing our attention from what is hosting a hole to what could be hosted by the hole. In the remainder of the book , we shall pursue thesematters more closely . We shall try to show that the Immaterial Theory allows us to deal with the hole problem quite nicely. But we shall also be interested in some theory-independentdevelopments. Thus, we shall begin by taking a closer look at the geometrical differencesamong the various types of holesdifferenceswhich we have already mentioned more than once and which are now starting to playa significant role in our investigations. Next we shall addresswhat it meansto fill a hole, and what sort of connectionscan be drawn betweena hole and its (potential or actual, partial or complete) fillers. Why and how holes are onto logical parasites will then become clear.

4

Ho Dows, Tunnels, Cavities, and More

Paradigms . Attempting to classify holes is an important task of our work. It is convenient to have a general idea of what holes look like and of the various forms they can come in. The adoption of a classification basedon circumscription of some core facts- and on a casuistry of some borderline facts- is not a form of cowardice in face of the threat representedby vague concepts. There is no doubt that the concept of a hole is vague, but that does not mean that holes themselvesare vague entities. Vaguenessis a matter of how We describe the world , but it does not carryover to the world itself. Then there remain the core cases. It could turn out to be awfully difficult to distinguish red shadesfrom orange shadesin some limit cases; it is, however , clear what a paradigm of red should be like. Consider the four objects in figure 4.1. Each of them could be used to host things. Theseconcavegestaltshave morphological affinities, but they differ in someimportant ways from one another. Among the elementary features of some of these gestalts are the presence of a discontinuity (an edge), the definition of a purely internal , encapsulated space, and the presenceof a perforation through the object. We can distinguish betweensimple depressionsand deephollows, betweenperforating tunnelsand internal cavities. As table 4.1 shows, there is a net of resemblancesspreading over thesegeometrical patterns. Of course, by combinatorial interpolation we could get additional patterns . Take, for instance, a completely internal hole, shapedlike a doughnut (figure 4.2). It is a cavity (for it is internal ), but we can also regard it as a tunnel (one can imagine walking through it ). Both options seem intuitively appropriate, as we find here some important features of both cavities and tunnels. There are also some interesting part -whole relations, made clear by some way of cutting the host, that introduce a sort of hierarchy between thesedifferent patterns. For instance(figure 4.3), we can sometimescut an object with an internal cavity so as to get two objects, each with a hollow ;

Chapter 4

hollow depression

=~.~ :0;-fii ..:~ng.Co 0.j 0~

tunnel

Fi88e4. 2 hole. An internal, doughnut-sbapeci

cavity

Fi88e 4.1 . Somebasictypesof boles

Hollows , Tunnels, Cavities, and More

and we can sometimescut an object with a hollow so as to get a piecewith a tunnel and a piece with a (smaller) hollow . A single cut may also be sufficient to produce a typical go-through tunnel from a doughnut-shaped tunnel (though we would need two cuts to get it from a standard, bubbleshapedinternal cavity). It is clear, however, that we can never obtain an internal cavity by partitioning hollows and/ or tunnels. Moreover, the difference between, say, a hollow and a depression is simply a matter of degree; the sharp discontinuity that goesalong with the former becomesneglectablein the latter. In the following we shall take as core casesthose patterns that are characterizedby the presenceof a sharp edgeor discontinuity on the surfaceof the holed object. Thus, we shall not explicitly elaborate on depressionsor other patterns where fuzzlnessmay introduce additional complexities. We think that most of our remarks will be sufficient for an understanding of those casesas well. In any event, we shall proceedcautiously, trying to gain new insights by applying different and sometimesnot obviously compatible ways of classification. Going through: On the differencebetweenhollows and tunnels. In certain cases, the difference between holes of distinct types is conveniently explained with the help of topological notions. Most notably , a topological perspective allows us to introduce from the outset a sharp distinction betweenobjects with hollows and objects with tunnels: an object with a hollow is to an object perforated by a tunnel just as a (solid) sphereis to a " " (solid) torus, i.e., what topologists call a handlebody. (In the following ' ' we shall often drop the modifier solid and simply speak of spheres, tori , etc.- meaning three-dimensional objectswith spherical surfaces, toroidal surfaces, and so on.) There are various ways of making this more precise. For instance, call a path any line segmentor curve that can be obtained from a line segment by elastic defonnation. A path cannot have any intersection with itself except possibly at the two end points, called the initial point and the terminal point of the path. If the initial and tenninal points of the path coincide, then the path is a closedpath, or a ring. Now , one can always draw a ring IXon a torus in such a way that any two points on the surface can still be connected by a path that does not meet IX. By contrast, every ring on a sphereseparatesit into two distinct regions, points in either of which cannot be connectedto points in the other region by a path without intersecting the ring (figure 4.4).

4 Chapter

" I \ \ \

I I ' ' \ J

Figure 4.5 Telling tunnels from hollows. The object on the right can be handled safelyby a string; the one on the left cannot.

Somewhat more perspicuously, we can also say that the crucial property that topologically distinguishes a torus from a sphereis the existence of non-reduciblerings. On a sphereany ring on the surfaceis homotopic to a point (i.e., can be reduced to a single point by elastic deformation). By contrast, on a torus this is not the case: a superficial ring that goesthrough the hole (like the one depicted in figure 4.4) cannot be deformed to a point " without " cutting the surfacesomewhere. The analogy is fairly suggestive.Intuitively , in the caseof an object with a tunnel we can imagine threading a string through the tunnel and tying the two extremities together so as to hold the object safely by holding the string; the tunnel defines a way of handling the object (a loop) that is by no meanspossiblein the caseof an object with hollows. A new glimpse at the prototypical examplesfrom figure 4.1 will help clarify this point - see figure 4.5. With all this, of course one cannot say that an object with a hollow is always sure to be homeomorphic (i.e., topologically equivalent) to a

Hollows , Tunnels, Cavities, and More

Fiaure4.6 A hollowon a weddingnng is topologicallyirrelevant.

sphere, or that an object with a tunnel is always sure to be homeomorphic to a torus; the object may have other topological properties that prevent such equivalences. For instance, a wedding ring is homeomorphic to a torus, owing to the tunnel in the middle, and the presenceof a hollow on it (figure 4.6) is surely not enough to make it homeomorphic to a sphere. One can, however, say that some objects with a holloware homeomorphic to a sphere, which is neverthe casefor an object perforated by a tunnel. To put it in another way: For any object x with ahollowy , there is a (possibly proper) part z of x such that (i) y is a hole in z and (ii ) z is homeomorphic to a solid sphere; every ring on the surfaceof z is homotopic to a point (or , equivalently, every ring on the surfaceof z separates it into two regions). Likewise, for any object x perforated by atunnely , there generally is a part z of x such that (i ) y is a hole in z and (ii ) z is homeomorphic to a solid torus; there exists rings on the surfaceof z that are not homotopic to a point (or , equivalently, somering on the surfaceof z is non-separating). Bewareof the knot. Although prima facie appropriate, the above criterion must be applied with some care, particularly with respectto tunnels. In three-dimensionalmanifolds the existenceof a non-reducible (or a nonseparating ) ring is in fact a necessarybut not sufficient condition for something to be elastically deformable into a solid torus. Take, for instance, the case illustrated in figure 4.7. Here the presenceof a knot in the tunnel prevents us from defining an elastic transformation of the object into a solid torus. Yet in this casethe existenceof a non-reducible ring is easily verified- and that is all we need to recognize the existenceof a tunnel. You can imagine entering the tunnel from the top of the cube, walking quietly through it until you come out at the bottom , and then completing

Chapter 4

Figure4. 7 An objectwith a knottedtunnel.

the loop by walking along the outside of the cube so as to go back to the starting point . In fact, in doing this you would not even notice that you " were going through a knot . The " knottedness actually resides in the object (i.e., the host), not in the tunnel. This means that the surface of the object in figure 4.7 has the same intrinsic topology as the torus, but its extrinsic topology (i.e., the way the surfaceis embeddedin three- dimensional space) is different. Thus, for the purpose of detecting tunnels the extrinsic topology can be ignored. All that matters is the intrinsic topology of the object- in fact, all that matters ' is the intrinsic topology of the object s surface. Using loops to count tunnels. It would be nice if one could rely on the above topological properties to answer not only the question of whether there are tunnels in a given object but also the question of how many tunnels there are. Indeed, one might be tempted to say that the number of loops admitted by an object- i.e., the different ways in which one can thread a string through a hole and tie the ends so as to handlethe objectdefinesthe total number of tunnels perforating the object. To someextent ' this may be true; however, one must be very careful about what it means for two loops to be equivalent- otherwise the number of tunnels could grow beyond any limit . First , the relevant notion of a loop should be properly qualified. Consider for instance figure 4.8. Here you can handle the object (atrefoil knotted solid torus) by either a, b, or c. Each of thesethree strings has been threaded through the hole and tied at the extremities in such a way that you cannot releaseit from the object by elastic deformation. However, here we would regard only a as a proper loop . The explanation for this is simply that loops are conceptually dependent on rings, and it must be

Hollows , Tunnels, Cavities, and More

Fiaure4.8 A properloop (a) and two improper loops (b, c).

" " possible to shrink a loop onto a ring on the surface. As is clear, this requirement is satisfied only by a; band c violate it . Second, there is no univocal (i.e., one-to -one) correspondencebetween proper loops and rings on a surface. In general, a loop can be mapped onto two or more non-homotopic rings, which meansthat the relation of equivalenceamong loops cannot be accounted for automatically in terms " of topological relations among " corresponding rings. A loop through the hole of a torus (figure 4.9) provides a simple illustration of this point . Here, band c are not homotopic . In fact, c is homeomorphic to a ring that surrounds the hole without going through it . (Just cut the torus along a meridian, twist one edgeof the cut, and rejoin .) Our suggestionis that the " intuitive idea of " shrinking comes in handy in this respect too: the relevant " " ring is the one onto which the loop can be shrunk. Thus, in the example, the ring correspondingto a is b, and c may be ignored. This criterion also takes care of a third difficulty that comesto mind as soon as we reason in terms of loops. In figure 4.10, is the simple loop on

Chapter 4

Figure4. 10 Difficultieswith findingproperloops.

Fi. - e 4. 11 Two tunnelsbut threetopologicall, irreducibleloops.

the left equivalent to the double loop on the right (or to any loop that goes through the hole several tirnes)? If it is not , we have two tunnels here. However, by the above criterion , the answer is sirnply that the loop on the right is not a proper one: becauseby definition a ring cannot intersect itself, there is sirnply no ring on the surfaceof the object onto which this loop can be shrunk. This does not rnean that the loop cannot be mapped onto a ring . In fact, this is possible; the result is a ring sirnilar to the one depicted in figure 4.9c but going through the hole twice. As is clear frorn that exarnple, such a ring is horneornorphic to a ring that surrounds the hole without going through it . More on counting tunnels. Homotopy of corresponding rings seemsto provide us with a precisecriterion for establishing whether two or more loops are equivalent for the purpose of identifying tunnels. It does not , however, provide a sufficient criterion if we want to count tunnels by counting non- equivalent loops. As a simple example, consider the caseof an object perforated by two parallel cylindric tunnels (figure 4.11). Three

Hollows, Tunnels, Cavities, and More

Fi88e 4. U How manyproper. non-equivalentloopsarethere?

topologically irreducible loops can be distinguished here, yet we would emphatically insist that there are only two tunnels. Common-sensephysics errs often in quantitative or geometrical respects, but in topological matters common senseis correct. As we see it, to account for this and other similar casesone would intuitively apply a simple criterion: A loop cannot define more than one tunnel. Hence, in this casethe loop in the middle- though topologically non-equivalent to the two loops at the sides- may be disregarded. It is also apparent, however, that this would be a circular criterion, for what is neededis preciselya meansfor counting the number of tunnels. What, in fact, preventsus from considering a third (scattered) tunnel, defined by the loop in the middle? A slightly more complicated caseis the following : Which of the loops in figure 4.12, if any, is a proper loop ? Is the 8-shapedloop in a equivalent to the simple loop in b? And what about c and d? T- - led holes. One possibleaccount would be that none of the above is a proper tunnel, though we certainly cannot sort this out by shrinking the loops. The suggestionwould be this: There is an X-shapedhole that is not an X-shapedtunnel(or , of course, an X-shapedhollow ), for eachof the four loops depicted contrasts with a certain intuition about tunnels. The intuitionis that going through a tunnel typically is going through one entrance and one exit - perhaps fewer, as is the case with an internal toroidal cavity, but not more. When you go through a tunnel, you do not exit it in the middle before entering it again and reaching the end. Hence, when you look at a loop you must be sure that it does not violate this in -and-out principle. Otherwise you are going to seea tunnel where in fact

4 Chapter

there isn' t any- or when there are more. (A similar account would apply to the caseof figure 4.11.) Denying the existenceof an X-shapedtunnel here would not imply that there is no tunnel in the object. Indeed, by the foregoing criteria the total number of tunnels would add up to 6 in this case: although none of them coincides with the global, X-shaped hole, we can certainly distinguish a v -shaped tunnel at the top and a A-shaped tunnel at the bottom , a ) -shaped tunnel on the left and a <-shaped tunnel on the right , and two diagonal tunnels- one I -shaped and one \ -shaped. (In addition to or perhaps instead of - the latter , one might also distinguish four smaller " tunnels: the four " legs of the X leading to the center of the object. However , that would mean that a loop may sometimesidentify more than one tunnel- and, as we already mentioned, we find that counterintuitive ). Regardlessof how exactly we explain these part -whole relations a problem that should not be underestimated, and one that will keep us occupied in much of what follows- the main difficulty with this suggestion lies in its cost. It doesjustice to one intuition about tunnels. But the question is not so much a question of whether the X shaped hole is a tunnel or what. The question is how many such holes there are. And translating our talk about ramified tunnels into talk about tunneled holes does not quite help solve that question. The initial idea was to rely upon the topological properties of tunnels not only to find them but also to count them. We cannot now deny tunnelhood to those tunnels that cause counting troubles. . The loops admitted by an object Trying to count tunnels by the genus ' reducible to the non rings that go through the object s perforation correspond (s). As there are circumstanceswhere this seemsto give rise to unsolvable complications, one could consider looking at the non-separating rings instead. Surely this is a viable criterion in the simplest case, the caseof an object with a single perforation . If there is only one tunnel, then there is no way ' we can draw more than one ring on the object s surfacewithout separating it . (There are in fact two non-homotopic rings that do the job - the one that goesthrough the hole, and the one that surrounds it - but one cannot draw both without introducing a separation.) The actual shapesof the object and the tunnel do not matter; we only needlook at this topological property . This is why we can regard the objects depicted in figures 4.7 and

Hollows , Tunnels, Cavities, and More

FIawe4.13 Four distinctobjects , eachwith onetunnel.

4.13 as being of the sametype, regardlessof the actual different shapesof the tunnels. Thus, in this casethe so-called genusof the surface(i.e., the maximum number of disjoint rings that can be drawn on the surfacewithout separating it ) coincideswith the number of tunnels. However, it is very difficult to generalizethis principle so as to count the number of tunnels in an object by counting its genus. Perhapsthat would provide a topologically coherent account, but once again it seemsthat it would also contradict our common-senseintuitions . The topology of the object can be misleading when it comesto classifying tunnels. To realize what the difficulty is, it is sufficient to move up one step and consider the caseof objects with surfacegenus2. The prototypical caseis that of an object that is homeomorphic to a solid double torus, such as a cube perforated by two parallel tunnels. As one can easily verify, any third disjoint ring on its surface would separateit into two distinct superficial regions. So here we have an answer to the problem raised above in connection with figure 4.11. But just look at the other casesdepicted in figure 4.14. They all have this property - they are all homeomorphic solids of genus 2! But surely we do not want to say that each of them involves exactly two tunnels. Partial tu Doeisand overlappinatu Doeis . The point is that object topology seemsto fall short of accounting for the complexity of the hole problem. Sometunnels can overlap, others seemto be related to one another by a part -whole relation, and theserelations are not adequatelyreflectedby the topology of the material object hosting the holes. Thus, as we seeit, in objects a and b in figure 4.14 there are only two disjoint tunnels (although in caseb the straight tunnel bearsan important

4 Chapter

e FIawe 4. 14 A bunch of topologically different holes.

equivalent objects of genus 2 (check that !) with sometimes quite

relation to the knotted one; were it not for the fonner, it would beimpossible to unravel the knotted tunnel simply by elastic deformation). The other casesare different. F or instan~ , in casec we are inclined to seea global, V-shaped hole (tunnel) along with three distinct tunnels, each of which partially overlaps with at least one of the others (figure 4.15). Perhapsthere is also sometemptation to consider additional "parts, such " as the three smaller tunnels .corresponding to the three legs of the V, each of which totally overlaps with two of the three bigger tunnels mentioned above. However, one should be careful in general not to be misled the actual geometric shape of the hole under consideration. We have by been speaking of a V-shapedhole. But that is of course purely indicative. A glimpse of figure 4.16 will sufti~ to illustrate that the V-like shapemay in fact disappear if the hole undergoesa slight topological deformation; topologically , the hole on the left and the one on the right have exactly the " same" skin. Similar remarks apply to the other varieties of holes and ramified tunnels that we have been discussing. For instan~ , the four tunnels of figure 4.13 have exactly the same basic skin, and this confirms our feeling that they are tunnels of the same type despite their differen~ in shape. By

Hollows, Tunnels, Cavities, and More

\0 FIawe4. 15 Tentativepart-wholerelationsfor a Y-shapedhole.

FiI8e 4. 16 Gettingrid of the Y in a Y-shapedhole.

contrast, someof the holes in figure 4.14 have topologically dif Terentskins, and that reflectsour conviction that they are different holes in spite of the topological equivalenceof their hosts. This meansthat the topology of the host does not completely determine the topology of the hole's skin. What we call the skin retains sometopological character traits of the host, but it goes beyond that. It introduces an important shift from the topology of the host to that of the hal~. Now , our suggestionis that this shift is pretty close to what we need in order to overcome the apparent complexity of the hole problem, particularly with respectto such taxonomic purposesas we have set for ourselves. We do, however, need a better way to characterizeit . We need a general criterion that tells us, given a certain hole, what its skin is. As such a criterion seemsto lie at the border of topology, we shall postpone it . But it seemsclear that until we have that criterion our account can only be very tentative. Are parts of tunnels tunnels themselvm? So we have implied . Now , this idea can be expressed by appealing to the notion of being a tunnel relative to some ( potential ) part of a given object : x is a tunnel relative to some

Chapter 4

part z of objecty if and only if x is a tunnel in z (you can thread a loop through it ) and every superficial part of y that is in contact with x is a superficial part of z. Referenceto superficial parts (actual superficial parts) excludesthe possibility that one might cut out a potential part of y along a tunnel-like shape. A rather unsurprising consequenceof this characterization is that a variety of holes (including non-perforating ones, i.e., hollows and internal cavities) have tunnels as their parts. The examplesconsideredabove with referenceto figure 4.3 are indicative of this. Conversely, a lot of tunnels have different types of holes as their parts (we shall come back to this in chapter 7, where these mereological issueswill be examined in greater detail). The hiddenside of things: A rule of thumb for counting cavities. Tunnels are only one type of hole. As they go through , they are conceptually located halfway betweenhollows, which pertain to the exterior, and cavities , which are hidden in the interior . The lack of contact with the external side makes it impossible to use loops or non-separating rings in order to ' characterize cavities, for a ring can be reduced to a point on a cavity s surfaceexactly as it can be reducedto a point on the surfaceof a tunnelless object. (Doughnut -shaped cavities are an exception, which is why such cavities also qualify as tunnels.) Nevertheless, there is a simple way to . disclosewhat is hidden inside a pieceof Swisscheese . An internal cavity of surfaces involves disjointness Complete internality has no contact with the external environment. And the presenceof a ' cavity splits the object s environment (i.e., what is spatially coincident with the object' s geometrical complement) into two parts (figure 4.17). Thus, a simple way to count the number of internal cavities hosted by an object is

Figure 4.17 An internal cavitiy splits the environment of the host into two disjoint regions.

Hollows , Tunnels, Cavities, and More

to count the object's disjoint surfaces. If n is the number of such surfaces, then there are exactly n - 1 cavities in the object. By this criterion , an internal doughnut - shaped hole Oike the one in

surfaces. But it is also a tunnel, for it satisfiesthe criterion of individuating tunnels by meansof drawing non-separating rings. It is, as we shall say, a cavity-tunnel. Moreover, this criterion gives a perfect explanation of the fact, mentioned earlier in this chapter, that one can never obtain an internal cavity by partitioning an object with hollows and/ or perforating tunnels (though the opposite is certainly possible). Cutting an object into two or more pieceswill increasethe overall number of surfaces, but not the number of surfacesof any single piece. Two ways of counting surfaces . We are back to surfaces, then. And our this time is their number. The question has amply beendiscussed: problem How many surfacesdoesa cube have? Six, according to one intuition ; just one, according to another intuition . One can draw a distinction here betweenthe faces of an object and its surface(or surfaces). A face is a part of the surface that is isolated from other parts of that surface, though not disconnectedfrom them. Thus, if an object has one isolated face, it has at least two of them: being a face is a relational property . (In this sense, spheresare facelessobjects, as opposed to prisms.) This is obviously not the case with surfaces. We may have objects with two or more surfaces, but the surfaceof an internal cavity is not isolated in that sense. Rather, it is insulated; that is, it is disconnected from the outer surface. More precisely, we say that an object x has an insulated surfaceif there are at least two superficial parts of x that are not superficially connected (thus, as with isolated faces, an object with an insulated surface is sure to have at least two of them). Our criterion for counting cavities relies not on the number of isolated surfacesbut on the number of insulated surfaces. The concept of insulation receivesanother interpretation if one considers that an internal cavity splits the environment into two disjoint regions , which becometo some extent causally independent of one another (even though this independenceis limited to certain types of causal interaction ). There is a sensein which the region inside the object is causally insulated from the region surrounding the object: It is inaccessible , and its

Chapter 4

content is protected against external influences(and cannot influence the external world ). Exteraallaoles. We have discussedtunnels and internal cavities, and we have seenthat they can be characterized in precise terms in a way that doesnot quite dependon a theory of holes. It now remains to characterize the non-perforating, non-insulated hole: the hollow proper. As we shall seein the next chapter, this cannot be done in topological terms, and it cannot be done without more explicit referenceto a theory of what holes are. Nevertheless, hollows can be characterizedwith the help of morpho logical concepts. Since these will also suit our criteria for distinguishing betweentunnels and cavities, the resulting account will then be complete, at least relative to the basic classification introduced at the beginning of this chapter.

lr Fillers andSkins

A plea for morphology. Topological conceptsmake it possible to distinguish an object with a tunnel from an object with no tunnel, and we can count the number of surfacesto tell an object with from an object without cavities. But how to characterizean object that has a hole that is neither a tunnel nor a cavity? We have seen , for instance, that from a purely topo logical perspectivethere is no way to differentiate a hollowed cube from a sphere. A plasticine die with a small, non-perforating hole (a hollow ) can undergo a slight elastic deformation that will suppressthe hole and liken the die to a spherewithout the die' s surfacebeing cut at any point . The foregoing is probably the simplest motivation for the introduction of morphological concepts into our inquiry . Both a hollow and a tunnel are holes. It seemsthat we needsomeother concepts, apart from topologi cal ones, if we are to be able to cope with hollows, depressions , and protrusions - that is, with everything that is characteristic of a holed but non-perforated object. We need theseconceptsin order to account for the feeling that cavities, hollows, and tunnels are aU parts of a single family . The network of such conceptsmay be termed a morphology. What are its distinctive features? 11Iemorphologiesl manifold of holes. The notion of morphology that we want to useis extremely broad. Call a morphologicalmanifold relative to a certain class of operations the universe of those objects that share the relevant features that are neededto explain how theseoperations can be performed on such objects. For instance, one morphological manifold is constituted by all the members of any class of topologically equivalent solids. Here the relevant operation is deformation without cutting . You do not need metric properties in order to account for deformation without cutting; topological properties suffice. Another manifold is the one whose membersare all the solids of the samevolume v. In this case, the relevant operation could be abstracted from, say, immersing the object in a container of liquid and seeingwhether the level of the liquid increases(relative

Chapter5

to the container) by a certain given measurez. The constancy of z is a criterion for the identity of v. Here topological properties becomeirrelevant. But there is no reason not to trace limits between manifolds in a way that cuts across and goes beyond geometrical and topological notions, provided the operation which definesthe manifold is sufficiently determinate . This is the case, we would like to suggest, of the manifold defined by the flu-in operation. The morphological manifold of fillable things is constituted by objects with holes. A fined bole is still a bole. Holes can be filled. Indeed this seemsto be an essentialproperty of holes. And we mean a property of holes- not of their hosts. You don' t fill a holed wall, really. You fill holesin the wall. In some casesperhaps this is not so apparent. " Your father got me drunk last night . He kept filling up the hollow in my glass." Surely nobody would say that ; rather, one would say " Your father kept filling up my " glass. But the reasonwe say that glassesget filled , rather than the hollows therein, is that glassesare somehow functionally parasitic on their hollows . Glasses(like cups, bottles, buckets, tanks, and so on) are defined by their function. Their function relies on the presenceof a hollow. And to fill the hollow in a glassis, in a sense, to fill the glass. There are of course other sensesin which a glass may be holed. For instance, if a glasshad a little perforation that would let the wine run out, one might say " Don ' t use that glass- it has a hole in it !" Certainly that would not be a hole on which the glasswas functionally parasitic. In that caseone would rather speakof the hole as an imperfectionof the glass. The main hole- the hollow - is now useless , as it is made imperfect by the production of the secondlittle hole. (Actually, things are even more complex : the perforation transformeda hollow into a tunnel. But we need not into the details here.) go Allow us to say that glasses(unbroken glasses ) have holes in our sense. Now , it is crucial to observe that a hole can be completely filled and yet remain a hole. If you fill in the hole in your wall with plasticine, you do not destroy the hole. For if you remove the filling, you do not create a new hole. You simply empty the original one. Thus, a filled hole is still a hole- which is why a full glassis still a glass. It is interesting, then, to seewhat sorts of relations can be drawn between holes and their potential fillers, and how we can get new insights about holes by studying the ways they can be filled. In fact, certain proper-

Fillersand Skins

ties of holes can roughly be characterizedin terms of the stuff one can use to fill them. For instance, hollows can typically be filled up with liquids, while most tunnels cannot (although some, such as Ushaped tunnels and internal cavity-tunnels, can). However, we shall ignore this aspect and concentratenot on the possiblefillings of a hole (the stuff that can be used to fill it ) but rather on the potential fillers of a hole. The notion of a filler we have in mind is a geometric, highly idealized notion. It is also a thoroughly dispositional one, though some qualification will be needed in this regard. We shall generally talk of potential fillers, that is, of objects that could be usedin order to fill a hole. In fact we ' ' ' shall use'filler and potential filler synonymously, unlesswe specifyotherwise ' ' . A hole s filler need not fill the hole; it s just that it might. Partial , complete, and perfect fillers. Holes can be filled in different ways. For example, one can fill a hole completelyor just partially . A complete filler entirely fills the hole (as in figure 5.la ), whereas a partial one does not (b, c, d). Note that a filler may carry a hole along with it . We shall discusslater (chapter 7) whether this hole is or is not part of the initial hole: it lies inside it and may even share some superficial parts " " with it , but the hosts are different. Also, a filler can completely hide the hole and still be partial (c, d). In particular , the fact that every superficial part of the hole is in contact with a superficial part of the filler does not necessarilymake the latter a complete filler (d). (We could say, rather, that it fits the hole completely, as opposed to a non-fitting filler such as that in c). Further distinctions can be introduced here. For instance, a partial filler may very well be scattered, as in figure 5.2a, or made of different materials, " " as in b. Think of a room full of furniture. The furniture is a scattered filler of the room - or , equivalently, the single piecesof furniture are all

"."-', d

Figure5.1 A completelyfilled hole (a) and three partiallyfilled ones(b, C,d) .

ChapterS

.2. Fia8e5

Morc waysof filling a holepartially(a, b) or completely(C,d).

5.3 ~ Two perfectlyfilled holes.

small, partial fillersof the room, and togethertheyfill the room partially. Casec showsthat somethingsimilar can alsohold for completefillers: a completefiller may consistof differentmaterials , thoughin that caseone of a of fillers that might speak pile partial togetherfill up the hole completely . By contrast, it is more difficult to think of a scatteredcomplete filler (apart, of course,from an arbitrary mereologicalfusioninvolving a non-scatteredcompletefiller). A rather interestingcaseis providedby a completefiller split into two partial fillers by afissured ). Now, imaginepartitioningthe classof all potentialcompletefillersof a hole(of a givenshapeand size) into the equivalence classes determinedby the relation " is madeof the samesort of stuff as.'' The greatestlower boundof eachsuchclass , relativeto thepart-wholerelation, maybecalled a perfectfiller of the hole: it fills up the hole completely , but without overflowing(seefigure5.3). And sinceall thesepotentialfillersaregeomet rically isomorphic,we may assumethat to eachhole therecorrespondsa canonicalperfectfiller that is representative of the entireclass. OD fdUDgand beiDain. Although there is an intuitive duality between partial and complete fillers, these notions are not exactly symmetric. A complete filler is any filler that includes a perfect filler as a ( possiblyim -

Fillers and Skins

Figure5.4 A complete , imperfectfiller.

proper) part; a partial filler is any filler that does not so include a perfect filler. But the latter condition is not equivalent to the dual requirement that a partial filler be any filler that can be extended(relative to the part whole relation) to a perfect filler . This is a sufficient but not a necessary condition for partial fillers. As examplesc and d in figure 5.1 indicate, our notion of a partial filler is compatible with the possibility that the filler " sticks out" of the hole. Something qualifies as a partial filler of a hole if has a and only if it (possibly proper) part that can be extended to fill the hole perfectly, or , equivalently, if and only ifit completely fills a part of the hole. Thus, putting a finger into a thimble is a way of completely filling the hole in it , while putting in the tip of a pencil would be a legitimate way of filling it partially . We find this to be the best way to make senseofa certain basic relationship between filling and beiNg in a hole- a relation to which we shall come back more than once in the following chapters. Our suggestionis that putting something into a hole is, in some way or another, filling the hole, just as one cannot fill a hole without putting something into it . And just as something can be said to be partly in a hole if some parts of it are in the hole, something can be said to partially fill the hole if some parts of it fill the hole (though not completely). Failing to fill a hole completely yields a partial filler , whether or not it can be extendedto a perfect filler. More on morphology vs. topology. A secondcaveat is in order, concerning the very notion of a perfect filler: As we introduced it , this notion dependson the notion of a hole. We have a criterion for something to be a perfect filler of a hole, but we do not have any notion of a filler in itself. Holes exist and are what they are prior to their potential fillers. This is why in the caseof figure 5.4 we can speakof the filler as complete but not perfect relative to the hole, which is defined not by its fillers but by

Chapter5

the presenceof a certain discontinuity in the surfaceof the host object. The intuitive idea operating here is that a perfectfiller heals a superficial discontinuity , but one cannot rely on the notion of a filler (perfect or not ) to identify any discontinuities. Of course, the fact that fillers are conceptually dependenton holes does not rule out the possibility that some interesting relations among holes be defined in terms of fillers. For example, one can say that a hole x is bigger than a hole y if and only if the canonical perfect filler of x is bigger (i.e., has a bigger volume) than that of y. Or one can assert that two holes x and y are isomorphicif and only if they have exactly the samecanonical perfect filler. However, one should be careful in generalizing this approach. It would be wrong, for instance, to say that two holes x and yare homeomorphic (i.e., topologically equivalent) if their canonical perfect fillers are homeomorphic. Obviously the perfect filler of, say, a perforating tunnel and a hollow may be homeomorphic (or even isomorphic). Yet we know that tunnels and hollows are all but topologically equivalent; i.e., an object with a tunnel and one with a hollow neednot be topologically equivalent. Likewise, a cavity might have exactly the same perfect filler as a hollow , though we know that cavities differ from hollows in that the former require two insulated surfaces in the host whereas the latter do not (figure 5.5). Moreover, the perfection of a filler is a relational property in the sense that one and the same filler can be perfect relative to some -hole - -- and imperfect (e.g., too big or too small) relative to some other hole. And , likewise, what is now a perfectfiller of somehole x may ceaseto be so later (e.g., if x gets bigger or smaller).

.a .-.

bI!

Figure5.5 A tunnel(a), a hollow(b), anda cavity(c) with the sameperfectfiller.

Fillers and Skins

As a consequence , it is clear that although some fillers can completely hole at the same time, no filler can do so in a perfect one fill more than way. A perfect filler servesonly one customer at a time. BatMan holes and reciprocal finers. That the difference among partial , perfect, and complete fillers is given purely in terms of the part whole ordering meansthat , according to our definition , even a perfect filler can have holes (though in a seasedifferent from that of figure 5.1). In fact this seemsto be quite intuitive . As an example, consider the two objects in figure 5.6. Here Bat has two holes, c and d (where c is part of d), and Man has two holes, a and b (where b is part of a). Man is a complete imperfect filler of both c and d; Bat is a complete and perfect filler of a and a complete (imperfect) filler of b. One can describe this pattern by saying that Bat and Man are two holed objects that fill (and fit ) each other completely . They are, as it were, reciprocalfillers; the holes they fill are in touch with one another. Of course one can think of many other patterns of reciprocal fillers, some of which are indeed very natural. Think for instance of an ice cube with a wedding ring frozen inside it (figure 5.7). The ring is a holed perfect Bat

Man Fiaure 5.6 A BatMan pattern involves complete fillers which can be more or lesscompletely filled.

Figure 5.7 An ice cube with a wedding ring frozen inside it .

arn

Chapter5

Figures . s

Reciprocalcompleteimperfectfillers.

filler of the toroidal cavity in the cube, while the cube is a holed but complete(imperfect ) filler of thehole(tunnel) in thering. Here, however , in contrastwith BatMan, the completeimperfectfiller of the ring's hole(the icecube) involvesan internalcavityand thereforehastwo surfaces . Figure5.8 illustratesanotherstrikingcase.Herea andb aretwo distinct objects- eachperforatedby a tunnel- that penetrateeachother. We can think of b asa complete(thoughimperfect ) filler of the holein a. But then, by the samepattern, a becomes automaticallya complete(imperfect ) filler of the holein b. As a generalprinciple, there are no two objectsthat are reciprocal , perfectfillers. The relationof filling a holepartially canbesymmetrical , in thesensethat thepartial filler of a holeh canin turn involvea holeh' that is partiallyfilled by thehostofh. (Justconsiderfigure5.8 andimaginethat both a and b involve an internal cavity just wherethey penetrateeach other's hole.) However, it is clear that the relation of perfectfilling is alwaysasymmetricandthereforeirreflexive.This is why snakescannoteat themselves . completely Walis, bridges, and other borderline fillers. There are also some cases where the above considerations seemto leave room for some ambiguity . For instance, take an object with a straight perforating tunnel and imagine " " building a wall of matter along the longitudinal axis of the tunnel so as to separateit into two halves, as in figure 5.9. Is that wall a partial filler of the initial tunnel? If so, a double torus should be regarded as a partially filled torus, a triple torus as a partially filled double torus (hence a partially filled torus), and so on- and all of that sounds somewhat counter-

Fillers and Skins

Fia8e5. 9 Buildinga wall in a tunnel. Is this a wayor partiallyfilling the tunnel?

intuitive. Shouldwe sayinsteadthat the wall splitsthe initial tunnelinto two newparalleltunnels? As wetakeit, the answerdependson whetheror not thereis a properfusionbetweenthe wall andthe object(i.e., on whether or not they merge- whetheror not the filler's matterdiffersfrom the

host' s). And, of course, the same would go for similar situations- walls and piecesof matter in internal cavities and hollows. Think of an hourglass . The glassdelimits an internal cavity, which is partially filled by the sand, but certainly the sand cannot be said to split the cavity into two cavities. ( This is linked to the fact that a filled hole is still a hole.) A related borderline notion is that of a bridge. Bridgesare different from fillers, but they servea function that is sometimessimilar to that of a filler: they restore a continuity on the surfaceso as to allow for a shorter path from one point on the edgeof a hole to an opposite point . They allow one to avoid the detour imposed by the presenceof the hole. However, insofar as a bridge is usually made of different matter than the hole' s host, building a bridge is usually not transforming a hollow into a tunnel. Finally , is an internal doughnut-shaped hole a cavity-tunnel, or is it a partially filled ordinary cavity? As we seeit, the question has now a perfectly " unambiguous answer. It just dependson whether or not the column " in the middle is one piece with the host. If it is, then the hole is a cavity-tunnel. Otherwise the column is really a distinct object partially filling aspherical cavity.

Why fdlen are so important. We are now readyto addressour initial questionof how we can get insightsinto holesby studyingthe waysthey canbefilled. Considera hole h with its perfectfiDerp. Let thosesuperficialparts of the fiDerthat are not in contactwith somesuperficialpart of the hole's

ChapterS

host be called free . And let a maximal free superficial part of pi .e., a free superficial part of p that is not a proper part of any other free superficial part of p itself- be called a free face of the filler . It can be verified that p is a perfectfiller of h if and only if it is a completefiller of h whosefree faces (if any) are all minimal (i.e., such that there is no smaller surface with the same boundary). From this fact, some interesting corollaries can be drawn. To begin with , we are now in a position to seethat the basic distinction among hollows, tunnels, and cavities, which in chapter 4 was formulated in topological terms, can be formulated exclusively in terms of fillers and free faces. An internal cavity has a perfect filler with no free face; a typical perforating tunnel has a perfect filler with at least two free faces(seefigure 5.10). This is in agreement with the criterion introduced in chapter 4, according to which there are cavities in an object if and only if the object has at least two insulated surfaces. This also suggeststhat, in the caseof the torus-shapedcavity-tunnel, the dominating character is the cavity, and the tunnel is receding(for it has no free faces). The fact that a tunnel' s filler has at least two free facesis also clearly ' " linked to the notion of a ring . In a sense, the filler s " web is a loop : one ' object s insulated surfaces: bow many?

/ """ ~2 cavity

<2

fillers

free faces :

how many?

/ ~ <2 hollow

tunnel

Fipre 5. 10 tree. : a classificatory Tellingholesby counting(sur)faces

Fillers and Skins

can imagine joining its free facesto each other by elastically pulling out the filler , as illustrated in figure 5.11. The filler 's web, the loop, can also be thought of as a sort of crystallization ofa plan for bodily motion . We sometimesplan how to move through objects and obstacles- and our movement follows an immaterial path. There may be somecomplications tied up with this suggestion. Take for instancea trefoil -knotted torus (figure 5.12). Topologically , it is equivalent to a torus (or to a cube with a cylindrical tunnel, just as in the previous example); but in this case it is not so immediately clear how we are to " " describewhat the web of its (perfect) filler would look like. Nevertheless, there is a preciseanswer: the perfectfiller of a trefoil -knotted torus consists of three solid tori glued together (it is therefore another caseof a perfect

Figure5.11 A loop maybethoughtof asthewebof a tunnel's filler.

Figure 5.12 What i~ the filler ' ~ web in a trefoil -knotted torus?

Chapter 5

filler with holes), and its web looks like a trefoil knot (or the projection of a trefoil knot on a plane). That a cavity' s filler has no free facesseemsto be another basic corollary of our account. In order to have a filler with no free faces, the object should completely envelop it . But the external surface of the object, by itself, would not suffice. Thus, for that purpose, the enveloping object needsanother surface, disjoint from the external one. It follows immediately that cavities, unlike hollows or tunnels, cannot be completely filled in an imperfect way: the only way to completely fill an internal cavity is to perfectly fill it . Infinite objects. An objection to the foregoing rules of thumb for counting holes presents itself in the case of objects infinitely extended in all directions of space(figure 5.13). An infinite object, like a hunk of matter " occupying the whole universe without discontinuity, can have no external " boundaries no " external" surfaces. , Consider now casesa, b, and c in figure 5.14. Can one conceiveof these entities in the infinite object as cavities, tunnels, and hollows, respectively?

Fia - e 5. 13 A portionof theinfiniteobject.

Fiaure5. 14 A ca~ity (a), a tunnel(b), anda hollow(c) in theinfiniteobject.

Fillers and Skins

We think so. A person could be trapped inside a, which therefore would qualify as a cavity regardlessof the infinitude of its host. Two persons could walk in opposite directions inside b until the end of time without ever coming to a wall or getting out; henceb would seemto be a tunnel in the infinite object. And peopleinside c could keep walking in one direction without ever coming to a barrier or to an opening: they would be in an infinite hollow , a hollow in an infinite host. The differencebetweena and c is that in the former case, but not in the latter, the walker will get back to the point of departure after a finite tour on a straight trajectory . To be sure, we are excluding here the caseof an infinite cavity. That is of coursea possibility, sincethe object is infinite. In that case, it is not true that the walker gets back to the starting point after a finite tour in a straight line, and this makesit impossible to differentiate an infinite cavity from a tunnel by the above criterion. If these intuitions are correct, then the rule for counting cavities by means of counting surfaces will not do; and the rule for individuating tunnels by counting rings will not do either. For an infinite object has no external complement, it has no external surfaces, and in casesa, b, and c the infinite object will have one surfaceonly . Thus, like b or c, a will not be a cavity - which is counterintuitive. Moreover, there will be no possibility of threading a ring through b, which will thus not be a tunnel- and this is counterintuitive too. One way out of the difficulty might consist in assigningto the infinite object an ideal external surfacelocated at infinity . Accordingly, the surfacesyou seein band c will be continuous with this infinitely remote external surface, and in the case of band c the infinite object will have only one surface. Not so in the caseof a. Moreover, in b you could thread a loop (albeit an infinite loop), thereby discovering that b is a tunnel. To pursue this line of objection, consider figure 5.15. Here a represents the longitudinal section of an infinite object with two parallel surfaces(a sort of thick plane extended in all directions). There are in fact two surfaces here, but no cavity. For if we regardedone of the two spaces- above or under the object- as a cavity, there would be no way to tell the cavity from the external space. According to our proposal, the object would on the contrary have just one surface, spreading to infinity , and a would presentonly two sidesof this surface. Caseb is dual (empty the full regions in a, and fill up the void ). Here there are two surfaces which seem to enclosea cavity. But as there are two distinct objects, there is no point in

ChapterS

FigureSolS Somemorepuzzle ! with infiniteobjects . discussing the presence of a cavity at all . This is an interstice between two objects , not a cavity . Of course , we could as well think that these mental experiments pose no new problem in comparison with other experiments that make use of the notion of infinity , and regard them as exceptions of minor importance .

Skinning fillers. Let us go back to our main theme. We have seenthat fillers allow us to take a complementary stance toward holed objects- a form of dual reasoning that can be quite illuminating . However, the importance of fillers does not lie only in their flesh, as it were; it also lies in their " skin," i.e., in their non-free surfaces. Figure 5.16 illustrates the basic behavior of the skinning map (labeled 0'), a function associatingevery hole with the skin of its perfect filler. Of course, the output of 0' is not topologically equivalent to the holed ' object. Rather, the mapping gives the topology of the part of the filler s surfacethat is in contact with the surfaceof the host (or , equivalently, of the part of the host's surface that is in contact with the surface of the perfect filler ). Indeed, the intuitive role played by 0' is more easily appreciated if we speak of a filler ' s skin as being the skin of the hole, meaning the skin associatedwith a hole (as we did when we first encountered skins in chapter 4). Furthermore, the classification of holes obtained by referenceto the skinning map 0' is congruent with the taxonomy introduced above, i.e., the classificatory tree based on the number of insulated/ fr ~ (sur)facesof the hole' s perfectfiller (figure 5.10). However, skins bring to light an important feature of our reasoningabout holes that topology alone could not dispel, and that the notion of a free face does not completely grasp either: they

alaqds

Fillers and Skins

..t'~ !

(

eJpe

(J ~

~ :

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Q

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I

ChapterS

put us in a position to get a first clear understanding of what goes on in the big family of all holes- of what is common to holes of different sorts. In other words, skins make us go back to topology with renewed and powerful taxonomic tools, but they also allow us to go beyond topology by regarding cavities and hollows as degeneratecasesof tunnels. If , in general, a tunnel with n mouths has a skin with n edges, then a hollow can be regarded as a degenerateone- edgedtunnel, while a cavity can be regarded as a totally edgelessone. (For more complex cases, the natural history of discontinuities requires more application .) Some additional remarks are in order. First , note that depressionsfit our scheme. It might be difficult to fix the exact skin of a depression; but once it has been conveniently fixed, it is easy to seethat it behaveslike a hollow 's skin. Second, note that talking about skins does not bind us to any hypothesis about the nature of holes. Someonecould say: 66Yousuggest that many properties of holes are explained by properties of their skins; why then insist that holes are immaterial bodies? Why not stick to their superficial nature?" This is an important remark- but the answer is straightforward . When we discusstopological properties of solid bodies, we need to talk only about the surfacesof such bodies. Yet this does not compel us to reduce- in any sense- solid bodies to their surfaces. Likewise , we use skins in order to sort and classify holes; but this does not credit the former with any onto logically significant role. The theory of skins and its importance. Skins give us such an effectiveand perspicuousdevice for a morphology of holes that all the theory could be obtained by a look at the above figures. The morphologicalcomplexity of a hole is the topological complexity of its skin. To appreciate this, look again at the variety of tunneled objects illustrated in figure 4.14. As we remarked in commenting on that figure, in generalthe topology of the object is not sufficiently fine-grained to allow a proper distinction betweenobjects having different holes. In this respect, object topology is far inferior to common-sensereasoning, which provides a clear and soundjudgment as to the nature and the variety of the tunnels of figure 4.14. Consider for instance the two objects reproduced in figure 5.17. Common senseand ordinary perception definitely say that in a there are two tunnels, each with two mouths, while in b there is only one tunnel: a ramified tunnel, with three mouths (IX, .8, ')I). On the other hand, object

Fillers and Skins

Fia8e 5.17 Two topologicallyequivalentobjectswith dift'erentholes.

topology saysthat there is no differencebetweena and b; the two objects are equivalent. Where does topology go wrong? The answeris simply that object topol ogy does not take seriously the structure of anything but the object. It keepsan eyeon the doughnut, as it were, but it neglectsthe hole. From the point of view of object topology , you can transform one object into the other. For instance, you can make b look like a simply by putting your finger through the opening and then pushing cSout of opening y. ( To visualize this way of moving tunnel mouths, take another look at figure 4.13.) One might object that when cSreachesy a border is crossed. This is true, but not from the point of view of the object' s topology . The only border you cross is the border of the hole: you cross the hole' s " surface," so to speak. This is' where we feel compelled to introduce the skinning operation, for it does justice to the idea that a hole can intuitively be provided with a surfaceof its own. This surfaceis now an object of proper classification, and the classificatory means is topology again- but this time in the right place. We have also said that the morphology of a hole is linked to a certain ' operation, namely filling. To describe a hole s skin is a way of keeping track of the operation of filling that hole. We wanted to link the hole to its perfect filler in a convenient way. In fact, we saw above that a description of the filler in itself, without further qualification, would not be sufficient to morphologically classify the hole, for a hollow , a tunnel, and a cavity ' might have the same perfect filler. Likewise, if we took the filler s global surface, its topology could be to a great extent irrelevant to the hole' s morphology (with the obvious exception of fillers for cavities, which have no free surface). By contrast, skins are just the non-free surfacesof fillers, which intuitively correspond to those parts of the host's surfacethat you

Chapter 5

cannot see once the hole is filled . And which surfaces are not free depends crucially on the relation between the hole and its filler .

Are there holesin Flatland? We have seenthat skins can do much, and they do what they can becauseof the edgesthey have. Now we can gain even more insight if we enhancewhat seemsto be a rather natural way of treating edges too (at least some of them) as holes, namely as holes in surfaces. A glimpse at figure 5.16 will suffice to illustrate this point . Skins becomeindependentobjects, and their intrinsic two -dimensionality immediately gives rise to a new array of questions: Does our theory of holes carryover to two -dimensional manifolds? Are there holes in Flatland? If so, how many kinds? Flatland is the imaginary world described by Edwin A. Abbott in his 1882 romance of that title (and of many dimensions). It is a flat world , inhabited by flat characters in the shapesof triangles, hexagons, and circles . We may think of Flatland as a two -dimensional world constituted either by a single surface or by several surfaces arranged on an ideal flat region (a world with no extension in the third dimension, although it might be extrinsically curved and admit of three-dimensional models: Flatlanders could set out on journeys and apply basic topological tools to find out , e.g., that their world is shapedlike a sphere, or like a torus). Are there holes in (or perhapson) such a world? Can the notions of tunnel and cavity that we inherit from a three-dimensional world be revised so as to make sensein such a pure two -dimensional environment? To begin with , let us make it clear that there is no such thing as a tunnel ' in Flatland if we consider Flatland s surfaceas the longitudinal section of a three-dimensional tunnel (figure 5.18). In that case there are two surfaces , not one, and hence there is no unitary, connected object to host a tunnel.

Figure5.18 Not a tunnelin (or perhapson?) Flatland.

Fillers and Skins

Second, the presenceof a cavity is linked to the presenceof at least two scatteredsurfacesbelonging to one object, but in Flatland no object can have two scatteredsurfaces. Here objects are surfaces. In Flatland , counting surfaces is- at the same time and by the same token- counting objects (if not worlds). Therefore, it seemsthat in Flatland there can be no room for cavities either. It becomesapparent that tunnels and cavities in Flatland would be ersatzentities. We can have tunnels only insofar as we can find a place for loops. And this is possibleif we consider a loop to be a ring that goesover two different sidesof a surfacewithout cutting the sameborder twice. This naturally leads to a requirement that is satisfied by the ersatz cavity namely, that the object have at least two distinct (scattered) borders. (In Flatland , a lower-dimensional world , linear borders play the role that in our world is played by surfaces: they define the boundaries of things.) The notion we need is in a sensea conceptual hybrid . We may term theseentities tunnel-cavities (to be kept separatefrom the cavity-tunnels encountered earlier). Tunnel-cavities (figure 5.19) are of a family close to three-dimensional holes. For they could be used to keep things at their place. In Flatland there can be fillers- plane fillers, as it were. Moreover, an object with a tunnel-cavity possess es two disjoint borders. This is the two -dimensional analogue of the three-dimensional property of having two surfaces- a distinguishing property of all cavities. Finally , note that an alternative criterion for identifying Flatlandian holes is also available- one that allows us to accommodate bidimensiona hollows in addition to tunnel-cavities. Holes involve concavities, we said. And the presenceof a concavity in an object x can easily be detected, as it dependson whether or not there exist two points in x such that the straight line connecting them intersectsthe surfaceof x. In chapter 2 we noted that in a three-dimensional environment one should not

loop

.i!a~vi tunnP

Figure5.19 Flatlandring, ring on two sides , loops, and tunnel- cavity.

Chapter5

infer the existenceof a hole from the presenceof a concavity: for instance, a valley is a concavity, but it is not a hole. Flatland is different. Here every valley is a hole- a hollow , to wit . In this lower-dimensional world there is much lessstructure than in ours. In particular , as we just saw, in Flatland linear borders play the role that surfacesplay in our world . If we take borders to play that role in the above criterion for concavities, we obtain a perfect one-to-one correspondence: for every hole there is a concavity, and for every concavity there is a hole. Flatlandian holes divide into tunnel-cavities and valley-hollows. Flatlands as infinite worlds. Corresponding to the three-dimensional infinite worlds introduced above, there is of course also a whole family of infinite Flatlands in which the various types of holes can be accommodated . Figure 5.20 showstwo examples. Here, as for the three-dimensional case, we could usethe convenient notion of a border located at infinity to cope with tunnels. In view of our earlier remarks on Flatlandian tunnels, in a two -dimensional world there is no interesting situation corresponding to the one depicted in figure 5.14b, where the longitudinal section of the tunnel yields two distinct surfaces. In Flatland every tunnel is a tunnel-cavity. What Flatlandian holes are. So far we have characterized Flatlandian holes from a purely formal point of view. But suppose that intelligent beings lived in Flatland (in Flatlandian surfaces, but not on them; the ' underlying speculative physics is left to the readers imagination). What special meaning would the notion of a hole have for them? There are, we think , two different possibilities. On the one hand, one could have it that the hole is in a surfaceand is something in which one

infinit ring :~=:) 0 ~-=

Figure 5.20 Holes on an infinite Flatland: an infinite hollow (left) and a tunnel- cavity with its infinite ring (right ).

Fillers and Skins

could be trapped. Accordingly, wherever there is a hole there is a portion of emptysurface, exactly as we could imagine that three-dimensional holes are localized at a portion of empty space. On the other hand, one could take the hole' s limits as the limits of space (as if , corresponding to the hole's limits , spacecould be at its end). The hole would then not be anything in which one could be trapped, for it is impossible to trespassthe limits of space. But such a speculation would lead us too far. In any case, it is easyat this point to envisagethe counterpart of immaterial bodies in Flatlands as the immaterial portions of the plane that constitute a Flatland. That is, they are the geometrical complement of any delimited plane. They lie on the abstract plane to which every concrete (material) Flatland also belongs. Holes vs. edges : More needfor morphology. The foregoing discussionwas prompted by the idea of studying skins as objectsof their own and treating edgesas holes. This treatment is rather intuitive and suggestsitself. However , there is some potential ambivalence in the last remarks: insofar as borders play the role of boundaries here, the identification of edgesand holes could lead someoneto question whether there is any fundamental difference between holes and boundaries tout court. As a matter of fact, one can generally deform a surfacein such a way as to exchangeinner and outer borders without affecting the surface's topology . As an example, consider the caseof a " pulsating" Flatland that keeps flattening and rising, shifting in shape from a plane disk (with an outer border and no holes) to a spherewith one edge(an inner edge, or hole, as in figure 5.16). Or consider the intruiguing caseillustrated in figure 5.21. Here we have a Flatland that keeps turning inside out , continuously reversing its outer edge(the " border" ) and its inner edge(the " hole" ). Now , one could rely on such examples to deny the existenceof Flatlandian holes. " There are no holes in Flatland ," one could say; " there are edges, and they mark the end of the world (the inner end, the outer endwhat differencedoes it make?)" We do not know and can hardly imagine what life on such a pulsating world could be like. But we certainly do not favor this way of looking at things. Surely a topological account must yield that conclusion. Yet that can hardly exhaust the matter. If every object in our three-dimensional world were made of rubbery stuff, then we would also be in a position to deform things in such a way as to turn them inside out. The hollow in a glasswould becomeits outside, and the outside

Chapter 5

Gi Q

-

~

7

~ u f

-

-

7

.

--.1-7W ~ [,-.~ 1 f-O

Fiaure 5. 21 A continuously reversing Flatland introduces some difficulties to the edgeJhole distinction.

wouldbecomea hollow. In generalthis wouldapplyto all sortsof hollows and tunnels, though not to internal cavities (which involve extra surfaces). But that is just one more way of realizing the importance of morphological concepts in addition to topological ones. The casesof the pulsating or reversingFlatlands are not much different. It all goesto show that we need morphology in Flatland too. Holes are edges, but once again topology fails to grasp some morphological distinctions that are important to the " hole theorist. (Of courseone could still object: What about the intermediate stages, when your edgesare halfway betweenholes and outer borders? How can you keep track of what is a hole and what is not in acontinu " ously reversing Flatland? Our answer is: We cannot except by appeal to some metric convention. We have a clear notion of what a clear case of a hole is, and we can tell when a given edge or pattern of curvature definitely is no longer a hole. But we cannot aim at completeness. Intermediate casesin continuous space-time are a seed of many difficulties. We encounteredthem before, and we shall meet them again.) Summary: Reasoningwith fillers and skins. This chapter was devoted to morphology. We argued that reasoningabout holes can to someextent be reducedto reasoningabout their fillers, and we tried to take the notion of filling in a most seriousway. There are quite a number of featuresof fillers, and of explanatory uses we make of them, that could be invoked to support the thesis that holes are immaterial bodies. We excluded that holes are just their own fillers;

Fillers and Skins

' still, we like the idea that the filler s ghost- the immaterial body spatially coincident with the filler - is the hole. Our reasoning about a hole is a ' reasoningconcerning the hole s potential filler . When it comesto holes, we take a complementary stancetoward the holed object: we consider it from the filler ' s point of view; we look for the place that is outside the object' s stuff but still inside the object; and we bestow life upon the filler ' s ghost. Much of this chapter was devoted to an examination of the outstanding ' properties of a hole s fillers and of their morphological interaction with the hole' s hosts. We also usedfillers to provide a first structured classification of holes, and to extend it to some limit cases(such as infinite objects and bidimensional worlds). But above all , a key role was played by the fillers' surfaces- notably by the non-free surfaceof the hole's perfect filler , which we called its skin. The topology of the skin, we saw, reOectsthe morphol ogy of the hole. In the next chapter we will explore some finer-grained interconnections.

6

De Natural History of

Do wehavea clearnodODof internalcracks? As we submitted,holesare . A crack is a discontinuitythat is very strictly relatedto discontinuities muchlike a hole. In fact, onecouldthink of a crackasa sort of verythin, purely bi-dimensionalhole; and whenthe crack is completelyinternal, it would be a sort of very thin, bidimensional, flat cavity. Do not think of somethin air layer insertingitself into the structureof an object. This would, in fact, presuppose the presence of a crack. Weshouldbetterimagine a crackasbeingdeterminedby two facesof a singlesurfacewhichare in touch with eachother (call themcontactfaces) and whichjoin at their borders."Join" is hereto betakenbroadly, moreor lessin the samesense in whichwesaythat two facesof a cubejoin alongtheedge. Again, surfacesgive rise to generalproblems. The notion of a crack (an exampleof surfaceparticular) defiesintellegibility, and is not easyto accommodate within the theoriesof holeswe haveintimated. On the one side, cracksare not just theseinternal two-facedsurfaces ; a surfaceis a surface , whereasa crack hasa surface . On the other side, cracksdo not seemto beimmaterialbodieslocatedbetweenthe two contactfaces ; there is no room for anythingbetweentwo facesthat arein contact. Cracksare germaneto holes, but their natureis evenmorepuzzling.Nevertheless , we shallseein this chapterthat an explorationof theseand other borderline superficialities mayshedlight on the studyof holesproper. Breakingpoints. We shall start by assumingthat an internal cut in an object(for instance , in the innermostpart of a marbleundergoingoverwhelming a newsurfacewhosetwo facesarein contact pressure ) generates " with eachother. Wherethe two contactfacesjoin, thereis a " filamentous crack: a breakingline. Likewise,theendpointof a filamentouscrackwill be " a breakingpoint. It is ratherintuitive to assigna " complexity magnitude to breakingpoints: the magnitudeof a breakingpoint x is the numberof breakinglinesoriginatingin x. ( Thisnotion of magnitudeis reminiscentof Brentano's notion of plerosis .) Thus, for example , a point at oneend of a

Chapter6

breaking line will be of magnitude 1, while a point somewherearound the middle will be of magnitude 2. In a similar way, a breaking line can be assigneda magnitude, corresponding to the number of non filamentous filamentous An isolated cracks that intersect in the line (see figure 6.1). crack is O-:breaking. Here, again, the notion is so strange as to bethreat ened by incoherency. One does not get breaking points by an operation resemblingthe deletion of a point from a continuous manifold; that would leave an open manifold, whereas here we tend to think of a breaking point as enveloped in a sort of punctual surface. (Using an atomistic, non- continuous space, such as a space with spatial quanta, will not do either, for that leavesunsettled the question of the edge betweenthe two contact faces.) By contrast, if one imagines blowing air into an internal crack so as to make something out of it - something clearly recognizable ' as a cavity - one can see that the crack s border turns into an obvious join, or into a line on the surfaceof the resulting cavity. Whenever two cracks intersect, they produce four (bilateral) comers, which meet at the 2-breaking line defined by the intersection. Three perpendicular cracks intersecting in one point define three 2-breaking lines (figure 6.2). In such a point , eight (trilateral ) comers touch one another. I -b~ hn ~ point

I -breaking line

2- breakingpoint

2-~ ~ n~ line

FJa8' e 6. 1 Breaking points and breaking lines (filamentous cracks).

6- breakiDgpoint

-

2-b~ - ki Diline

Fi88e6.2 cracksprod~ a 6- breakingpoint Th~ intersecting

The Natural History of Discontinuities

The number defining the complexity magnitude of our entities is independent of the number of their spatial dimensions. We could generalize and say that the relative magnitude of any entity is the number of objects of a given dimension of which it is a limit . For example, on this account both the endpoint of a line and the face of a cube have magnitude 1; a surface that cuts a solid, with both sides open, has magnitude 2; and a point (any point ) on a surfacehas infinite magnitude, as does any point in a solid. On the other hand, if we defined the magnitude of a point as the number of geometric lines going through it then every point on the surface of a crack would have infinite magnitude. By contrast, our restricted notion of magnitude allows us to introduce the somewhat dual notion of a 0breaking point , which is nothing less or more than an isolated pointlike filamentous crack. The question then ariseswhether theselimit casescorrespond to any reality . We are inclined to think that they do. The process whereby a crack is formed must begin with the formation of a O- breaking point or of a O- breaking line. Moreover, the two endpoints of a O- breaking line are I -breaking. We can imagine that the line shrinks into a point : the I -breakingnessof the two endpoints collapsesinto the O- breakingness of a single point . Conversely, a O- breaking point can split and its two I -breaking halvesgeneratea O- breaking crack. Moblus cracks. The notion of the magnitude of a filamentous crack is open to further qualificati.ons. Take a crack in the shapeof a Moblus strip (intuitively , a one-sided surface obtained from a ribbon by applying one end to the other after rotating the latter through 180 degrees). In the presenceof such a pattern (figure 6.3) one cannot speak of two distinct faces touching each other, but one can speak of a single surface with a single face, two different parts or sections of which are permanently in touch. Note the difference from, say, the case of a ribbon in which two parts of the samefaceare in touch. In a Moblus crack two opposite parts of the same face are permanentlyin touch- they cannot be separated. In this case, the nature of the edgebecomessurprisingly unclear: according to one intuition , the edgeof a Moblus crack must have magnitude 2; on the other hand, according to the definition its magnitude has to be 1. Perhaps one can generalizeand say that an edge has magnitude 2 if it divides either two facesof the samesurface(as with the edgesof a cube) or two parts of the sameface(as with a Moblus crack). In this case, however,

Chapter 6

Fia8e6.3 Whatis the maanitudeor theedgein a Mobluscrack?

-

-

crack

~

(ftUed )I~ vity

Fipre6 . 4 A closedcrack (spherical ],and a filamentouscrack(circular). ODe caD DO1t get rid a fare -

of the notion of an edgeby appealingto the notion of

Thus, questions of magnitude must be settled by recognizing that magnitude is a relative concept: onceyou choosewhat it is relative to , and only then , you get a definite answer .

aosed interal cracks . Dd interal holes. A closed internal filamentous crack (that is, an internal fdamentouscrack whosetwo breaking endpoints are made to coincide) definesan internal edge. Linear cracks, if you make them shift, can generate cracks and can limit the cracks so generated ). Call these (imagine the shift of a nylon thread used to cut soft cheese line-generatedcracks. (Seefigure 6.4.) When a crack reachesthe external surfaceof an object, it becomeswhat we call a fissure (seefigure 6.S). Imagine now that we have a closed filamentous crack (a filamentous edge) completely on the surface of the object : a surfaceedge. By generating a closed fissure (i.e., a fissure with no other boundary line than a surfaceedge), a surfaceedgecan line-generate a hole: a hollow , to be precise, and a perfectly filled one. Likewise, two

cl

The Natural History of Discontinuities

point edI8

fl18meat0 U8crack

1Urfaceedi8 ft88UN

hollow

tuDD8l d8i8D8ratetunnel (. d8l8n8ratehollow)

tuDDe1 degenerate (. d8i8Der8te cavity)

internal ft88Ure

~ vity

internal crack (and backto a point)

Fiawe6.5 Frompoint edgesto internalfilamentouscracks:a briefhistoryof discontinuities .

Chapter 6

non-intersectingsurfaceedgescan line-generatea tunnel, while an internal edgecan line-generatean internal cavity . The interesting thing to observe is that by the suggestedpattern one , as figure 6.S illustrates. A can also line-generate all intermediate cases contained in it can linenot surface a surface edge and breaking point or generatea degeneratetunnel (something betweena tunnel and a hollow a this makes a cavity). The presenceof an isolated surfacebreaking point I - degeneratetunnel, or a degeneratehollow ; the presenceof two such points makes it a 2- degeneratetunnel or hollow , or a degenerateinternal cavity. . As an internal fracture has two facesand one surface, Other Umit eases it must be considered a cavity, albeit a degenerateone. In fact an object , thereby satisfying the fundamental with an internal fracture has two surfaces criterion for the presenceof a cavity (casea in figure 6.6). Suppose you completely close the internal fracture by joining its borders (as in figure 6.4, left). The surfaceit definessplits'into two surfaces, corresponding to what were previously the fracture s two faces (its faces become ' surfaces). Now there is still a cavity, but it is perfectly filled. Don t be misled by the fact that there are now three surfacesin the scenery, because one surface does not belong to the object any longer: once the fissure is completely closed, there is no longer oneobject; there are two, the host and the filler , and one surfacebelongs to the filler. In the samefashion, caseb ' in figure 6.6 depicts a fracture that reachesthe object s external surface. It is in fact a kind of (very thin ) hollow . When the fracture is closed, as in case c, there is another hollow taking the place of the previous one. The two

FJa8e6.6 . Otherfactsaboutinternalcracks

The Natural History of

facesof the fracturesplit into two surfaces , one of which now belongsto the hollow's filler. The elementof magicin the generationof a newhole and the elimination of anothershouldnot surpriseusany morethan theelementof magic in creatingtwo new objectsby splitting a former one. This is consistent with the ideathat what is a fracturein b is no longera fracturein c. In c there is somethingdifferent- not a fractureseparatingtwo objects , for thereis no suchthing. A fracturebetween two objectsis a non-entity. Internaleraeksandyolumeless eavities . We are inclinedto favor the following intuitive principle: A holedobjectand a hole-filler haveto be two distinct entities, though their surfaceshaveto be in contact. Therefore , whenevera closedinternal crackgenerates a hole, it doesso by creating two objects:a hole-hostand a hole-filler. Now imaginean internalcrackdefinedby a concavefaceand a convex face, as in figure 6.7, left. ( The drawing could be misleadinghere; the sectionof the crack is shapedlike a U, not like a C.) Doesit generatea hollow? If it does, thena singleobjectwill be the (complete , but of course imperfect ) filler of a holein the object. That is, hostandfiller will coincide. But we couldalsoconsiderthesetwo facesastrappingan internalcavity, a perfectlyvolumeless cavity. In this casethereis no filling at aU. Though commonsenseseemsto suggestthe first alternative(the crackgenerates a hollow), wefavor thelatter; only in this waycanweaccountfor the possibility that the two facesof the crackmoveawayfrom eachother(thereby generatinga canonical , voluminousinternalcavity). Likewise,if the internal crackis shapedlike an annulus,or evenlike a Moblusstrip (look again at figure 6.3), we can speakof a volumelesscavity-tunnel: blowing it up

internal crack

Fiaare6.7 A hollow-shapedcrack(left) anda Klein crack(right).

internal crack

Chapter 6

would generate an internal hole in the shape of a doughnut (which we already know qualifies as both a cavity and a tunnel). Imagine now an internal crack in the shapeof a Klein bottle (figure 6.7, right). Intuitively , a Klein bottle can be obtained by passing the narrow end of a tapered tube through the side of the tube and Oaringit out to join the other end. ( Theself-intersection is disturbing, but there is no other way to embed such a surface in three- dimensional space.) We get a peculiar bottle, with only one face. How many objects do we have here? Again the answer prima facie seemsto be: one object, which is the complete and imperfect filler of a hole in the object itself (viz. , the inside of the bottle). But the foregoing remarks extend to this caseas well, and suggestthat we may also take this as an internal volumelesscavity. In fact intuitions get fuzzier. Yet this account becomesevenmore compelling, in view of the fact that a Klein bottle can be obtained by gluing the edgesof two Moblus strips. If we regard internal Moblus cracks as cavities, albeit of a very degeneratesort, then the result of gluing up two of them must also yield a cavity. Finally , how many faces does a sphere with an internal crack have? Recall that if the crack is closed there are two objects, one (the host) with two facesand the other (the guest) with just one. If the crack is not closed, then the sphere has three faces; in this case, the perimeters of the two touching internal faces coincide with the 2 breaking closed filamentous The . their them crack that generates sphere has two faces if the edge) ( has two faces if the closed it . And non-closed crack is Moblus shaped crack is Klein shaped.

7

Parts and Holes

An old children' s riddle tells the story ofa fellow who dug two large holes in four hours of hard work. How many holes did he dig in one hour? " " " Fooled you if you answer half a hole - so goes the answer- for there is no suchthing ! He dug a hole: a small one (smaller than the one he would end up digging), but a wholehole." What would you have answered? To be sure, the key to the solution is not the dubious existenceof such a thing as half a hole, but the impossibility of digging one: one can only dig an entire hole, not a part thereof. Be it as it may, the story is instructive, for it is indicative of a more general problem that has somebearing on our study. Do holes have parts? If they do, how can we account for the relationships among holes and their parts? For instance, can you take away a part from a hole? If you do, what remains? Theseare legitimate questionsthat we have already come across at various stagesof our study (for instance, in our discussion of partial and overlapping tunnels in chapter 4). We shall now addressthem more directly - if not to come up with a complete account, at least to impose some order on this apparently difficult issue. Mereology (from the Greek ' ' jJ P~ , part ) is the theory of the part -whole relation. One of our main tasks is to see how- and how satisfactorily- the previously discussed theories of holes interact with me ol.oglcal facts and principles. ~ ~ ? There are two distinct ways of approaching What is the mereologyof 1Io1 the problem. On the one hand, we can reason about part -whole relations as they apply to surfaces, for holes are in an important sense superficial particulars. On the other hand, we can also consider parts of holes that do not reduce to surfaces. Indeed, the very notion of a partial filler immediately brings to mind the concept of a voluminous part of a hole. Qearly this secondapproach does not fit the conception that holes are nothing more nor less than superficial particulars, but it becomes appealing as soon as we think of holes as immaterial bodies.

Chapter7

Let us begin with the first account. It is evident that reasoning about surfacesallows us immediately to answer in the affirmative the question " " Do holes have parts? Surfaceshave parts ( potential parts, at least), and theseparts in turn have parts; therefore, if we think of a hole as a superficial particular (i.e., as a superficial part of a material object), then the parts of that part are by the same token parts of the hole- at least, potential parts. Moreover, from this perspectivethe important question of what are the basic principles governing the relationships among holes and their parts also appears to receive a fairly clear answer, for the mereology of superficial parts (i.e., the theory of the superficial-part-of relation) reduces to a great extent to a simple mereology for two -dimensional objects in general. In chapter 2 we noted that there are various complications stemming from the very notion of a surface, but these are issuesthat do not depend specifically on the fact that we are interested in the surfacesof holed objects (although they could have some bearing on the theory of holes one favors). As long as we give surfacesa geometrical (rather than physical) character, it seemsquite natural to consider the relationships betweenparts of surfacesas representedby standard mereology for two dimensional objects. Note incidentally that a mereological perspectivesuggestssome interesting links betweenthe notion of holes as constituted by superficial parts that is, holes as superficial particulars, parts of surfaces) and the only ( Ludovician conception of holes as hole-linings. Surfacesare parts of the objects they bound. A hole as a superficial particular is therefore a part of all hole-linings that , according to the Lewises, constitute a particular hole: it is the least upper bound, relative to the part -whole relation, of the equivalenceclassof all hole-linings constituting the hole. Now consider the second and more important possibility mentioned above: holes have parts that are not, or do not reduceto , superficial parts. Here it is interesting to introduce a further distinction betweenthe notion of a part tout court and that of a part which is itself a hole. Such a distinction could also apply to the discussion of mereology as exclusively concerned with superficial parts, but in that caseno specific puzzlesseemto arise. (Some puzzlesdo arise, as we shall shortly see, but they seemto be common to all theories of holcs.) By contrast, the distinction becomes interesting and somewhat more problematic on the view that holes are immaterial bodies.

Parts and Holes

If we consider parts tout court (i.e., parts of holesthat are not necessarily holes themselves ), then again the relevant account seemsto go in the direction of some standard mereology- in this case, the mereology of three-dimensional bodies. In other words, immaterial bodies have immaterial parts, parts that are immaterial bodies, and the class of these presents no mereologicalpeculiarities. If , on the other hand, we are interested in those parts of a hole that are themselvesholes, a different treatment is in order, for clearly holes are immaterial bodies of a certain type and may therefore have parts that are not themselvesholes. We have already pointed out that not every immaterial body is a hole. The question is: Which parts of a hole, if any, qualify as holes? A caveaton " standard" mereology. Before going any further , let us clarify ' what we mean when we say that the notion of a hole s part tout court does not presentany mereological peculiarities, whether we look at holes superficially or as immaterial bodies. We are in fact suggestingthat some basic principles that have been set forth in the construction of axiomatic theories of parts and wholes (such as Lesniewski's " Mereology" or Leonard and Goodman's " Calculus of Individuals" ) continue to hold for holes, though not necessarilyfor parts that are themselvesholes. Actually it seemsthat holes are even more standard , for they approximate to , mereologically, than material substances " volumes of space. To illustrate, let us consider the relation is a (proper) " " " part of (we shall omit the qualification proper unlesswe need to rule out that the part coincide with the whole). Then the following are general mereological principles whose validity seemsto be preservedwhen x , y , and z are allowed to range over holes: (a) Everything is part of itself. (b) Ifx is part ofy and y is part of x , then x = y. (c) If x is part of y and y is part of z, then x is part of z. (d) If every proper part of x is a part of y, then x is part of y. The first three of theseprinciples assert that the part -whole relation is a partial ordering: a reflexive, antisymmetricand transitive relation. The fourth is an independent thesis asserting that the relation is projective in the sensethat inclusion of all parts entails inclusion of the whole. This

Chapter 7

Fipre 7.1 Ahollowy ) with a smallerhollow(x) asa properpart.

already implies that we take it as possiblefor a hole to be a proper part of another hole. For instance, a discontinuity in the lining of a hole can, under certain circumstances, qualify as a hole in its own right (though one that is also part of a bigger hole). A paradigmatic example is depicted in figure 7.1, where we take the square part at the bottom of the hollow to qualify legitimately as a hollow itself (a hole in a hole, as it were). We shall come back to this and similar patterns shortly . Other interesting consequencesof the above principles are worth mentioning. For instance, principle e below immediately follows from principles b- d: (e) If x and y have exactly the sameproper parts, then x = y. This principle corresponds to the set-theoretic axiom of extensionality, and one might be tempted to regard it as a natural identity criterion for holes (as long as one takes " part " and " proper part " in a broad sense; surely the criterion would be inadequate were the relevant notion of part to range over holes only , for that would , e.g., equateall holes that have no holes as proper parts). We shall come back to this and related issuesin chapter 9, where we shall fully addressthe problem of identity . For the time being, what is to be noted is that most of these " standard" mereological principles- and - hold unproblematically only from an extensional their consequences perspective(that is, only if parthood is interpreted relative to a fixed instant of time t, and ' = ' as the relation of spatio-temporal coincidenceat t). This is a caveatthat many authors have put forward or would put forward when considering the mereology of material bodies. Think of Heraclitus' remark that " you could not step twice in the sameriver" (fragment 41), or think of the ship of Theseuswhich " the Athenians were constantly repairing " , as Leibniz put it (New Essays, II -xxvii-4). From a purely extensional

Parts and Holes

~ ~ ? ~ 00 -@ ~@ ~ 00 -E IE time

~

Figure 7.2 Two holes in the same piece of rubbery stuff , both with the same shape and volume , moving in opposite directions .

' perspective, using such principles as a- e above, it becomesdifficult to account for the fact that such things survive through time- that they are continuously changing (growing or getting smaller, losing some parts or acquiring new ones) and yet remain the same. Well, the sameapplies when we come to consider immaterial bodies such as holes. Think of a spinning ' whirlpool in Heraclitus river. Or - to remain closer to our customary examples- think of one of the holes dug by the fellow in our children's riddle. Did he actually dig one hole, making it bigger and bigger until he got the desired size, or did he dig a sequenceof pairwise different holesone at eachshovelful- each bigger than its predecessors ? Surely we would stick to the first, common-sensealternative. Just like material bodies, immaterial bodies can develop. They can changesize and shape- or one can changetheir sizeand shape. Insofar as they are immaterial bodies (bodies that can be completely penetratedor occupied by other bodies), it is conceivablethat two distinct holescoincide (i.e., occupy exactly the samespatial region) at someinstant in time. Consider two holes, h' and h", both with the same shape and volume and both in the same piece of rubbery stuff, moving along the same trajectory toward each other. (This example expands a made by Kit Fine in conversation.) Eventually the two holes are going to meet; in particular, there has to be a time t when h' and h" are perfectly co-localized (i.e., overlap completely with each other) beforecontinuing on their way. Mirabile dictu, that is made possibleby their immateriality . Yet that does not prevent the two holes from preservingtheir identities (figure 7.2). (One might perhaps say that at time tl < t h' and h" are two distinct holes; that at (or around) time t h' and h" ceaseto exist as they together, giving birth to a new, single hole h; and that at some time t2 > t

Chapter7

00 tl

@ ' 2

e ta

00 tn

~

Figure7.3 of superficialparts? An exchange

h splits into two distinct , new holes, ho and hoo. We find this alternative view tenable too, but lessintuitive . For how could we then account for the fact that , say, the hole in a doughnut does not ceaseto exist as soon as we take the doughnut into the Mont Blanc tunnel? It is misleading to think that holes behavelike drops of water; they are, rather, like ghosts- immaterial bodies that can be completely penetrated and can penetrate one another). The foregoing provides a good test casefor a comparison betweenthe Superficial Theory, according to which holes are nothing but superficial parts (theory 1, according to the labeling introduced in chapter 3) and the Immaterial Theory, that has holes as immaterial bodies (theory 3). Consider the behavior of superficial parts (figure 7.3). It seemsthat by fusing together and then splitting again, the holes exchanged their respective superficial parts. This is simply impossible according to theory 1, which can hardly account for this situation (one would hardly speakof two holes bumping into each other and bouncing back, or anything like that ). By contrast, the problem does not arise in theory 3. In this case one may perhaps be worried about the fact that two bodies penetrate each other. But then again, they are immaterial bodies, and we must take this fact seriously.

. We . Let us then focuson extensionalhole-mereology Parts andfiBers of some that havesaidthat holescan haveparts, and we havesuggested , a hollow can havea thesecan be holesin their own right. For instance featureof holes most a . Since a as distinguishing subhollow proper part to want now we first the investigateis is their fillability, one of things matters by looking at the partwhetherwecangainanyinsightinto these whole relations of the fillers i.e., at whetherthe mereologyof holes ) fillers. (perfect mirrors in somewaythe mereologyof the corresponding We can 7.4. in 7.1 of Take againthe pattern figure , reproduced figure the be . Let a is a x perfect maintainthat x and y areholes, and part of y

Parts and Holes

.

-

c part of (J

% part of y

b part of (J

... part of y 'i

Fiaure7.4 Doesthemereologyof holesmirror the mereologyof hole-filleR?

filler of the entire hole y, and let c be the perfect filler of x. Clearly, c is a " " part of a, and so is the remainder, b. But what exactly is b? On theory 1, it is not clear how the relation betweena and b can be taken to mirror a corresponding mereological relation betweeny and a part thereof. b is a partial filler of y, just as c is. Unlike c, however, b cannot be said to fill a part of y; whereasevery non-free superficial part of c is a superficial part of a (whence we can infer that x is a superficial part of y), some non-free superficial parts of b are not superficial parts of a. (Recall that a superficial " " part of the filler is called free if it is not in contact with the host's surface.) Thus, it seemsthat on theory 1 we can restore a fair harmony between part -whole relations among fillers and part -whole relations among holes only by restricting the domain to those parts of a hole's filler that have actual superficial parts; thesesuperficial parts would then define holes that are proper parts of the main hole (i.e., the hole relative to which the filler is defined- y in our example). This restriction is also useful on theory 3- according to which holes are immaterial bodies- when we set ourselvesto account for the mereological relations betweena hole and those parts of it that are themselvesholes. However, on this theory these are not the only " legitimate" parts a hole can have, and we therefore have a perfectexplanation of what b is: It is just a partial filler of y to which there correspondsa proper part of y, albeit one which is not itself a hole. This account can easily be generalized. Consider for instancethe following " argument: The dashed regiony in figure 7.S is not a hole, but it may become one if we partially fill hole x while leaving exactly that part " empty. If we are concerned only with superficial parts (theory 1), the argument is plainly incorrect: that region is not a part of the hole but a ' potential part of the hole s (perfect) filler (i.e., a part of its potential filler ). On the other hand, and for the very samereason, the view that holes are

Chapter 7

Fiaure 7. 5 A hole (x ) with one of its potential parts (y).

Figure7.6 littleholeinsidea biggerone.Is thefonnerpartof thelatter? immaterial bodies (theory 3) allows us to say that y is a part of x , albeit a potential one. It is not a part that qualifies as a hole in its own right , but . it is a perfectly legitimate part nevertheless At this point one could, of course, ask further questions: Is it appropriate to speak of y as a potential hole? What prevents us from seeingpotential holes in every part of an actual hole? However, thesequestions tie in with some more general issuesconcerning the conditions under which we seeor have the impression of seeinga hole. We shall treat them diffusely in chapter II , which is devoted explicitly to hole detection. Nested boles. Let us now consider more closely the view that holes can have holes as proper parts. This view is suggestedby our rudimentary intuitions concerning the notion of a hole as connectedto a discontinuity and to certain superficial gestalts, and is further supported by our reasoning about holes in terms of partial and complete fillers. As we mentioned at the beginning, however, some puzzles immediately arise from it . Is a given hole part of another hole? Does a given part of a hole qualify as a hole itself? Let us begin from the first question. Imagine a hollow x in an object a, and supposeyou introduce an object c in it that has ahollowy of its own, as in figure 7.6. (Think , for instance, of putting a hollowed chunk of

Parts and Holes

Gruyere inside a bigger hollow in a bigger pieceof Emmenthaler.) We can " " speak of thesetwo holes as nested holes. Can we say that one is a part of the other? That is, is y a part of x? If we consider only superficial parts (theory 1), then the answeris simply and quite definitely in the negative: Althoughy lies inside x , it is not a part of x , for on this reductive theory a necessarycondition for a hole y to be part of a hole x is that every superficial part of y be also a superficial part of x - a condition that in the present case is plainly violated. On theory 3, however, the situation is more complex. It is clear that the space of which y is made is part of the spaceof which x is made, and that object c is in hole x. One suggestionis that , sinceholesare immaterialy is simply part of x. Alternatively , we can maintain that , for the very same reason (holes are immaterial bodies and can therefore be penetrated by other bodies, be they material or immaterialy is not a part of x ; rather, it spatio-temporally overlaps a part of x. Let us look at some consequences of thesealternative accounts. Does the introduction of c in x modify the shape and the size of x? Yes, for there is less room left for operations inside the hole; no, because we still want to say that a filled hole is a hole (c partially fills x , but that does not affect x ). If you take the first alternative and accept that x gets modified, you will find it understandable to say that y is part of x. The form of the immaterial body x is now such that it has a small indentation: y . If you take the secondalternative and leave x unmodified by the presence of c, then parthood vanishes; nesting and parthood becometwo independent notions (as they usually are, e.g., in the caseof material objectsthe table is in the room, but it is not a part of the room ). By denying that y is a part of x , we do not mean to deny that by filling we would also fill part of x. Just as c is a partial filler of x , the result of y filling y would be a bigger such filler . It would be a scattered or a nonscattered partial filler of x , depending on whether the new piece (then a filler of y) is made of the same stuff as the old one (the host of f ). The mereological relations among holes are representedin the mereological relations among their fillers, but not vice versa. Supposenow that c is not still but rather floats inside the hole of object a (figure 7.7). As c moves, its hole, y , moves as well. Can we say that there is a part of x that moves? Take the first account again (x modified by c's ). Insofar as there is genuine parthood , y is part of x and henceis presence a movingpart of x. But how is this moving part related to other parts of x?

Chapter 7

a a time

~

Filure 7.7 Is holey a movingpart of holex?

Figure7.8 How doesthe fusionof two holedobjectsaffectthe two holes?

One answer could be that y coincides with gradually different parts of x. That is, the movement of y actualizessome potential parts of x. But what if we now take c (along with its hole y) out of hole x? Shall we then say that a part ofx has floated free and acquired a life of its own? Shall we say that hole x is now a scattered individual , one part of which has left the main body (perhapsforever)? If we consider theseproblems, then the alternative view seemsto provide us with a better (or perhaps more natural, less exotic) account. On this view, y is not a moving part of x - it is not a part at all. It is simply a hole that overlaps (spatially, not mereologically) some part of x. And the fact that it moves is simply explained by saying that it occupiesdifferent parts of x at different times, just as c penetratesdifferent parts of x at different times. We can make our example even more complicated. Supposethat objects a and c are made of the samestuff (for instance, they are both pieces of Emmenthal cheese ), and imagine that at a certain point they meet and fuse together as illustrated in the leftmost diagram in figure 7.8. Can we still deny that y is a part of x? Well , as the good theory has it , the answer is that we cannot. Now y is a part of x ; it becameone when its host fused

Parts and Holes

with x ' s host. In fact the problem is related to the one discussedin chapter 5 and illustrated in figure 5.9. In caseof fusion with the hole' s host, a filler ceasesto be a filler and becomespart of the host. Well then, if the filler has a hole, this hole becomespart of the host's hole. Suppose, however, that c slidesdown as indicated in the middle diagram in figure 7.8. The outcome is a perfectexampleof a hollow 's being part of another hollow (as in figure 7.1), and sure enough we can easily endorsethat: y is a hole that is a part of another hole. But of which hole is it a part? Of x (our initial x , that is)? Has x becomebigger, acquiring a new part, as it were? Or shall we say that we now have a new hole, z, which is the mereologicalsumof x and y? And what if , instead of sliding down and disappearing, crises like a cake, as illustrated in the rightmost diagram? Is y still a part of x? Is x actually still there? At this point , it appears that we need more instruments to deal with theseproblems (and with related ones: e.g., what if c starts rotating clockwise , while a rotates counterclockwise?). We need some theory of the movement of holes, and someway to consider the role played by causality in the movement of objects. On pain of leaving some questions unanswered , we therefore postpone our analysis to the next chapter. A refutation of bole-monism. The above remarks on nesting and parthood , although incomplete, suggestan argument against the position that can be labeled " hole-monism." That position can succintly be describedas the theory that there exists exactly one hole. On that theory there are plenty of objects, and many of them seemto involve one or more holes, but in reality there exists only one big hole; everything else (you, the planets of the solar system, the totality of all material objects) is but an assortment of partial fillers of that hole- or, if you prefer, they all add up to a scatteredpartial filler of that hole. (Recall that we talked about scattered fillers in connection with figure 5.2. ) Now , what is wrong with this theory is simply that regarding every object as a partial filler (or part of a scattered partial filler ) of a big hole doesnot make the latter the only hole. Someobjects involve holes, and we have seenthat thesemay be viewed as true holes even if they are entirely located inside some other (bigger) hole. Whether or not we take them as parts of the big hole or as independent holes that coincide spatiotemporally with parts of the big hole, still they exist as holes. Hence, the theory that there is only one hole is false.

Chapter 7

An intuitive example will further illustrate this point . A " model" of the hole-monist theory is provided by a cave inhabited by people. The cave they live in is the Hole; everything else inside the cave is there to fill it partially . This may very well be the case. But if an inhabitant of the cave has a hole in her pocket, is that anotherhole? Surely that little " hole" is, or coincides with , a part of the big hole. But is it also a hole? It is. And the proof is simply that the day our friend finds her way out of the cave she will not ipsofacto lose the hole in her pocket. Problems with location. The preceding example indirectly introduces us to an additional problem that we must addressexplicitly in our investigation of part -whole relations. This problem relates to our earlier remark that a hole is always in something. You have a hole in your pocket, and you thereby have a hole in your trousers. But as you hang up the trousers in the closet, you do not thereby get a hole in the closet. Holes are superficial entities. In order to be in the closet, a hole has to affect the surfaceof the closet. But its affecting the trousers' surfaceprevents it from affecting the closet' s surface. By contrast, the hole is superficially linked both to the trousersand to the pocket, which explains why in that casethe transitivity of the in -relation is preserved: A pocket shares a part of the trousers' surface. Now , one can also think that it is not the superficiality of holes that per se blocks the transitivity of the in -relation; it could simply be the fact that what matters hereis parthood. The pocket is a part of the trousers, and this is why a hole in the pocket is a hole in the trousers. The pocket is not a part . of the closet, and this is why a hole in the pocket is not a hole in the closet. Or so one could argue. But something more intricate is going on here. The pocket is a part of the trousers. Now the trousers are in the closet. Is the pocket also in the closet? Surely the closet has no pocket- but there is a pocket in it. (If the ' ' ' ' ' ' example sounds a bit artificial , read airplane , wing , and hangar for ' and 'closet' ' ' trousers' ' , pocket , respectively. Hangars don t have wings, but there can be wings in a hangar.) The relations that holes bear to spatial conceptsbelong to a no man's land between mereology and locative structure. A hole in John' s ann is a hole in John. Is this becauseJohn' s ann is part of John, or because John' s ann is located in John? The relevant principles here seemto be the following :

Parts and Holes

( 1) If x is located in y and y is part of z, then x is located in z. (2) If x is part of y and y is located in z, then x is located in z. (3) If x is part of y and y is part of z, then x is part of z. ( This is the transitivity of the part -of relation.) Our conjecture is that these are indeed the basic principles relating parthood and location. Every other thesis that can be obtained from any ' ' ' ' of them by any replacementof located in for part of , or vice versa, is false. On being in a hole. Another question left open by our preceding discussion of holes in holes is this: We examined the case of a holed object located in the hole of another holed object, but exactly what i.s it meant by the expression'in a hole' ? The insertion of an object into a hole modifies the hole: the immaterial solid body has now a cavity in it ( perhapsa hollow or a tunnel- we' l come to that in a moment). A cavity in a hole? That surely sounds bizarre! ' ' But that is exactly what gives the proper intended meaning of in in the ' ' expression in a hole . Nothing can be more in, after all , than something that is in a cavity - and cavities in immaterial solid bodies are on the same foot as cavities in bodies proper. Now , something is in a hole, properly, when it is surrounded by the immaterial body that is the hole. But things can be in holesevenif they are not properly in holes. (An elephant can be on a table even though it is not properly on the table, i.e., even if a part of the elephant extendsbeyond the ' limits of the table s top .) A glimpse a~figure 7.9 will illustrate the point . In the leftmost picture, object a is definitely not in the hole. In the rightmost

a

Fipre 7.9 aboutbeingin a hole. Reasoning

100

Chapter1

picture, a is properly in the hole. And there are some intermediate cases where we might still find it appropriate to speak of a as being in the hole,. though perhaps not properly so. One could take this to be a matter of vagueness , but let us try another route. We know that holes are functional entities (they keep things, let them go through, etc.). Our suggestionis that sometimesan object is in a hole when it is in the range of activity of the hole's function. This could be a general principle of the scienceof practical location. A pencil is in a hole - not properly , for it is much bigger (along one axis) than the corresponding immaterial body; but it is inside the hole nevertheless , for the hole keepsthe pencil at its place(relative to somepossiblemovement). We then add to the proper in -location a different, functional way of being in a hole. Surely the two are related, for the exerciseof the function is made possibleby someintimate spatial relation betweenthe hole and the object. Formally , we can also give a morphological characterization and say that an object a is in (or penetrates, or occupies) a hole x just in casea is a partial filler of x , that is, if and only if a is a complete filler of a part of x. And if this holds for every part of a, then a is properly in x. (Of course, the relation of " being in " is different when we think of a hole in an object than when we think of a filler in a hole. The host/ guest symmetry that we used on other occasionswould be misleading here.) Parts as holes. Up to here we have focusedon whethera given hole is a part of another hole (or when it becomesone). Let us now turn to the question which parts of a given hole qualify as holes themselves. In this case, too, our policy will not be to look for a general, definite answer; rather, we shall consider some puzzlesfrom which an account will gradually emerge. We are looking for parts of holes that are holes- not for parts that could be holes, or that could have been holes, or that could be or could have beentreated as holes. This meansthat the samecriteria are relevant here that apply in the more general caseof looking for holes tout court. Thus, part of the issue pertains to the general problem of hole detection rather than to mereology. As an example, consider figure 7.10, where a simple hollow is depicted along with someof its parts. In the caseillustrated on the left, we imagine a continuum of smaller hollow -shapedparts, one inside the other, aligned

Parts and Holes

- -- -~~.." , / ......... / ~ '...= 7 \' ..', ...-; ...-._.-../ V .---. . . \ . -)

~

~ o V ' =7 ~~

FiIDre7.10 Lots of hollow-shapedparts, but only onehollow.

Edges and fillable parts. Roughly, our conjecture here is that , given a hole-pattern or a holey superficial gestalt, any closed edge definesa hole or a part thereof (although the conversedoes not generally hold: depressions , internal cavities, and doughnut-shaped tunnels are all examplesof holes defined by non- edges ).

102

Chapter7

CQ ) Figure7.11 Edgeinclusion(left) andedgeintersection(middle, right).

Figure7.12 Differentfillable parts of a hole: (a) canonical . (c) non-canonical

(non-privileged );

We have seen that this applies particularly to hollows: to count as a hollow is to possessone edge. We can also refer to this criterion to explain why we acceptthat we have two hollows, one of which is part of the other, in caseof edgeinclusion (as in figure 7.11, left): there are in fact two edges, one corresponding to the big hole and one corresponding to the smaller hole at the bottom. Another simple consequenceof this criterion is that in the presenceof an intersection of edgeswe cannot have two distinct hollows but must have three or one (a limit casewould be edge tangency). Thus, for instance, in the middle picture in figure 7.11 there are three hollows: a big one and two smaller ones, the latter being proper parts of the former. A similar criterion seemsto hold true in the presenceof internal cavities, as in the right picture; here the intersection of two cavities yields a (new) cavity having two distinct hollows as proper parts. The existenceof edgescan also be usedas a criterion for analyzing holes and their parts in terms of fillers. With referenceto our figure 7.12, we can introduce the following distinctions: a is a canonical, privileged tillable part; b is a canonical but not privileged tillable part; c is a non-canonical tillable part . Our suggestionis that we can generally speak of a hole' s part as being itself a hole only if it is a canonical tillable part . Hence, in accordance with our previous remarks regarding this pattern, only the part corresponding to a would here qualify as a part that is itself a hole.

Parts and Holes

~ ~~ ==() ( ~ ) A-AAAAAAA I I

103

\Jgv

Figure 7.13 An " accordion" tunnel and two " dumbbell" holes.

Accordion tunnels and dumbbell holes. Further puzzles are depicted in figure 7.13. How many holes are proper parts of the accordion-shaped tunnel on the left? What about the dumbbell cavity in the center diagram? According to our criteria , the latter is in fact an internal cavity, for two surfaces can be distinguished within it . But when you are inside it , it sometimes looks to you as if it were a tunnel. Your perception is well grounded, for part of the cavity looks like a tunnel, with one entranceand one exit, and this is the part you are sometimesaware of. However, this perception is not truthful . What here looks like a tunnel does not in fact allow you to handle the object by threading a loop through it , and hence it does not satisfy one of the most basic, distinguishing properties of tunnels . (At most we could say that there is a tunnel through a potential part of the object.) A similar caseseemsto arise in connection with other " dumbbell" patterns, such as the one depicted on the right in figure 7.13. Nonetheless, here there is a substantial difference: the tunnel doespassthe loop test. Hence, it seemsmore appropriate to regard this hole as consisting of two hollows connected by a tunnel, i.e., a tunnel with two hollows as proper parts. As to the accordion-shaped hole on the left, such a pattern cannot be regarded as involving internal parts that are cavities, for that would contradict our general criterion for distinguishing cavities by counting the number of surfacesof the host object. External or perforating holes can never have cavities as proper parts; one can climb up a hollow or passthrough a tunnel, but one can never get out of a cavity. As a general rule, we suggest that every time a doubt arises as to whether a certain hole is a hole of type '" or a hole of type f/J, one can find a (possibly improper) part of the hole that is a '" or a f/J(respectively). A note on dissectivity . We have seen that a hole can have holes as proper parts . But we have also seen that this does not by itself imply that the part - whole relation is conservative with respect to the properties that de-

Chapter 7

104

1 11 / 3 1 \ 1 / 3 / 3 1 . . ~ I I ii i . \ t a . a . . -.-.e-.-.-. - .I!i a ,i.+ r .-.-.-.-.!1 '.-.-',..i.--.q ,!I!i8 CAVITY 2 hollows (a )

HOLLOW 1 tunnel + 1 hollow (a) or 2 notches(3)

-TUNNEL CAVITY TUNNEL 2 tunnels (a ) or 2 2 hollowsor a -shaped grooves (3) or 1 tun- grooves(a) or 1 hollow + 1 tunnel (3) or 2 tunnel and 1 notch(1) nets(y)

Figure7.14

Basic partitions obtained by cutting a holed object with a plane.

termine the classification of holes in tunnels, hollows, cavities, etc. For instance, a cavity can have parts that qualify as tunnels or hollows (relative to someinternal surfaceof the host object), though it is apparent that no cavity can have a cavity as a proper part : being a cavity is not a dissectiveproperty . ' Is there any easyway to infer the type of a hole s subholesfrom the type of the hole itself? The operation of cutting (by a plane) provides a useful method for investigating this further. What relations can be drawn between an object x with a hole of type t/Jand the two (or more) objects that can be obtained by cutting x with a plane? We discussedsome examples at the beginning of chapter 4 (seefigure 4.3). A more extensive(but still very incomplete) taxonomy is schematizedin figure 7.14. The schemais incomplete insofar as we confine ourselvesto some core caseswhere both " the host and the hole have a " canonical shape. (The host is cubic; the hole " " is round and perfectly unfilled. We could have worked as well by cutting the corresponrling skins.) However, it is already indicative of the complexity of the problem and of the variety of mixed casesthat we may get when we start considering real holes in arbitrary , real-life objects. The important thing to observe j8 that , although certain cuts do not even yield holes proper (but notches, grooves, or other non-convex superficial particulars), the outcome is always an object involving a discontinuity of some sort. As an example, consider the result of cutting an object perforated by a tunnel. If the cut does not intersect the mouths of the tunnel (cut (Xin the third example of figure 7.14), both pieces will have a tunnel; if it only

Parts and Holes

105

Fipre 7.15 A groove can shrink and develop into a hollow.

intersects one mouth (y), one piece will have a tunnel and the other a notch; if the cut goesthrough the tunnel lengthwiseso as to intersect both mouths (.0), there will be two pieceswith grooves. Notches and grooves are interesting types of superficial particulars and would deservesome treatment on their own, if only to provide further points of comparison with holes, cracks, and other types of superficial particulars we have encountered so far. Grooves, for instance, resemble holes in many respects; e.g., they refuseto exist halfway (there is no such thing as half a groove), although one can certainly cut them into parts or join them together. But grooves also possesssome features that distinguish them from holes. For instance, a groove can entirely surround an object, whereasa hole cannot. Theseinteresting attributes are difficult to describewithin a framework basedexclusively on topological and morphological concepts. Sometimes a groove and a hollow can be transformed into each other: take a cube with a groove surrounding it along the vertical faces, slide the groove upward and then shrink it to obtain a @-shaped(or partially filled ) hollow on the top face (figure 7.15). On the other hand, supposeyou start with a slightly different (but topologically equivalent) object, say a ball of plasticine. A groove along the equator is a groove, but a groove along a small parallel line may qualify as a @-shapedhollow . Parts, holes, and ontological dependence . We have discussed several mereological features of holes, including various puzzles, but we are still far from a general theory of parts and holes. In particular , we have said very little concerning the interaction betweenthose mereologicalfacts and the fundamental fact that holes are onto logically dependententities. This is a rather complicated and unprecedentedissue, although the literature

106

Chapter7

on the connection betweenmereology and ontology - particularly ontol oglcal dependence- has a very respectablepedigree. Some basic principles are nonethelesseasyto single out. Every hole is a hole in (or through) something, and we of course suppose that this something is not itself a hole (or a part of a hole). Thus, the parts of a hole are not in the hole in the samesensein which the hole is in its host. Moreover, the way a hole is in its host is not a relation of part to whole. This was somehow in the background of our discussion so far. Supposeyou put something in a hole, say a coin in the hole of a doughnut; you would not say that the coin has become part of the doughnut, nor would you say that it occupies part of the doughnut . Supposeyou bend your hand so as to form a concavity; then you open your hand flat again and the concavity disappears. Are you creating a new part of yourself? Are you then destroying it? Certainly not. Acquiring holes is not acquiring parts- in fact sometimesit is losing them. Keeping this in mind , the first basic principle we wish to set forth (to be added to the general mereological principles a- e discussedearlier) is that a hole is not part of its host. More generally, this can be strengthenedto the principle that holes do not overlap their own hosts: (f ) Holes have no parts in common with their hosts. (Strictly , this does not mean that a hole may not overlap the host of another hole; if we allow arbitrary mereological sums, the mereological ' sum of the hole in John s pocket and the slice of Emmenthal cheeseon 's Mary plate is a scatteredindividual , partly material and partly immaterial , hosting every hole in that piece of cheese. But thesecaseshave little interest of their own; everything dependson the fact that the sum includes an individual - in our example the slice of cheese- that is itself a host of its own holes.) Our second principle states that hosting a hole is having some proper part that entirely hosts the hole in other words, there is no minimal host for a given hole. This principle can also be strengthenedto a more general proposition : the underlying intuition is that any two hosts of a hole must share a proper common part that entirely hosts the hole. Formally : (g) If x is a hole in y and also in z, then x is a hole in some proper part of both y and z.

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This has the consequencethat a hole cannot have two discrete hosts. Moreover, it follows that the differenceof two hosts of a hole is not itself a host of that hole, and more generally that the difference between an object and the host of a hole is not a host of that hole. Another simple consequenceof (g) is that mereological atoms (i.e., things that have no proper parts- if one is willing to countenancethe existenceof such things) are necessarilyholeless. And , of course, we can also .use this principle to deduce that if there exists such a thing as the " universal" individual , including all other actual or potential individuals as proper parts, then it is surely not a hole- a mereological manifestation of the parasitic nature of holes. A third important point concerns the relation between the hosts of a hole and the parts of that hole: (h) If x is a hole in y and z is a hole that is part of x , then z is also a hole in y . In other terms, to host a hole is to host any proper parts of the hole that are holes themselves. This is, in a way, complementary to (g), though of course it does not hold relative to parts that are not holes. Alternatively, we can see (h) as expressing an interesting form of monotonicity that characterizesthe mereological background of the host-hole relation; this is a natural way of embeddingthe transitivity of the part -whole relation in the present context without violating the thesis that a hole is not part of its host. We may speak here of left-monotonicity, to distinguish it from the following principle of right -monotonicity: (i ) If x is a hole in y and y is part of z, then x is a hole in z. Thus, an object whose parts include the host of a hole is itself a host of that hole, which implies that a hole in an object is also a hole in any mereological sum involving that object. To be sure, in the absenceof restrictions on the underlying general mereology, this principle is not free from exceptions; z could be a mereological sum involving hole x , and by (f ) we have ruled out the possibility that a hole overlaps its hosts. In its most general form , (i) can be reformulated as follows: ' (i ) If x is a hole in y and y is part of z, then x is a hole in z- provided z and x do not overlap.

Chapter 7

Finally , the following is a basic fact, not directly related to the ontolo gical dependenceof holes, that seernsnonethelessto capture an irnportant feature of their rnetaphysicalrnakeup: (j ) Every hole has sorneproper part . Thus, no hole is atornic. Here the underlying rationale is, quite sirnply, that you can always fill a hole partially . On our view this rneansthat you can always fill a proper part of a hole. However, this does not rneanthat a hole's proper parts needbe holes thernselves . In fact, we already know that this is an undesirable requirernent; otherwise it would follow that every hole correspondsto a pile of infinitely rnany gradually srnaller holes- an absurdity which we already ruled out in our foregoing discussion, particularly in connection with figure 7.10.

Resume . One could pursue the topic of hole-rnereologyfurther; certainly there are rnany aspectsthat we have barely broached. The reader will find additional rnaterial in the appendix, where sorne basic facts concerning these rnatters are spelled out more geometrico. However, our discussion gives us grounds for sorneprelirninary conclusions. The rnereology of holes (the theory of the relationships arnong holes and their parts) is an intricate rnatter, and there are rnany puzzles left open. Yet it is not such a cornplicated businessif the true theory of holes is, as we argued, that they are irnrnaterial bodies. In that case, the peculiarity that distinguishes holes frorn other objects, such as tables or drops of oil - the fact that they are immaterial- does not have any drastic consequences relative to rnereological issues. Part of the problern lies in the difficulty in reasoningwith bizarre shapes. But frorn a certain point of view holes turn out to be even rnore standard, rnereologically, than rnaterial substances , for they are quite like vol urnesof space. Only a few serious problerns seernto arise that do not arise with rnaterial bodies. Theseproblernsconcern the account of part -whole relations in the presenceof rnovernent, and their treatrnent seernsto dependgreatly on causalconsiderations. We shall addressthern in the next chapter.

8

Causality, Shapes , and Solidity

Two encounterswith the invisible man. There is a claim by Berkeleyas to the inseparability, either in perception or in imagination, of color from extension: Every time we perceiveor imagine a body extendedand moving " , we must withal give it some color or other sensiblequality which is " acknowledged to exist only in the mind (Principles, I , 10). Is this thesis ' ' correct? If we take imagination in a broad sense, we seemto be able to conceiveof an object that has no color. It would be a perfectly transparent object. But conception is a much too abstractive ability . Could we perceive or depict to ourselves(the proper senseof ' imagining' , in this context) an extendedbody that appearedin an entirely colorlessfashion? Supposethat, while walking along a street, you come acrossthe following strangephenomenon: Besideyou, on the ground, human footprints are being impressedone after another, accompanying your path. There are at least two explanations available for this curious event: either the surface of the road is undergoing a continuous autonomous modification , for instance by contraction of some underground regions (the immanent explanation ); or an invisible being is walking next to you (a relational explanation . Other conditions being equal, neither the immanent nor the ) relational explanation seemsto be privileged. Accordingly, it seemsthat the hypothesis of an invisible being is at least plausible. In this case, you could therefore say that - oddly enough- you perceivedan invisible man (or a part of him: his feet). Here is another example. Supposethat an invisible man is swimming in a pool full of transparent strawberry syrup. You perceivethe man a kind of bubble going around in the syrup, but the shape of the bubble is so telling that no one could deny that you perceive an invisible man. Of course, one could adopt a parsimonious strategy and say that what one perceivesis only the medium, and that the medium is seen to possessa shapeapt for hosting a man (like a Pompeian cast). But sinceyou perceive bubbles, you disregard this minimalist strategy. A by-product of this example is a criticism of Berkeley's inseparability thesis. There is a conceptual link betweencolor perception and extension

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or form perception, but it cannot be so tight as Berkeley would have it . The following seemsto hold: To perceivethe color of (a part of ) x implies to perceive (a part of ) the extension of x ; but to perceive the extension of x implies either to perceive the color of x or to perceive the color of ' an objecty (different from x ) that constitutes part of x s immediate background. As we shall see, this has some important consequencesfor hole-detection. An analogouscasefor two -dimensional configurations is the perception of a line separating two differently colored surfaces. One can be sure that the line is not colored and say that it is nonethelessperceivedin virtue of ' the perception of the two adjacent regions(seeagain Peirces puzzle, figure 2.1). . The case of the invisible man suggeststhat we Holes and dispositions the should loosen conceptual link betweenthe ability to perceivethe object and the ability to perceive its secondary qualities. But it leaves untouched the kind of causal link that relates guest and host, holed object and hole-filler , in those caseswhere the filler is causally responsiblefor the hole' s form. Thesecausal relations individuate a dispositionalpattern that concernsboth the object to which the disposition is normally ascribedand the hole itself. Typical examplesof dispositional properties are the fragility of a glass and the solubility of sugar. Even if sugar in a spoon is not dissolved, it has the disposition to dissolve. Likewise, we say that holes can be filled and emptied, that they can entrap things and let them pass through . Let us explore the possibility of regarding these as dispositional properties of holes(and not , or not only , of holed objects). It seemsthat , if an entity has a certain disposition, this in turn dependson the fact that the entity has certain non-dispositional properties that ground the disposition. Sugar is soluble (disposition) becauseof its chemical structure (ground). Now , holes are not made of anything material. Will the dispositions of a hole be ungrounded, or will they be grounded in other properties pertaining to the hole' s guest(or perhaps to its host)? We seemunable to make senseof ungrounded dispositions, where the ground is always describedin terms of some non-dispositional properties of a stufTthat is, in the final analysis, a bare space-filler. This is, at least, the common-sensenotion of a dispositional property, for there is always, in the scientific picture of reality, the possibility of dissolving an apparently

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monadic property into a dispositional- hencerelational - network. Now , bare dispositions seem to be expelled from the naive conception of the world (if we take such a conception to also include explanatory elements). And there is at least a clear paradigm of bare monadic properties- color properties in visual space- that provides us with an example of what a non-dispositional property could look like. Thus, if the ordinary conception is correct and there are no ungrounded dispositions, a problem arises naturally for any conception of holes as provided with dispositional properties: There is simply no matter of which the hole could be composedand which could account for the dispositional properties we ascribeto the hole. Fillability is the most striking case. If any property of matter could account for the fillability of an entity composed of that matter, then the presenceof that very property would imply that matter is not impenetrable, contrary to our assumption. Thus, either we reject the idea that holes have dispositional properties or elsewe reject the idea that there cannot be ungrounded dispositions. Is the dilemma real? Considering again the example of the perception of the invisible man, we could reasonanalogically and propose the following principle : x has dispositional propertyD if and only if either (i ) x has a property G that constitutes the ground for D or (ii ) x is metaphysically or causally dependent on an objecty , different from x , such that y has a ' property H accounting for x s havingD . H need not be the ground of D, in the sensein which G would be such a ground. Thus, holes are fillable insofar as they metaphysically depend on objects that have a certain shape, but such a shapeis not a ground for the disposition of the hole. It seemsneverthelessthat we should establish whether the above principle is useful in casesother than holes. Think of the casein which a person has adispositionD to remember certain things becausesomething else- the person' s brain - has a certain property H that accounts for D. According to one version of this theory" people remember becausetheir brains host memory traces. This would satisfy the seconddisjunct of the principle - but somequalifications are in order. H , the property of the brain, is the basis of another disposition, D', which can be described in purely physical or physiological terms (e.g., as the disposition to host certain chemical patterns). If we explain remembering by saying that certain patterns are activated, we could be inclined to say that D and D' are one and the samedisposition describedindiffer -

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' entways . H accounts for D becauseit accounts for D , which is identical with D. Needlessto say, such an account must be taken with some care, for it is a limit caseof an acceptableexplanation. The examplethus dit Tersin one important respectfrom the caseof holes and their properties. The hole' s dispositionD , fillability , is accounted for ' by the object s having a certain form. This form is certainly a basis for other dispositions D', D", . . . of the object to which the form belongs, such as the disposition of interacting in a certain way with liquids, stutTs , and other entities. But thesedispositions are not identical with D. It thus seems that the above principle would be clearly applicable only to holes, and therefore this seemsto be an utterly ad hoc solution. It remains the fact that , even if we do not want literally to credit holes with dispositions, we can accept that dispositions are attributed to holes in an indirect way. And the principle for such an attribution is connected to the one mentioned above: There must be somespecificproperties of an object in order for it to be possible that we attribute fillability or other dispositional properties to holes in that object. Someparticulars are not bearersof dispositions . The fact that there is no literal ascription of dispositional properties to holes does not per se make holes queer entities, for there are other spatio-temporal entities that are not the bearersof dispositions. This is, for instance, the caseof events. Of course, events are not spacefillers, and are not constituted by any space fillers, which hinders ascribing dispositions to them. Substancesand stuffs only , we would say, bear dispositions, and substancesdo so partly because the stuffs they are composedof do so too. In a sense, however, events differ from holes as to the impossibility of dispositional ascription. In order to be the bearer of adispositionsome thing has to endure. This is a necessaryrequirement, for the application of a counterfactual expressingthe disposition requires re-identifiability of the entity the counterfactual is about. No event could satisfy the requirement, for eventsdo not passthe test of time. But holes do passthe test- and we suspectthat this is a good reasonwhy it is not so hard to credit them with somedispositional properties. Holes are made of space- all problemssolved. If spaceis a kind of basic substanceor stuff (according to the theory discussedin chapter 3), then it

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must have some dispositions, albeit very basic ones. It must have modifi ability or qualifiability (by some property at some time), at least. And one disposition that superveneson theseis fillability . If a zone z is empty and then gets qualified by some quality , it will not be improper to say that z has been filled. Therefore, if holes are made of space, they have no difficulty in being the subjectsof thesevery primitive dispositions. Do holes have to be in material objects only? As we have put it , the materiality of the host accounts for the presenceof those dispositional properties we can attribute to holes. Now , matter comesin different states. And although the core caseof holed matter is solid matter, softer possibilities are worth mentioning. Here we cross the border into a new territory where geometrical or morphological conceptsalone prove insufficient for orientation . Somephysical descriptions will probably help. Think of a universecomposed of an infinite magnetic field. In such a universeit would be possible for somebodyto produce a hole by creating a (non-magnetic) stone somewhere . The stone would also be the filler of the hole. It could even happen that the distribution of forces in a magnetic field determines a closed surfaceenveloping an empty magnetic pocket (figure 8.1). Objects inside the pocket would be subject to no magnetic force, and the pocket would move around in the field. Another exampleis provided by the eyeof a hurricane, which definesan area of non-circulating air amidst an area in which the air is in motion . Supposethat the eye got smaller and its projection on the surface of the

;-\ .~ ;:~

Figure 8. 1 A hole in a magnetic field.

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Earth eventually reduced to a point . We would then have an internal filamentous crack (seefigure 6.5) in the hurricane. Even softer examples may be mentioned here, although their peculiar nature suggeststhat we postpone their discussion to later chapters. For instance, a regular pattern printed on a piece of fabric may display a sudden irregularity ; this would be a true hole in the pattern, though of course not in the fabric. Or a non-denoting term or an expressionlacking a definite meaning may be thought of as introducing a hole in the referential structure of a sentence, thereby yielding a truth -value gap. We shall come back to thesecasesin chapters 11 and 12 respectively. Is a bubblea kind of hole? Weare not thinking of soap bubbleshere; their charming iridescenceand the way different physical forces harmonize to mold them with exquisite geometrical perfection invite us to regard them as something more than mere internal cavities in soapy hosts. Our concern is, rather, with gas bubbles- bubbles in gaseousliquids like champagne , or in smoldering lava. Are they holes of a kind? They are. The differenceis that in order to explain why bubbles have a certain shapeone must take both the state of the filler 's stuff and the state of the host's stuff into account. (The forms of ordinary holes are sometimes accounted for in the same way, but they need not be.) Accordingly, bubbles can becomeordinary holes. It is sufficient that the filler 's stuff ceases to have any causal role as to the hole' s form. Thus, all the gas bubbles in a stream of lava are transformed into holes once the lava solidifies. (Bubbles turning into holes, and holes turning into bubbles, bear somesimilarity to ice turning into water. If we take ice and water to be two different stuffs- ice being constituted by water and co-localized with it - a hole in ice turns into a bubble in water when the ice melts- that is, as soon as the water ceasesto constitute the ice. But the notion of constitution , when applied to two categorially homogeneousentities, is yet quite unexplored and involves difficulties that we need not fully addresshere.) Also, sometimesthe responsibility for a hole's shape is entirely on the filler 's side. For instance, a ship on the seacarries a hollow along with it , and when the ship is standing still the hollow is also there. (An interesting, fact that would require a detailed description is that when the ship is moving it producesa slightly bigger hollow , which always expandsinto a wave.)

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...-.~ ~ II -_ :-I.T " , . . _ ~ " ' ~ " " ' ' ~ . ' . , t I.J,-~ ..-~ ; \ : ,w : . ; '.:~ ..':",,.;o ; .'o . " J "' ,'c.",.-,: f\'"-,;."-"~ "-:/.~ Figure8.2 A closed , expandingunderwaterwave.'

Bubblesof a special kind. Figure 8.2 depicts a closed, expanding wave such as an underwater explosion would produce. We could consider it a special kind of bubble- special insofar as the filler continuously receives stuff from the host and thereby increasesits surface and volume, and special also becausethe notion of surface involved here (that of acom pletely permeablesurface) is special. Here the surfaceof the bubble is just the place at which compression is maximal. The bubble is causally sustained by its filler as well as by its host. And when it gets to the surfaceof the sea, the wave becomesa sort of superficial bubble, belonging to the samecategory as ordinary hollows. It seemsthat a more ambitious consideration could be advanced here. In the caseof the underwater explosion, both the inside of the bubble and its outside are made of the samestuff (water) in the same liquid state. By contrast, in the caseof gaseousbubbles in champagne(samestuff, but in different states) or in the case of oily droplets in water (same state, but different stuffs) there is an important qualitative or state discontinuity , respectively. A general principle could then be that a bubble whose inside is of the samestuff and in the samestate as its own host can live only as a dynamic entity . It must expand and conquer stuff, or elseit must contract, implode, and lose stuff. Or it must keep someexchangeof stuff in balance, as a whirlpool does. Causality and shapes . Most of our discussionof fillers required a notion that could be characterizedpurely in geometrical terms. In short, a thing z is a filler of a hole x if , and only if, z is in x and every superficial part of z is in touch with a superficial part of the host of x. But of course there is more to it than that. An actual filler is, typically , a material, solid object (it might well also be a portion of some liquid or gas, as we just saw in our discussionof wavesand bubbles), and once it occupiesits position in the

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hole it has a much more constrained possibility of movement. Nails and keys are essentially connected to this limitation ; the filler is kept in the object by the hole. Another important interaction feature is linked to the dimensional reduction of an object's surface, consequentupon the filling of its holes. (For convenience, we are taking the aggregatemade up by the object and its fillers as one causal unity .) If you had to make a ball roll on the surfaceof an object, it would be in your interest to eliminate any fastidious hole from the surfaceby perfectly filling it . It seemsthat these are the two most important interaction patterns connected to holes: keeping objects (fillers and pluggers) in place, and hindering the movement of other objects on the surfaceof a holed thing . The presenceof a distinctive interaction pattern with the inanimate world is a primary mark of objectivity . This might not by itself be sufficient to ensureobjectivity to holes, becauseit is clear that the interaction patterns describedhereare to be attributed primarily to certain shapes:at the same time, shapesare not causally efficient in or by themselves , but only insofar as they are shapesof some material object. This is trivially true of holes as well, for there are no holes an sich but only holes in material objects. (Holes are de re dependenton their hosts.) Indeed causality is often connectedto materiality , but that is not the point at issue. The point is that the explanatory burden of why certain causal transactionstake place along certain lines is supposedto lie on the possession of certain shapesby material objects. This holds true particularly of those local causal transactions that only pertain to a proper part of the object. Thus, the shapeof the part of a key that we hold onto firmly when we insert the key in the lock is not responsiblefor the interaction between the key and the lock ; such a part could have been shapedin a completely different way, and yet the key could have interacted with the lock in exactly the sameway. The shapeof a hole is the shapeof a proper superficial part of its host. The hole' s responsibility in a causal transaction is in some way tied to the causality exerted by that superficial part . (Shapesare not the end of the story, though. A square-shapedperfect filler will be kept at its place by its square hole; it will not rotate. But a circular perfect filler will rotate in its circular hole. One would need here to investigatethe frictional properties of the stuff of both hole-filler and hole-host.)

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Holes and hosts: Causal interactions. Think again of the invisible man swimming in the strawberry syrup. We can describe what goes on (or at least part of what goes on) by saying that there is a filled internal cavity moving around in the syrup. This suggeststhat , in general, the movement of a hole does not imply the movement of its host. Conversely, it is also possibleto envision situations where the host can be moving without there being any corresponding movement of the hole; for instance, we can imagine that the invisible man stays still while the syrup goes down the drain and is gradually replacedwith new syrup. Moral : Although the tie linking a hole to its host in causal interactions is very tight , the movement of the one does not generally imply the movement of the other: ( la ) movement of the hole + movement of the host ( 1b) movement of the host + movement of the hole. Now , it seemsthat an entire family of idiosyncrasiesfollow from these facts. For example, although (2a) and (3a) hold ' (2a) movement of the hole --. movement of someof the host s stuff (3a) movement of the hole --. movement of some parts of the host it is apparent from (4a) and (Sa) that they cannot be strengthened by ' ' ' ' replacing some with all . ' (4a) movement of the hole + movement of all of the host s stuff (Sa) movement of the hole + movement of all parts of the host On the other hand, the conversesof (2a) and (3a) are clearly not valid: ' (2b) movement of some of the host s stuff + movement of the hole (3b) movement of some parts of the host + movement of the hole (you can certainly remove stuff or a part from a doughnut without affecting the hole); but it is not clear whether their strengthened forms, corresponding to the conversesof (4a) and (Sa), should be taken to hold unrestrictedly: ' (4b) movement of all of the host s stuff --. movement of the hole? ' (5b) movement of all of the host s parts --. movement of the hole?

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Likewise, it seemsevident that the host may changeform without there being a corresponding changein the form of the hole: ' ' (6b) changein the host s form + changein the hole s form (You may deform a doughnut or even cut it into piecesand at the same time leave the hole intact.) The converse, however, is not as clear: ' ' (6a) changein the hole s form -+ changein the host s form? At first sight, it would seemthat this is a valid implication : the movements of the invisible man affect the overall form of its syrupy host; and when a hole gets bigger, small.er, larger, or narrower, the form of the host changes correspondingly. But theseexamplesdo not provide sufficient evidenceto support (6a) in its generalform. The samegoesfor (4b) and (5b): When you move an entire doughnut from one placeto another, you thereby move the hole that is in it . But there is more to it . Suppose the movement of the ' doughnut s stuff (parts) is obtained by spinning the doughnut clockwise. Does the hole rotate too? Does it move? Let us addressthe matter more closely. The caseof the rotating hole. The rotation of homogeneousspherical (or round ) entities around their axes (or around their centers) poses philo sophical problems. First , it is difficult to tell whether these entities are rotating or not ; their homogeneity hinders the distinguishability and re to characterize it is difficult identifiability of their constituent parts. Second, the movement itself onto logically . " " D . Arm strong (in Identity through Time ) puts the onto logical problem in the following way: Imagine removing a section of a uniformly rotating homogeneous sphere, somewhere close to its surface, thereby producing a hole. As a result, the rotation of the sphere will involve the circular movement of the hole. But by looking at the scenewe could as well consider that a hole is circularly moving around the center of a homogeneous , still sphere. What would make us reject the second possibility? What would make us think that it is the sphere that is rotating, rather than taking the sphereas still and the hole as moving around by annihilating one portion of the sphereand at the same time creating another one behind it? The preferability of the first solution is linked to the central role immanent causality plays in determining the structure of the world . The spatio

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temporal stagesof the portions of the spherethat are now adjacent to the skin of the hole are preferably understood as being counterfactually dependent on and nomologically related to the spatio-temporal stagesof the ' portions of the spherethat were sometime ago adjacent to the hole s skin (when the hole was at another place in its movement). No such counterfactual or nomological dependenceis available in the caseof creation and annihilation - that is, we have no idea of what sort of laws would be causally compatible with alleged facts of creation and annihilation . And surely a hole cannot create anything. Let us now think of a disk perforated by a round tunnel (or of a sphere with aspherical cavity inside it ), as illustrated in figure 8.3. Supposethat the geometrical center of the tunnel is at the same time the geometrical center of the disk. Suppose further that the disk is rotating around its center. Is the tunnel also rotating ? Intuitions diverge. First , consider an argument in support of the idea that the tunnel is not rotating : " What a curious statement! Look at the hole; don' t you seethat it is perfectly immobile?" Of course we have the impression that we seeit immobile, but our concern here is not epistemology. We admit that it might be impossible to find an effective way to establish whether or not a circular hole is rotating around its center. Still , we need a better argument in order to make the onto logical distinction . Here is one such argument. As it links the movement of the hole to its causaleffects, we label it the argumentfrom causalrelevance. Take two disks (figure 8.4), one perforated by an elliptic tunnel (cocentered with the disk) and one perforated by a circular tunnel as previously described. Each disk rotates around its center. Now supposethat we project light through their tunnels, and look at what happensto the shadows that the two disks cast on a wall. The oval perimeter in the middle of the former shadow (a) rotates, whereasthe round perimeter in the middle

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Fi. - e 8.3 Is the holein a rotatingdisk rotating?

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4

.

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of the latter (b) looks immobile. Now - so goesthe argument- if the oval tunnel rotated, this fact would be causally relevant to the rotation of the oval perimeter of the corresponding shadow. On the other hand, if the round tunnel rotated, this fact would be not causally relevant; it would have no causal bearing on the state of the round perimeter in the shadow. Ergo, the first tunnel (oval) is rotating, whereasthe secondone (circular) is not. To this argument we respond that it is not even clear whether the internal rQund perimeter of the shadow cast by b (call it shadow b) is not rotating, as the argument requires. If we allow that it is not, then we have to accept that shadow b is not rotating . But now look at shadow a. We agreethat a good deal of it is rotating the minimal circular area enclosing if the elliptical hole surely is. But we accept that shadow b is not rotating , then we must say that the external edge of shadow a is not rotating either. Thus, shadow a would be composedby at least two parts: a rotating part and a non-rotating one. This seemsbizarre, but we are not in the position to say whether the oddity comes from reasoning about shadowsor is a consequenceof the argument. (If you prefer, you can think instead of some device for watering flowers through rotating holed disks and consider why a-like devicesdo not, and b-like do, let the most peripheral flowers dry; you will come to similar conclusions.) One way of escapingthesedifficulties is to credit shadowswith momentary existenceonly . That is, shadowsdo not move, just as cinematographic pictures do not move; what looks like the movement of a shadow (or of ~ picture in a movie) is a sequenceof static patterns. Even if we assumethat

middle

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holes are momentary entities like shadows, the problem of hole-rotation will be left untouched, for we still have to distinguish betweena seriesof holes whose members are all in the same position and a series whose members occupy slightly different positions. The fact that one could reconstruct a hole's movements cinematographically does not imply that one can tell whether the seriesone getsis composedof elementshaving the samespatial situation or of elementshaving different situations. A better statement of the argument from causal relevanceis contained in what may be called the argumentfrom the movementof thefiller . Simply put, it assertsthat the movement of an object involves the movement of a hole in it if and only if the movement of the object would involve the movement of the potential perfect filler of the hole. Here the movement of the tunnel is causally relevant to the movement of the filler in all caseslike a (where the tunnel is not circular ), but it becomescausally irrelevant in caseslike b (where the tunnel is circular). The circular tunnel in a rotating disk could contain a filler which is indifferently rotating or immobile relative to the disk. Ergo, the tunnel could indifferently be rotating or not rotating . Since there is no reasonfor taking it as rotating , we should take it as immobile. We believethat there are reasonsfor resisting this line of argument. One of these is expressedby the argumentfrom relativity of movement . Let us the conclusions of the from the movement of the filler ; that grant argument is to say, let us concedefor a moment that a tunnel h in a rotating disk b could be immobile. Supposenow that an observer had a referencesystem integral with b. We should say that , relative to him, disk b is immobile, whereash rotates. But if this is a possibility, then nothing could prevent us from taking every hole in a non- rotating object as rotating . Imagine that space is absolute, and that the only movement of the Earth relative to absolute spaceis rotation around the Earth' s axis. Miss Anscombeleft her wedding ring at the North Pole. Have a look at it from some fixed point on the surfaceof the Earth (figure 8.5). Would you say that the hole in the ring is restlesslyrotating, while the ring itself is quietly resting motionless, rotationless relative to you? This is plain nonsense.Therefore, it is idle to take holes in rotating objects as immobile. Nevertheless, this line of argument could still be unconvincing, for we could imagine that circular holes and ( they only) neverrotate, which fact would be compatible with both the ' immobility of Miss Anscombes wedding ring and the motion of our holed disk. Together with the speedof light , the non-rotation of circular holes

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Fipre 8. S Is a hole at the North Pole still, while you rotate with the Earth? In that caseit would be rotating relative to you.

would be one of the few universal invariants. Incidentally , this solution seemsto recommend itself on the ground of some uniformity with the most economical solution to a similar problem arising for closed cracks and fissures. Suppose the hole in Miss Anscombe's ring is filled with its perfect filler f , and that, while the Earth rotates clockwise, f rotates counterclockwis . What then is the direction of rotation of the fissure separating f from the ring? Is the fissurerotating at all? Or imagine an elastic host b rotating counterclockwise and a rigid elliptic filler f rotating clockwise. The form of the fissurefollows the form of the filler , but we would not say that the fissure moves. Would we say that the hole moves? Here the hole is, in a sense,moving together with the object, becauseits superficial parts are tied to the object's surface. But the fissuredoes not move. A fissurehas no superficial parts- it is geometrically defined by the internal surfacesof the object hosting it , but it has no surfacesof its own. We simply cannot accept that only the surfaceof an object can move, gently sliding over the entire object, while the object itself remains still . Hence, we cannot help accepting that the hole could be rotating together with the surface in an otherwise completely still object. Let us try another argument; call it non facit saltus. Disk b rotates; h is an elliptic , centeredhole in b. Being elliptic , h rotates at an angular speed which is the same as that of b. Suppose now that we were to gradually equatethe vertical and horizontal axesof h. This processwould not reduce h's angular speed. Why should h' s angular speedbe brought at once to 0

Causality, Shapes, and Solidity

~

123

~

Figure8.6 A paper-towel roller (a) with a toilet-paper roller insideit (b). Here the two rollers are . spinningin oppositedirections

when the two axeseventually have the samelength? This would imply a saltus, not clearly acceptablein this context. Our provisional conclusionsseemto point in the direction of accepting that the hole in the rotating object moves together with its host, even if that movement does not imply any movement of the filler and even if we could accept that the fissure betweenhost and filler is not moving. But holes, as immaterial bodies, are plastic. We have another account at our disposal. Suppose you have a pot made of rubbery stufT. You can deform its mouth by turning a stick all around it. The deformation rotates; the mouth and the pot do not. The new account draws precisely on this fact: Some entities are subject to elastic deformation and can therefore absorb motion. Let us go back to our circular hole in a rotating object. Imagine deforming the hole and making it elliptical . The object rotates; the hole' s lining is rigid (i.e., is not subject to elastic deformation). We saw reasonsto say that the hole rotates. But now we can also say that the hole simply stays still and gently changesits form. As a matter of fact, the unacceptability of a saltus proceedsalso in the reversedirection. Supposethat h is a perfectly circular hole, and that it doesnot rotate. Supposenow that we gradually disequatethe vertical and horizontal axesof h, so as to make it elliptic . If the hole did not rotate, why should it start now? As a further illustration, consider the puzzle discussedby the Lewises and mentioned at the beginning of chapter 3 (objection 3). Take a papertowel roller , spin it clockwise, put a toilet -paper roller deep inside it , and spin it counterclockwise (figure 8.6). Then something (the tunnel in the paper-towel roller ) would spin in one way when a part of the very same thing (the tunnel in the toilet -paper roller ) spins in the opposite way. The

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Ludovician reply was that the little hole inside the big hole is not a part of it . And on this we agree. This was in fact a problem in our discussion of the mereology of holes, and we found it appropriate to describethis situation by saying that the hole in the toilet -paper roller happensto coincide spatially with a part of the bigger hole in the paper-towel roller , for holes are immaterial bodies and can therefore be penetrated by other (material or immaterial) bodies. Still , the puzzle is not solved by this. How can two bodies perfectly overlap each other (in this case, the little hole and the corresponding part of the big hole) and yet behavein opposite ways? The solution we ofTeris that they just do not behave in opposite ways. Holes, as immaterial bodies, are plastic; how could it be otherwise? At a closer look they never rotate; at most, if their rotating host is not circular , they gently changetheir form. Thus, imagine that our two overlapping holes are elliptical . Their hosts rotate in opposite directions, and we are tempted to say that the holes also rotate accordingly. The truth is that they are still. They do not rotate but they are continuously deformed by the rotation of their hosts. And this solvesthe problem. Likewise, by assigningplasticity to immaterial bodies we can cope with the caseof the rotating object with a coaxial circular hole. All things considered, the hole is still , as when you rotate a glassfull of wine the glassrotates but the wine stays still . An interlude on empty spaceand immanent causality. Empty spaceis a delicate matter. We will confine ourselvesto pointing out a very peculiar, philosophically interesting feature of empty space that is implied by the conception we favor here. Think of Ann strongs problem of the rotating homogeneussphere. Supposethat you do distinguish betweenaspherical chunk of rotating empty spaceand a similar but immobile chunk. How can you make the distinction if both chunks have parts which are indistinguishable from one another? In the caseof non- empty space, the suggestionwas that there is atracing back of certain parts to earlier parts, where the track is guaranteedby : if some spatio-temporal parts of the object counterfactual dependence had not existedat tl in place PI ' they would not exist now in place P2; and if place PI is different from place P2' then the object is rotating . But can parts of empty spacefulfill the same constraint? If they can, then immanent causality is really of another nature than ordinary (static or dynamic) causality, for empty spaceis causally inert and its casewould provide an

Causality, Shapes, and Solidity

125

display - - in~ - immanentcausality . without at the same time having anything to do with ordinary causality.

example of something

. Let us now turn to the general issue of Causally connectedaggregates causality and shapes. There is an important question, left open from our earlier discussion, that our subsequentreasoningin tenDs of homogeneus matter might have further concealed. Consider a hole in a wall: a missing brick. But consider also an irregular perforation produced by a cannonball. Its boundary crossesthe boundaries of several(adjacent) bricks. It is a hole- a tunnel through the wall. But this seemsto contradict the idea that holes cannot cross fissures, that a fissure or a crack by itself produces a hole. Hence, that idea must be corrected. The possibility of an object having holes independently of its being a unitary , simple substancebecomesimportant . At the same time, we would not have holes everywhere(at least, not in totally unconnected ). You do not create a tunnel by simply putting your bent fingers aggregates in contact with one another, nor do you create a cavity by putting a reversedglass on the top of a table. In thesecasesyou create something that looks like , but is not , a hole. To mark the difference we are aiming at, it is useful to introduce the notion of a causally connectedaggregate. Such an entity is, in the first place, an aggregate:somethingwhoseparts could exist both independently and disjointly from the particular they are parts of, and something whose parts thus consideredare actual parts. An aggregateis causally connected relative to some operations if (and only if ) it behaves uniformly under certain interaction patterns, detennined by those operations, with the inanimate environment. The speciesof this connection correlate with the kinds of causal interaction at stake. A table and a tablecloth laid on it constitute a causally connected aggregateunder certain operations that one can perform on it ; e.g., if one movesthe table, one nonnally makesthe cloth move as well. Thus a simple criterion for causal connectednessof aggregatesis the movement of somenon-pushedpart as causedby a push in another part . In general, causalclosure is marked by the corresponding closure of the by-relation holding between certain operations one can perfonn on aggregates ; thus, a door , the aggregateconstituted by a wooden board and a handle, is causally closed under the operations of pulling and pushing (by pulling the handle you pull the door, and vice versa). Note that the by-relation in question could be antisymmetric, as it happensto

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be in many superficial operations. (You wash a door by washing its surface , not vice versa.) Causal closure is a matter of degree. Relative to a certain operation, aggregatescould be loosely causally closed(a set of tenpins), but they can also be very strongly closed(a tightly constructed tennis racket). A totally causally closed aggregate approximates the paradigm of a philosphical substance. (Even though a piece of marble is composed of miriads of atoms, it behavespretty much as a unitary being under many operations.) In the same way in which aggregateshave lesser onto logical dignity than homogeneoussubstances , holes in causally connectedaggregatesare somehowlessconspicuous. We will come back to this problem in chapter 10 when discussingsome basic ways of making holes in artifacts. We then need to find an adequate interaction pattern. In the case of tunnels, we already know that this is the hang-by-a-loop pattern. If you can hang the object by threading a loop through what you would take as a tunnel, then you do indeed have a tunnel. This holds for ordinary (simple . But it holds for aggregatestoo. Here the hang-by-a-loop ) substances interaction pattern would be describedby saying that if you pull the loop you do not disjoin parts of the aggregate. The aggregateis causally closed under the hang-by-a-loop operation: by acting on some part of the aggregate , you act on someother parts as well, and possibly on the whole aggregate . (Is a net- constituted by knotted ropes- a holed object? According to our criterion , it is: you can thread loops through it and hang it up. Natural understanding, however, would accept that there are holes in a net only if a knot is missing. This is probably a relic of the underlying presenceof the purely processual concept of a hole as something previously occupied by matter and something you can make things pass through .) One could object that a hole entirely enclosedin an actual part of an aggregateis not a hole in the aggregate, for you can pull a loop in a (tu;nnel entirely enclosedin a) brick and thereby pull the brick out of the aggregate . But this would not constitute a problem. In this caseyou would have a hole in the brick that would not be a hole in the aggregate. If , on the contrary , the pulling of the brick were a pulling of the wall , then the hole that is relevant for the pulling would indeed be a hole in the aggregate. We have here two possibilities. On the one hand, in a causally connected aggregate one can eliminate fissures simply by welding the two adjacent surfaces. (It is possible that a trace of the fissure remains in the

127

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form of a solid join .) On the other hand, it is possible that there still are fissures, and that the aggregateis causally connectedbecauseof the particular form of the fissures. After all , this is exactly why bolts and nuts are so important in daily life. The caseof the explodedtorus. Consider now an apparently more complicated case. A torus is a holed object; it is perforated by a tunnel. Imagine exploding one as in figure 8.7. Is the tunnel still there? Certainly we have the impression of seeinga spacein the middle of this aggregate- a space which is in some sensedifferent in nature from the small spaces separatingeach of the facetsof the exploded torus from its neighbors. But is this impressionjustified ? Again, some contextual principle could be invoked. But a positive answer to the question is possible if we take into account a relevant causal context. We only have to take the exploded torus as somehow causally connected- as a solid unit , for instance, such that dislodging anyone of its separatedfacets would produce a movement of the overall structure. And now imagine a sufficiently thick loop , made of a rope whosediameter is bigger than the space separating any two neighboring facets of the exploded torus. It would be possible to handle the exploded torus by letting this big loop cross the big spacein the middle- the latter would be a real hole.

~ ~

J ~

=

te ~ f

~

~

Figure8.7 Anexploded torus. What happensto the hole (h) inside it?

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Chapter 8

We invoked someextremely contextual considerationsin order to make senseof the question. But it would be a mistake to consider this a marginal problem. The caseof the exploding torus is but a macroscopicanalogueof what happensat the microscopic level for (ordinary) matter. Causal closure seemsto be quite a pervasivefeature, and it plays an important role in explanation. A puzzle about flaures. Imagine a world in which matter is homogeneous . Suppose only the EifTel Tower exists in that world (or , better, something that looks just like it ). Such a tower is built up entirely of components of size 0'. Theseare slabs and bars, connectedto one another by meansof bolts and nuts. The slabsand the bars that make up the tower can be functionally reduced to either bolts or nuts. The solidity of the tower (i.e., the fact that it doesnot disintegrate into the scatteredsum of its component parts) is guaranteedby the particular form of the fissuresbetween the bolts and the nuts. It is in fact the tower' s having these fissures - thesebolt -shapedand nut -shapedfissures- that causally accounts for its overall solidity . Now imagine that every single component of sizes of the Tower is itself built up of smaller component parts, of sizes/ 2. Again theseare slabsand bars, connectedto one another by meansof bolts and nuts. And again the solidity of suchcomponentscan be accountedfor by the form of the fissures betweenthe relevant bolts and nuts. In fact, by the samepattern one can imagine that for every n > 0 each component of size sIn is built up of component parts of size s/ 2n: an endless series of smaller and smaller slabsand bars connectedto one another by meansof smaller and smaller bolts and nuts. A fractal fracture. An infinitely ramified web of fissuresspreading into every part of the tower. Is that a solid tower after all?

9

Samenessand N on- Substaoce

On being tbe same bole. Take a chunk of Emmenthal cheesewith a round hole h in the middle. Supposeyou move this chunk and replace it with someother chunk of Emmenthal cheese , also with a round hole in the ' middle. Supposethat this new hole, h , is now located in the very same ' place where h was located before. Now , h and h might very well have exactly the samesizeand the sameshape. Yet they are not the samehole. The fact that holes are not made of ordinary matter emergesforcefully from this example. On the one hand, we can certainly imagine that the air that occupied h is the sameair that now occupiesh' ; on the other hand, we can imagine filling the samehole with different stuff at different times (the air that is now hosted by h is different from the air that was initially hosted by h). Thus, we cannot ground the identity conditions of the hole on those of the matter hosted by the hole; otherwise we would have the absurd consequencethat holes cannot possibly be filled. Moreover, holes are not regions of space, as we saw in chapter 3. For instance, you can move a hole, but you cannot move a region of space. Thus, we cannot provide adequate identity conditions for holes exclusively in terms of the identity conditions for regions of space. Likewise, we may insist on the fact that holes are superficial particulars, but an adequate identity criterion for holes cannot simply exploit the intuition that the identity of a superficial particular is ensured by the identity of its superficial parts. For instance, a hole could have exactly the same superficial parts as a color patch- and yet the hole and the color patch would be two distinct (superficial) particulars. Identity of superficial , but in no way a sufficient, condition for the identity of parts is a necessary the objects possessingthe superficial parts. Consider another case. The surface of a cube has the same superficial parts as the cube (that is, of the object it bounds), but clearly it is not " identical with the cube. In his paper " Volume and Solidity , David Sanford introduced the notion of a material plane: a sort of two -dimensional solid which has no volume and is neverthelesscausally interactive. One

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can construct pseudosolids out of material planes. A pseudo-cube would be a hollow solid constructed by juxtaposition of six square material planes. Of course, the two casesdiffer in that the cube is not a superficial particular relative to another particular . Yet this was the condition in terms of which the above criterion was supposedto provide a definition of the identity of superficial particulars as dependenton the identity of their superficial parts.

Kant' s cushion. " If I lay the ball on the cushion, a hollow follows upon the previous flat smooth shape; but if (for any reason) there previously exists a hollow in the cushion, a leaden ball does not follow upon it ." (Critique of Pure Reason, A203/ 8248- B249) Here Kant is probably guilty of confusion about the terms of the causal relation, which are neither material objects (the sphereand the cushion) nor properties (the correspondin forms). This granted, Kant overlooks that the presenceof a hollow could explain why the sphere is in that place rather than, say, on the comer of the pillow . That is, in a sensethere is also an explanatory path from the hole to the object, not only from the object to the hole. But we could take Kant as saying that the responsibility for the shaping of objects depends on what other objects are (or are made of ), while this shaping itself bears no responsibility for what other objects are (made of ). Alternatively , we are just apprehending that a hole can survive its filler which is what we will expand on in the following . Identity and spatio- temponl continuity. The identity of a hole over time will have to be traced to somedelicate interplay betweenthe identity of the host and the identity of the filler. Let us start from the latter. Consider a continuously changing object: the ship of Theseus, for instance, which we already encounteredin chapter 7. One day a wooden plank is cast ofT and replaced; on another day, another pieceis replaced. The replacementsare exactly like the old pieces. Every day a new piece takes the place of an old one, until no part of the original ship is left. The result, we may imagine, is a ship that is exactly like the one we started with . Is it the same ship? Philosophers have been puzzledby this for a long time. But let us leavethe question open. Suppose that the initial ship had a hole in it (figure 9.1), and supposethat throughout the successiverepairs of the ship Theseuskept an unchanging per-

Sameness andNon -Substance

filledhole Fipre 9.1 ' The identity of the filler ensuresthe identity or the hole evenif Theseus ship changes . continuously

b'

a

' t

~

Fiaure9.1. A filled holeexperiencing thedisintegratiolland creation of its host.

fect filler constantly inside the hole. Did the hole survive the continuous changes in its host? Our suggestion is in the affirmative. Regardlessof whether or not we find it problematic to speak of one and the sameship persisting through time, the identity of the hole seemsto be ensured by the . identity of the filler . Consider a secondexample. Supposethat a machine capable of disintegrating and a machine capable of creating (philosophical tools envisaged and usedby Arm strong and Shoemaker, among others) are pointed at the samespatio-temporal region, which happensto be occupied by an object, b, in which a hole h is perfectly filled by an object a (figure 9.2). The two machines are made to act simultaneously. However, only the region occupied by b will undergo disintegration and creation; a will be kept alive, while b will be substituted by an indistinguishable b' . In this casetoo , the permanenceof filler a seemsto ensurethe permanenceof hole h, evenif the host is no longer the same.

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Spatio- temporal continuity is sometimessufficient for identity. If we perform the disintegration-creation experiment on people or tables, it seems that spatio-temporal continuity does not prove sufficient for identity over time. The existence of later stages of people and tables has to depend counterfactually on the existenceof previous stages. But this requirement seemscompletely vacuousin the caseof holes. What matters is the persistence of a certain functional capacity of the hole, which in turn dependson the persistenceof some shapeand sizecharacteristicsof the object. Think of the nostrils of Lot ' s wife after she was transformed into a statue of salt. Moreover, though the identity of a filler in the full exerciseof its filling faculties (i.e., without any spatio-temporal interruption in the geometrical career of the filled object) is sufficient, it is in no way necessary . What we needhere is just the spatio-temporal continuity of the filler. With reference to figure 9.2, supposefor instance that we simultaneously operate a disintegration and a creation on filler a, replacing it with an indistinguishable filler a'. Then h will stay the same. As a matter of fact, we could simultaneously intervene both on the host and on the guest, and h would be left untouched. Supposeyou take holes to be a kind of accident. The feasibility of the destruction-creation experiment would suggestthat accidents could migrate , contrary to somerather uncontroversial theories. Take, by contrast, the view that properties are individuals, or tropes (as the individual shape or color of this table). It is important to remark that holes can survive the destruction of the individual properties they are dependentupon. For the friend of individual properties, the individual shapeof this table would be destroyed in the destruction-creation experiment; other properties of other objects would seemsidle to say that h has beenreplacedby an indistinguishable h'. Holes are ephemeralentities, but sometimesthey die hard. Survival of entities of a kind. If spatio-temporal continuity of the guest and of the host is a criterion for the identity of a hole, then we have an important consequenceas to the survival of holes through changes in kind. A hollow can becomea cavity, which in turn can becomea tunnel. It is one and the sameentity , but this entity undergoestransfonnations (as figure 9.3 illustrates). This brings us back to the natural history of discontinuities. Moreover, it squaresperfectly with our hypothesis that a hole' s essenceis given by the property of being fillable: what makesthe difference among the various sorts of holes is the way in which the filler is kept at its

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Samenessand Non -Substance

time

.

Fipre 9.3 ofthekindofhole . modification Spatia-temporalcontinuityensuresidentitythrough

tl

..

FiI8e 9.4 Survivalof a holein a temporallyscatteredobject.

place. Independent of gravity and friction, in an object with a hollow , a (deformable) filler could escapein only one way; in an object perforated by a tunnel with two mouths, the filler could escapein two distinct directions, corresponding to the two openings; and in an object with an internal cavity, the filler could escapein no way. Is spatio - temporal continuity neceaary ? Consider figure 9.4. The destroying - and - creating machine acts at tl and at t2 . But at t2 something unusual happens : exactly the same object a is re- created as it existed before it was replaced by the indistinguishable object b. Object a is a peculiar kind of object : a temporally scattered object . By our previous argument , h would be the same from start to finish . We would thus have a surprising situation in which a hole after a while re-encounters its original host (in the same way in which a table can be owned by the same person in two distinct phases of their respective biographies ). Now , imagine that we skip the intermediate phase between t 1 and t 2 by just canceling b from the

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9 Chapter

universe. Nothing exists in the universe during that interval. Then a will be the samescatteredobject, but should we think of h as the samehole? In other words, is spatio-temporal continuity necessaryfor the survival of h? It does not seemthat there is any clear answer. But the problem could be completely reducedto the problem of the status of temporally scattered entities in general, for there is surely an element of magic in the existence of a as a scatteredobject over time. Thus, a definite solution could safely be postponed to the moment in which the status of the temporally scattered object itself has been made clear. Here it is sufficient that some law ensuresthat the phaseafter t2 be counterfactually dependenton the phase before tl and not , say, on the phasebetweentl and t2' One should not be confounded by a bad analogy here. Imagine that a comesinto and goesout of existenceseveraltimes very quickly . It would be impossible for us to tell a properly persisting object from an object like a, which is temporally scatteredbut rapidly oscillating in existence(in the sensethat a continuous processis cinematographically representedby a successionof static frames). If a had a hole in it , a properly persisting object could be kept there. It seems, then, that the continuity of function would be ensured, and that this would settle the question of the hole's survival. But the analogy is misleading, for a and the hole in it would be unable to keepat its place another temporally scatteredobject that existed at the times at which a did not and that moved relative to a. One could even have the impression that thesetwo objects go through one another. The function (e.g. of holding an object at its place) will not be performed ' by the hole in this case, and the hole s survival will be questionable. Male branche-holes. Independently of the options one can take as to the nature of holes (be they immaterial bodies or parts of surfaces), one can hold either that they survive over time or that they are momentary entities . This alternative position would not be interesting hereif one held that substancestoo are momentary (ordinary objects would be sequencesof phases). However, supposethat substancesare not momentary but holes are. Holes would then enjoy the kind of existencewe attribute to shadows or to the image of JeanneMoreau on a movie screen(the kind of existence ' objects have in Male branche s world , as we learn from his Treatise on Nature and Grace). They would live for an instant only , being at once replacedby someother momentary and quite similar thing, and so on. In this case holes would not - in any sense- move. (Of course, there are

135

Sameness and Non - Substance

t

t

'

" t

-

-

+

9.5 Faaure FivedistinctMalebranche-holes.

some differencesbetween shadows or colored shadows like the image of Jeanne Moreau on the screen , on the one hand, and holes on the other hand. For instance, shadowsdepend on processes, while holes do not.) In particular , note that on this theory the puzzle of the two holes floating inside a piece of rubbery stufT(figure 7.3) would receive a perfectly straightforward account. The holes are new and different at each instant of time. Thus, we have here two holes hi and h2 at time t , a new and different hole h3 at time t ', and two more new holes h4 and hs (different from h3 as well as from hi and h2) at time tn, and so on. Seefigure 9.5. A modal analOgue . The problem of identity over time has a modal analogue : Could this samehole have been in another object? Could this colander have different holes than (but in exactly the samespots as) the ones it has? ' " Supposethat holes h and h are co- localized in two different possible worlds but are possessedby two different objects at theseworlds - objects that happen to occupy the same positions. Are h' and h" the same hole? For instance, suppose Mont Blanc were in the place now occupied by Monte Leone. Would the tunnel of Mont Blanc be the Simplon tunnel? (Of course, the problem does not arise on the view that objects in different " possible worlds are never identical- though they might have counterparts " . In the caseof holes, such a view is related to the theory, introduced in the previous section, that hole- counterpartsare the modal analoguesof Malebranche-holes. If we subscribe to this view, much of what follows makes no sensewhatsoever.) Supposewe answerthe above question in the negative. Supposethat the Mont Blanc tunnel could not have been the Simplon tunnel were Mont Blanc in the place now occupied by Monte Leone. It follows that the fact

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that a hole is existentially dependentdoes not imply that it is rigidly de re dependent. Its dependencyon a material object is generic. This is coherent with our remarks in chapter 2. But there is an asymmetry here with respect to the non-modal case, where spatio-temporal continuity is sufficient: the nostrils in Lot 's wife are the same as the nostrils in the salt statue into which she is transformed. All this is related to the basic question of what it meansfor a hole to be " or to " in an object- what it meansfor an object to " possess have" a hole. On the Ludovician theory that holes are hole-linings, the relation between a hole and its host would be a mereological one: a hole is just a (potential) part of its material host. Hence, on that view the question of whether a hole could be in a different object pertains to the issueof whether an object could have different parts in different worlds- a rather controversial issue in itself. For instance, Roderick Chisholm set forth a form of mereological essentialismto the effect that if x is a part of y then it is part of y at any time that yexists . If we subscribeto this view, and if we identify holes with hole-linings, then evidently every holed object could not but have the holes it has. The samewill be true, trivially , if we take holes to be parts of their hosts' surfaces. But if the true theory is that holes are immaterial bodies, then they are not parts of their hosts, as we pointed out in chapter 7. Does that mean that one object could have different holes? In his lectures " Naming and Necessity," Saul Kripke maintains that if this table is made of this matter then it is so made in every possibleworld. If he is right , then the relation of that holds betweenthe matter and the table has the modal strength of an identity relation. Thus, our question becomes: Does the relation of that holds betweenan object and " its" hole have the modal strength of identity ? Once again, the answer is obviously in the affirmative if we seeholes as hole-linings. But if holesare immaterial bodies, the answer seemsto be in the negative. If you can change the host and leave the hole intact - as the experiment of the destroying-andcreating machines suggests- then there is no reason to suppose that a hole must be hosted by the same object in every possible world. Once again, holesare onto logically dependenton their hosts; they are not , however , rigidly dependent. The doubtful host. We have seenthat holes do not necessarilydepend on their hosts. But they do necessarilydepend on having actual hosts. You

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cannot take any empty region of spaceand say that there is a hole there, insofar as you can imagine a potential filler. Nor can you take any full " " region of spaceand call it a filled hole. Counterfactual reasoningshould not allow you to seea hole where there isn' t any. Holes needsurfaces, and surfacesare actual. In the same fashion, we can argue that counterfactual reasoning does not allow one to seepossibleholeseverywhere. You can point at an empty " " region of spaceand say Here there could be a table. And you can point " at a table and say, very reasonably, Here there could be a hole." But it would be awkward to utter the latter sentencewhile pointing at any region of space. It would be awkward, for instance, to go back to the initial region of space, where you just said that there could be a table, and say " Here there could be a hole." Of course there could be a hole there- but only insofar as there could be a table (or some other object) and insofar as the table could be holed. The oddness of this situation seemsto derive from the fact that the possibleexistenceof a hole is always counterfactually linked to that of its host. Accordingly, where no host is specified, the possibleexistenceof the hole is two worlds away, as it were, not one. This is why the awkwardness disappears when the host is specified, and a certain symmetry between material and immaterial objects is restored. It doesmake senseto point at some hole and say " That hole, which is there, could be here." Holes do not migrate. Consider now the situation illustrated in figure 9.6. Someonemight find it natural to describeit as follows: " Look , there is a

9.6 Fiaure Ajumping hole?

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water @

Figure9.7 A bubblemovingfrom a waterlayerto an oil layer.

hole here, call it h, that is literally jumping from an object (a) to another " (b). We would not accept that. Even in the case where a filler actually leavesa and migrates to b (leaving no hole behind, and making one where it lands), the filler does not carry its hole along with it . Holes are existentially parasitic on their hosts and cannot live without them- they are like fish out of water or shadows out of light . In fact one cannot even take a hole out of its host. One may think of a hole being in something as a case of perfect fit : a hole and its host fit each other perfectly, like piecesof a three-dimensional puzzle. But one cannot undo the puzzle- one cannot remove a hole from its host. (This is why, for instance, one can easily imagine a destroying-and-creating machine acting on material objects, but not a similar machine acting on such immaterial entities as holes. That is, one cannot think of a machine that keepsdestroying and creating a hole in a fixed object, so that at each instant of time t the host is still there but the hole is different from (though perfectly similar to) the one that was there at t ' < t. Suppose the object in question were a metal disk, and supposesomebody turned off the machine after a few minutes. Then the hole in the disk would disappear, leaving a perfectly filled disk - and that is absurd.) The same applies to other situations where migration would perhaps seem intuitively more acceptable- for instance, when a stone sinks through many layers of pudding, carrying its cavity (i.e., the cavity it fills ) with it , or when a bubble risesfrom a water layer to an oil layer (figure 9.7). In the latter case, the correct description involves two holes. We first have a bubble bl (a cavity in the water layer), then the same bubble opening into a hollow hi plus another hollow h2, and then the second hollow alone, dignified as a bubble in the oil layer (b2). tp-boles. Of course a certain feeling remains that we are confronted here with a single entity . Seeingthe entire scenewould most likely produce an

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impression as of a single entity slowly moving from one layer to another. To accommodatethis natural feeling, we introduce the auxiliary category of " cp-holes" (of which the thing one may believe one seesin the example of figure 9.7 is an instance). cp-phenomena are known to scholars of perception, who sometimes mention the example of two lamps blinking alternately at a certain speed and being seenas a single moving light . There is no real movement in the perceived situation , and the illusory movement is called a cp-movement. Likewise, cp-holes are composedentities that look perfectly unitary . They are typically composedholes, but their essentialproperty is that they can survive the death of their components. Typically , but not essentially, cpholes can undertake migrations resting on the survival of their fillers (e.g., the portion of gason a bubble like the one depicted in figure 9.7); typically , but not essentially, the component holes could on the one hand be where they are just by chance(compare the lamps of the standard ~ -movement, which could switch on and ofT by sheer chance); on the other hand, the filler itself could changein one of the countlessways fillers do, and this has no necessaryconsequencesfor the identity of the cp-hole. The notion of a cp-hole becomesparticularly important on the view that every hole is a Male branche-hole, for in that caseeverything that we now regard as an ordinary hole (enduring over time) would be a series of instantaneous holes and hence a cp-hole. (The converse does not hold: cp-holes neednot be made up of Malebranch e-holes.) Cartoon boles. The notion of a migrating, removable hole is so intriguing that it fits perfectly the magic world of cartoons. A famous scenein Who " Framed Roger Rabbit? features some fellow taking a " ready-made hole ' from a box it looks like a vinyl record, but the label HO LE S' on the box informs us of the difference- and sticking it onto a wall in order to escape. The sameidea may be found in many other cartoons (in YellowSubmarine, for instance, a hole in Ringo Starr' s pocket is used to free SergeantPep" ' per s Band). These strange holes, which we may simply call cartoon " holes, are a sort of object-independent two - or three-dimensional portable patches that can be manipulated and which interact with material objects much as solid or rubbery stuffs do (sometimes they seem to be corrosive). You can, for instance, take them out of an object, move them, sometimesdeform them elastically, and of coursestick them onto a wall to produce a real hole (typically a tunnel). They fall in the samecategory as

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detachablecolor patchesand floating shadows- like the shadow of Peter Schlemihl in Chamisso's wonderful story. All these- cartoon holes, detachable patches, object-independent shadows- are the results of a processof reification. They wereessentially immaterial entities, upon which we have illegitimately imposed a thingish character. To be sure, a world in which theseentities exist is nomologically very distant from ours; but there is some conceptual and perceptual affinity that makes these entities a cherished object of literary and cinematographic fantasy. There are also some relations among theseother-worldly entities. Take a moving, independentshadow. You can make it coincide with a hole. The shadow is in the hole, but without occupying it as a filler does. This case could be considereda re-qualification of the spaceinside the hole (though matter here cannot account for the requalification).

10

Ways of Holemakiog

How do holes come into existence? There are several different ways of creating a hole- of transforming an object into a holed object, or a holed object into an object with a hole of a different type. Digging comesto mind first. To make a hole in the ground is typically to dig a hole. You can do that with your bare hands or with the help of some tool or machine: a scoop, a shovel, a spade, a hoe, a dredge, a bulldozer, or what have you. Furrowing is also a form of digging, though this processis typically related to the creation of grooves. Drilling is another process. Usually you do not dig a hole in a wall; you drill one. But you can also drill into the ground, as when boring an oil well. Carving, gouging, and the like are also typical hole-making operations , as when you hollow out a trunk to make a canoe. All of the above are extremely complicated ways of holemaking, involving removal of stuff and repeatedcutting and gluing of the surface of the object that is being dug or drilled . A simpler processis continuous deformation of the object' s surface, consequent(for instance) on the exerciseof somepressureupon it (figure 10.1). This is a relatively smooth process, and one that does not involve any cut. Still a different way of bringing holes into existence, or transforming a holed object into a differently holed object, is through piercing, punching, or puncturing (as when you make a hole in a.balloon). All theseoperations can be seenas forms of cutting, and typically they have the effect of creating a hole without any removal of stuff. The outcome is usually a tunnel or , as in the example of a punctured balloon, the transformation of a cavity into a hollow. . There are some important differences among Topological guidelines theseways of making holes which tie in with our criteria for distinguishing among different types of holes. For example, even if we can do without cutting or gluing in the caseof hollows, the samedoes not hold for tunnels: the making of a tunnel always involves a perforation (a breaking through

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Figure 10.1 Digging involves removal of stuff; deforn1ationdoes not.

of somewall or surface), or elsea gluing (a joining of edges). This seemsto hold irrespectiveof the processwhereby a tunnel is brought into existence. For instance, one digs a tunnel by first digging a hollow and then breaking through to the other side, or by digging two hollows that eventually meet and fuse (think of the British and the French digging under the English Channel). The caseof drilling is perfectly similar. Thus, in general we may describethe making of a tunnel as a sort of by-product : one createstunnels by digging hollows. The qualitative difference, the magic moment of perforation, comesonly at the very end. The creation of tunnels is always accompaniedby an abrupt changein the topology of the object- a spherebecomesa torus, for instance. Usually the creation of a hollow is not so characterized, but there are exceptions . For instance, take an object with an internal cavity . As we know, in this case two surfacescan be distinguished: the external surface of the object and the internal surface, which delimits the cavity inside the object. If you drill a tunnel so as to break open the cavity, thesetwo surfacesare brought into contact with eachother; the topology changes, and the result is a hollowed object with a single surface. More generally, the relevant rule here is that the creation of hollows or tunnels in an object with n surfacesalways yields an object with m ~ n surfaces(i.e., possibly a decreasebut never an increasein the number of the ' object s surfaces), whereasthe creation of internal cavities always yields an object with m > n surfaces. The natural history of discontinuities (or at least a conspicuousportion of it ) is obtained by reiterated application of this simple rule.

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Instantaneoustunnels. The caseof a hole drilled to break open a cavity inside an object is intuitively a very safe one, but it gives rise to some puzzles. From a certain point of view, you reach the internal cavity by " " drilling a tunnel that connectsthe two surfacesof the object: the external and the internal. But - mirabile dictu- this tunnel ceasesto be a tunnel as soon as you finish drilling it . It becomesa proper part of the resulting hollow as soon as it comesinto existence. We could say that such an " instantaneous" tunnel is no tunnel at all: there was a cavity, and now there is a hollow , but at no time of the enlargement of the cavity did a tunnel exist. Yet intuitions diverge. Suppose the object in question is much bigger. Supposeit is a large planet, and suppose inside that planet there is a large cavity. Suppose further that there are people living on the planet' s outside surfaceand also populating the internal cavity. And supposeone day somebodyfrom the outside surface digs a deep hole into the ground and eventually reachesthe internal cavity . Two different worlds come into contact. Two different peoples meet. Now there is a tunnel connecting one world to the other. Why do we have such a strong tendency to speak of such casesas involving tunnels? Our answer is similar to the one we offered in chapter 4 when we discussedthe related question of whether parts of tunnels are tunnels themselves. We take them as tunnels becausein a sensethey are tunnels: they are not tunnels in the object, but they are tunnels relative to a (potential) part of the object. The aforementionedtunnel is not a tunnel ' through the planet, but it is a tunnel relative to the planet s crust. A hole through a wall connects a room (an internal cavity) with the outside, yielding a hollow. It is therefore not a tunnel through the house, but it is a tunnel through the wall. If you separatethe wall from the house, then even the hang-a-ring interaction pattern works. (The pattern does not work if you leave the wall where it is, and that is why it seemsinappropriate to regard the tunnel as suchas a (potential) part of the hollow. Compare the caseof the " dumbbell cavity" discussedin chapter 7.) How can cavitiesbe created? A first possibility, of course, is that we form an object with a cavity by fusing together two objects with facing hollows (figure 10.2). In that casethe total number of surfacesremains the same before and after the creation of the cavity. But that is no exception to the rule suggestedabove. Here a cavity is created by creating a new objectone that has.a cavity inside it .

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Fiaure10.1 . Fonninga cavityby fusingtogethertwo hollowedobjects

But can a cavity be produced inside a given object? And can a cavity be produced from the inside of the object, without any cut in the external surface? The fact that a cavity has to be produced from the inside of the object is not in itself a problem. A hand need not have introduced itself through any tunnel, nor need it have existed inside the object before the cavity was produced (as a sort of worm eating its apple from inside); indeed, we do not need any such invisible hand. Imagine one possible physical processwhich could lead to the production of a depressionin a surface: The matter inside the object moves in such a way as to withdraw (for instance, by increasingits density in someinternal part). At the surface a hole would appear. On the samelines- one could argue- it is possible to account for the increasein volume of somecavities (though not for their origin) by saying that the stuff all around them withdraws. One is reminded hereof the humorous discussionof the issueoffered by Kurt Tucholsky in his 1921sketch " Where Do the Holes in CheeseCome From?" Some parties think that holes are due to the contraction of the cheese , others to its expansion- but of course that is not the point . It can well be the samephenomenon under two different descriptions. The very contraction of the cheesewith respectto the center of a hole may be due to the uniform expansion of the cheesein all directions. That is the idea behind the intuitive model of the expanding universe. If the universeis like a huge, expanding plumcake, the inhabitants of each raisin may have the impression that the other raisins move away from them- that they are at the center of an expanding universe. Likewise, if the universe is a uniformly expanding chunk of Emmenthal, an inhabitant hanging at the center of a cavity would have the impression of a cavity that gets bigger and

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1

F~

10.3

Fusion of a hollowed object and a tunneled one, yielding a hollowed object.

seemsto be withdrawingfrom him, while in reality it is bigger: the cheese . expanding Beit asit may, couldthecontractionor expansionof thestuffbeall that mattersin the caseof a cavity's formation? To this question,our answer is in the negative : In order to createa cavity whereno discontinuity exists,onemustcut. Cutting morphologicallyalignscavitiesand tunnels. Evenif cavitiesdo not haveedges , their birth cannottake placewithout Caesareansection(starting with a pointlike internal fracture), and the sameappliesto tunnels. Note incidentallythat wehavespokenof creatinga cavityby fusingtwo hollowedobjects. Referenceto hollows is important here. If one of the initial objectshada tunnelinstead,asin figure10.3, the resultof thefusion would be an objectwith a hollow, not an internal cavity. By contrast, if eachobjecthasa tunnel, then the resultcould be an objectperforatedby a tunnel; but it couldalsobean objectwith an internal, doughnut-shaped tunnel-cavity (think of joining two V -shapedtunnels, eachmirroring the other). Not surprisingly, thesemodesof combinationreflectsomebasic patternsthat we havealreadymentionedin our mereologicalremarkson hole-dissectivity : Fusionis in somesensedual to splitting, andfor this very reasona taxonomicaccountof dissectivitycanbealsoilluminatingfor the es(seeagainfigure7.15). studyof hole-makingprocess . Whenyou aremakinga holewith a drill, you areat Makingfilled Iaoles the sametime filling it ; the drill 's point actsas a filler of the hole that is . Indeed , it typically beingdrilled. This is not suchan unusualphenomenon occurswhenyou makea holein a liquid. By dippinga woodencaneinto wateryou producea completelyfilled hollow (as in figure 10.4a), and by

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.

c

FIawe10.4 Makingfilledholes: a hollow (a) and two cavities (b, c).

throwing in a stone you automatically create a perfectly filled internal cavity (as in b). If you throw a hollow billiard ball (c) rather than a stone, you make an internal cavity as well. In the last case, however, it is a partially filled cavity. It is in fact a cavity that is partiaIly overlapped by another cavity: the cavity inside the baIl. The host of the latter (the material the baIl is made of ) becomesa partial filler of the cavity in the water. Holes in liquids cannot survive unlessthey are filled, with the possible ' exception of spinning liquids and of the Red Sea under Moses supervision . We could even imagine a magnetic liquid in which we could produce a depression; in this case, the filler of the depressionwould presumably be a magnetic field. Note that in the case of holes in liquids the host stuff is irrelevant to the form of the filler - unless the filler is itself liquid , as in the case of an underwater bubble. ( Note also that if we consider the theory that spaceis bare matter we now have the elementsfor saying that creating holes is either to dequalify or to requalify matter, depending on whether the hole is empty or filled.) A related question is: Can there be cracks or fissuresin liquids ? It would not.seemso. But think of two wavesin water, one sliding over the other in the opposite direction; maybe that does generate a fissure in water. Or think of the bottom section of a whirlpool , which becomesthinner and thinner and in the end vanishes into a filamentous crack. Prefabricated and ready-made holes. Before moving to the difficult topic of holes in artifacts, we shall mention two ways of creating holes by assembling , where the assembledpi~ fuse perfectly so as to produce a connected , unitary object. The first way offers what we call prefabricatedholes: you have a holed object, and you fuse it with a (bigger) non-holed object. As a result, the

Ways of Holemaking

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-

)

~

-

7

~

Fil8e1 aprefabricatedtunnel Oeft) and a ready-made one (right~ Using latter is now an object with a hole. For instance, in the left portion of figure 10.5 a prefabricated tunnel is attached to a squareobject to produce a tunneled object. The secondway offers what may be called ready-made holes. It consistsin adding non-holed parts to an object in such a way as to produce a hole by fusion. A typical example is a handle. The result of attaching a handle to an object (as shown on the right in figure 10.5) is always a tunneled object. We do not have much to add on thesenotions, but it is useful to keep them in mind in our discussion of hole-making techniques, for they are indicative of the fact that holed objects can sometimes be produced by adding something to a non-holed object- i.e., by joining and fusing rather than cutting, compressing, or removing stuff. With prefabricated holes we have in some sense a more powerful, general-purpose notion : Although one can have prefabricated holes of all types, it is rather unclear whether ready-made holes include anything but handles. From a certain point of view, one should perhaps say that the only way to ready-make hollows or cavities is to use prefabricated holes (e.g., a prefabricated tunnel can be used as a ready-made hollow ). But one could also take the opposite stand and argue that virtually everything can qualify as a ready-made hole. For instance, one can think of using a flat tile as a ready-made hollow , as long as the relevant piecesare attached in a suitable way. Or one can even think that wrapping a blanket around a stone is a way to ready-make a perfectly filled cavity (or a perfectly filled hollow or tunnel in casethe wrapping leavessomeopening), just as throwing a stone into a lake is a way of making a real ( perfectlyfilled ) cavity. Or , again, one can perhaps ready-make holes by separating two portions of spacewith a diaphragm. Be it as it may, both ready-made and prefabricated holesdiffer from real holes in one important respect: They are removable. They account for the only reasonablesensein which one can speak of a hole as of a detachable

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thing - the only reasonableway in which one can speak, loosely, ofa hole migrating from one object to another. They are- one could say- the only available portable holes. Diaphragms. A diaphragm is but a material object or a part of a material object. But the notion of a diaphragm is conceptually tied to hollows, perforating tunnels, and internal cavities. A diaphragm separatestwo empty portions of spacein a material object, and is conceptually dependenton holes; wheneverone thinks of a diaphragm, one thinks of a hole. A question then arisesas to the relative dimensions of diaphragms and holes, and this is a matter of proportions : the dimensions of the hole determine the acceptabledimensionsof the diaphragm. This is also related " " to a question discussedabove: Are diaphragms and other walls partial fillers (in which caseone could also question whether they would be diaphragms any longer), or do they split holes into two halves(thereby generating two new holes)? (We sometimesimagine diaphragms as localized in tunnels, though this is not quite correct. Think of a diaphragm separating the two mouths of a tunnel, as opposed to a longitudinal diaphragm. As soon as the diaphragm is created, the tunnel ceasesto exist, yielding two hollows.) . Now , does the existenceof diaphragms So much for conceptual dependence depend on the existence of holes? Consider an analogy with hammers. Imagine a world where there are no people left. Do hammers remain hammers? In a sense,yes. The essenceof a hammer is dispositional, and it has to do with the origin of the hammer: Hammers are things whose perfection lies in their being used as hammers (and which were manufactured for this purpose). A hammer is, qua hammer, dependent on the performability of certain operations. In the samefashion, one could maintain that diaphragms would even survive the elimination of holes from the universe. But if the essenceof diaphragms werejust dispositional, then any object that could be used as a diaphragm would be a diaphragm and that seemshardly tenable. Any object, to wit , is a potential diaphragm, but that does not make it a diaphragm. After all, not every potential hammer is a hammer. Diaphragms have an irreducible artifactual flavor. In the previous chapter we suggestedthat counterfactual reasoningdoes not allow one to seeholes everywhere. Now we are suggestingsomething similar with respectto diaphragms. But note the difference. One can point at a table and say " There could be a hole here." That is reasonable, we

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said , whereas it would sound awkward to point at an empty region of " " space and say There could be a hole here, because this would take us two worlds away . Now consider the symmetric case. Suppose somebody " " points at a coin and says This could be a diaphragm . Why does this sentence sound more acceptable than the previous one in spite of the fact that it takes us, so to speak, three worlds away ? The reason - we fear - is simply that we have a certain preference for material reality , after all .

Holes, artifacts, and mock holes. Artifacts introduce severalcomplexities into our account. As we saw in chapter 8, there is a clear notion of a hole relative to someexample of causal closure. This has consequencesfor the theory of holemaking. If one adopts a strict view on the matter, one will say that we never create a hole merely by assembling pieces. Fusion is essential. Likewise, we do not make a hole by tying a loop in a rope. The proof is also perceptive: You do not perceivethe would-be hole as a discontinuity , as the discovery of something missing. You do not tend to intuitively restore the continuity on the surface. If one is lessstrict about causalclosure, one will acceptthat the drawing on the left in figure 10.6 representsa hole in an aggregate. As it is not a hole proper, we may term it a mock hole, although the modifier " mock" contains too much understatement. Mock holesare not just a sort of imitation , in the sensein which a black spot could be said to imitate a hole. Mock holes do most of the things that holes do; the difference is that in their case the host is not a unitary ,

Fiaure10.6 Somehole-shapedsuperficialities arenot realholes.

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_

IIII

~

Fia8e10.7

A mockhoUow(left) canbeclosedto obtaina mockcavity(rilht ).

FiI8e 10.8 Buildingmockholesfrom realholes.

connectedobject but an aggregate. Just as causally closedaggregatesmay " be regarded as " mock substances , holes in such aggregatesare mock holes. However, the host of a mock hole need not be an aggregate; a loop such as the one depicted in figure 10.6 qualifies as a mock hole too. (For another example, think of two arms of a unitary host joined in such a way as to form a mock tunnel.) We can also have, of course, mock hollows and mock cavities of the most different sorts. In figure 10.7 we give some additional self- explanatory examples. We only insist on one point : What gives mock holes their hole like character is some functional property . They behaveas holes in certain appropriate circumstances, under which the holed aggregate is causally closed relative to some operation. Moreover, something can at the same time be a real and a mock hole, though not necessarilythe same type of hole. For instance, in the left diagram in figure 10.8, h is a real hollow in a but it is a mock cavity in the aggregatemade up of a and b. The right diagram shows a similar case, where a hollow h fuseswith a mock tunnel h' in an aggregatecomprising the hosts of h and h' as a proper parts. Unlike the case on the left, the host of this caseshows in addition that what counts as a hole of a certain

Ways of Holemaking

0 0

0 ~ 0 0 0 0

Figure10.9 A foldabletunnel?

kind (here a hollow , h) may very well becomepart of a mock hole of the samekind. (Compare this with the caseof prefabricated holes introduced above.) There are, finally , some rather dubious cases. Supposeyou make a tunnel through something which turns out to be a folded piece of cardboard (figure 10.9). After unfolding it , you realize that you have made many tunnels, not one. Are these parts of the initial tunnel? Indeed, what happened to the initial tunnel- is it a sort of scatteredtunnel? Can we speak of it as a tunnel in an aggregate? We shall leave these questions to the reader' s imagination, noting only that by perfectly filling up the initial tunnel you also fill up the smaller tunnels (at least, until you unfold the cardboard) and that by perfectly filling up the eight tunnels you do not fill up the initial tunnel unlessyou fold the cardboard and produce a fusion of the eight fillers. At this point one might object: " Look : the world around us is entirely constituted by little particles, which are taken together to form ordinary " , are they not? objects. So in the end all of theseobjects arejust aggregates The answer is that it does not matter much. If they are, then every hole in an ordinary object will be a mock hole. But this does not diminish the fact that we understand the concept of a mock hole by referenceto the concept of a hole. Hole destruction. Holes die hard, we said. But how do they die, if ever? There are various cases. For instance, the matter that surrounds the hole may contract and simply close up on itself. Or we can fill up the hole. In general, as we know , to fill a hole is not ipso facto to destroy it . If the matter used to fill the hole (the guest) is different from the matter surrounding the hole (the host), and if the two matters do not fuse or

IS2

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8 .J,

. by a big bole produced by a big meteorite

becomehomogeneous, the hole continues to exist as a filled hole. At most we can say that betweenthe host and the guest there is a fissure. ( This is another way to createfissures.) On the other hand, if the matter of the host and that of the filler do fuse or become homogeneous, then the hole is destroyed. There is a somewhat vaguedifferencebetweendestruction and enlargement . Think of the surfaceof a planet. A little meteorite falIs down, making " a hole in the ground; then a much bigger meteorite falIs down onto that samespot, producing a much bigger hole (figure 10.10). We would say that the first hole is destroyed upon creation of the second- not that it is enlarged. This is even more evident if we imagine that the big meteorite falls onto an area marked by severallittle holes, which are thereby wiped out. This does not mean, of course, that one cannot enlargea hole without destroying it . It meansthat a small hole can be replacedby a new, bigger one. Annillnadon. Digging is the result of first creating some fissures) and then removing an isolated part . It is, therefore, conceptually linked to fissuration- and this leads us back to the natural history of discontinui ties. A related operation, though of a different nature, is annihilation of some parts of an object. Of course we need not face directly the question of whether there is real annihilation in the actual world . But we do accept the logical possibility of annihilation . The metaphysical issuesare these: Does creating a hole by annihilation amount to creating a new solid body? Does it amount to creating new surfaces? These issuescould be discussedindependently from the hole problem. Supposeyou annihilated a layer of the matter constituting a prism so that

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the result of your operation was a smaller prism. Did you create a new individual , or did you just changethe old prism? Are the old and the new prism one and the same? In the caseof holes the question becomes: Does an object remain the sameafter you make a hole in it? Some casesare rather easily decided. If a plasticine prism is changed into a plasticine cone by defonnation, we say that the same matter was first a prism and then a cone. If a prism undergoesa continuous replacement of matter but retains its shape and its dimensions, we say that it is the sameprism and that its composition has beenchanged. And if a prism is slightly gr~oved on its surface, thereby loosing matter and changing form, we refer to some notion of degreeand try to settle whether it is still the same prism. We accept, for instance, that it still has some superficial parts in common with the initial prism. But if only some of the original matter survives, are we entitled to say that the prism survives? Consider the suggestion that it does not. An amoeba does not survive splitting; rather, some matter was first one amoeba and is now two. Prisms do not survive breaking; rather, some matter was first one prism and is now two. We could object that there is no unit of matter that wasfirst one prism and is now two, becausethere are now two units of matter, and no single unit of matter survives splitting . However, we cannot accept this suggestion unlesswe also accept that units of matter could be scattered. Final remarks. We began this chapter with the question of how holes may come into existence, and we naturally ended it by discussingwhether and how one can get rid of them. Do we really needto operate on holeless objects in order to make them holed? In a sensewe do not. The world could have been transfonning since its beginning without any alteration involving the holed status of its inhabitants. Just imagine a universeconstituted ab initio by an immutable chunk of Emmenthal cheese . Nevertheless , operations taking us from holes of a certain kind to holes of another kind are not futile. They servethe purpose of perspicuity. They show how a certain classof physical operations could be seenas internally connected in a way that mirrors our conceptual mastering of the notion of hole. This and the precedingchapter allowed us to explore broad philosophical issues,such as those concerning the nature of aggregates , of substance, and of causality (the cement of the universe). Unexpectedly, holes provide an unusual viewpoint on much more solid entities.

11

Hole Detection

A realist' s desire. A realist about holes would be pleased if there were some harmony between those basic properties that he found reasons to ascribeto holes and the major featurespresentedby our experiencewhenever this is linked (causally, in some standard way) to those regions of space where a hole is or is expected to be found. For instance, a realist should welcome the fact that some morphological features of holes are satisfactorily representedin perception; just think of the intimate relation holes have to certain shadow patterns (that is, in the end, to light). We find here two small difficulties and a genuine philosophical problem. First , the basic properties of a region where a hole is expected to be found are, to some extent, discontinuity and concavity, but the variety of acceptablediscontinuities and degreesof concavity is somehow unclear. This is not, in itself, a problem- or , better, it is not a radically new problem . It arisesevery time we analyze the ascription of vague concepts, and we will not pursue any investigation of it . Second, some inaccessibleholes are somehow distressing for the hole detector. Theseinclude internal cavities, which becomeapparent only upon dissection of the object, thereby losing their status as cavities; tunnels that hide some of their mouths and are therefore taken as simple hollows; hollows whoseedgesare gently slanting and which are difficult to tell from their environment; and other results of interbreeding of hollows, tunnels, depressions, and cavities. These facts do not present philosophical difficulties , though they do present technical difficulties. The main philosophical problem arisesfrom the fact that holes do not fit one very intuitive and reasonable account of perception. As Locke noticed (Essay, II -viii -6), if causality has to do with materiality , and if holes are immaterial, then a causal theory of perception does not apply to them. Is there any way out of this problem other than abandoning the causalfactor in perception? This will be our concern in the first part of the chapter. In the secondpart we will study the perceptual conditions under which we have the impression that we perceive a hole. We shall mainly

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concentrate on vision, though we shall not ignore other forms of perception (such as touch and hearing). Perception: The causalobjection. It seemsthat a philosopher who accepts that we perceive holes is like a philosopher who thinks that we perceive empty space. Have a look at a clear night sky. Spaceis everywherearound you. It is mostly empty. The big black spot is but empty space. But there " is no causal flow from it to you. Our philosopher will say: If holes are immaterial, then they could not possibly be known. They could not even be perceived, for in order for an entity to be perceiveda causal flow must proceed from that entity to our sensory systems. But no causal flow can " originate in anything immaterial. Joined with the thesis that we do perceiveholes, this reasoning would imply that holes are not immaterial. Consider an answeralong the following lines: " Why should holes bother you? We know of plenty of abstract entities. We could even perceivesomeof them, as in the caseof the North Pole, or as in the caseof the center of mass of a heavenly systemconstituted by two planets which rotate one around the other (if we float in such a center, we feel that gravity is equal to 0). Abstractions are immaterial. Thus, if we do perceive holes, and if they are immaterial, can they not " enjoy of a status similar to that of abstractions? It seemsthat we cannot hold at once that ( 1) holes are immaterial entities, (2) we perceiveholes, (3) a causaltheory of knowledge and perception is true, and (4) no causal flow can originate in anything immaterial. Thesis 2 is true- we do perceiveholes- and thesis4 appearsto be true too. Let us then start from thesis 3. At a first glance, we do not have much to say about this. We think that , even if in some casesa causal theory of knowledgecould be of no philosophical use(as, for instance, in the caseof our knowledge that 2 + 2 = 4, or that modusponensis logically valid ), it seemsclear that we had better acceptit for entities which are in spaceand ' time around us, as holes are. Still , causality s place in perception and its place in knowledge are far from precise. Let us consider abstractions for a while, for it seemsthat perception of them too is a rather odd affair. Not all abstract objects are perceivable; and if we perceiveabstractions at all , we do not perceivethem directly but only via the perception of somematerial object or entity (which instantiate them or with which they might be co-localized). Thus, we perceive the North Pole by perceiving, say, someicy region. Would this line of reason-

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ing, when transferred to the perception of holes, representa problem for our theory? Holes would be perceivedby virtue of the perception of holelinings (and sometimes of fillers, as in the case of a pink oil bubble in transparent water). And would this not resuscitatethe old idea that what really matters are hole-linings and hole-fillers, and that holes, after all, can be dispensedwith? It would not. From the fact that all you could directly perceive are hole-linings or fillers, it does not follow that you cannot perceive holes. Neither does it follow that holes can be dispensedwith , or that they just are hole-linings or fillers. We made a casefor the perception of abstractions only in order to show that it could be of help in the developmentof a theory of the perception of holes. Consider now the hypothesisthat holesare not abstractions, but are similar to abstractions at least insofar as they are not material. To perceive an immaterial entity would be to perceiveit mediately, through perception of some material entity on which it depends(e.g., the host) or to which it is spatially linked in somerelevant way (e.g., the filler ). You could perceive the hole by perceiving the hole-lining or the hole-filler , but the fact that your perception of the hole would be so mediated does not make it a counterinstanceto a causal theory of perception. It follows that thesis 1, according to which holes are immaterial entities, is not inconsistent with the other three theses. In the caseof the perception of abstract objects, the difficulty with the causaltheory of perception could be of a very generalnature (among other reasons, becausethe place of causality in a causal theory of perception is far from clear). If the only type of causation one allows is eventcausation, and if perception is to be construed as a causalrelation in such a way that the referentsuggestedby the content of the perceptualevent is the external term of the relation, then the only entities we could possibly perceiveare events. And this is implausible, for we surely perceivetables, stones, and other non-eventlike objects. At any rate, we have to weaken the causal condition in a causal theory of perception so as to make room for perception of things and properties and states of affairs. Now , in all thesecases one has easyaccessto the servicesof mediation; one can perceivematerial objects by being causally-perceptually related to some events involving them, perceive properties by perceiving material objects exemplifying them, and so on. The difficulty with holes is yet of another kind : Even in thesemediatedaccounts, it is difficult to explain their perception, for holes

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(unlike properties) are not strictly co localized with material objects. (Recall we perceivedby perceiving something whom the caseof the invisible man, else: his environment.) Then where does the causal flow we need originate? Thus, we probably should insist on a very weak causal requirement in the theory of perception. This would take the form of a counterfactual: if a certain fact in the world outside the perceiver were not the case, then a certain mental event would not be a perception. The fact in question would be the presenceof a hole in an object, on which the perception of the hole is counterfactually dependent. Here we no longer require that a causal flow originate from the object of the perception- that there be a sort of mechanismthat produces the perception starting from the object. And the details of the story that provides an account of the perception of the hole (e.g., the fact that we perceivepart of the surfaceof the object at which the hole is localized) now have no influence on the satisfaction of the counterfactual. Figural conditions. The foregoing were the philosophical implications of a theory allowing for the perception of holes. But what are the conditions under which a certain pattern prompts in a perceiver the specific responseswe would explain by crediting him with the perception of a hole? When does a perceiver have the impression of perceiving a hole ' ' (independentlyof a hole s really being there)? What , in Nelson Goodman s form of words, makes us classify certain perceptions (or certain pictures) as hole-perceptions? Obviously the appearanceof veridicality of a perception is by no means a guaranteefor the existenceof what we believe to perceive. But here we might (like the phenomenologist) allow ourselvesnot to care about veridicality . We might only consider the more modest purpose of discovering when the impressionof perceiving somethingfinds its way into our mental landscape. Figure and ground: White spots? Black holes? Take the visual casefirst. Gestalt psychology teachesus that the minimal conditions for the perception of any entity whatsoeverare those that involve the unity of the visual its isolation from its region of spacewhere the entity is to be perceivedand ' ' to see' and ' to use shall perceive lazily surroundings. (In the following we ' ' ' as abbreviations for to have the impression of seeing or to have the

IS9

Hole Detection

case for and

' veridical and exisimpressionof perceiving; wearenot makinga other factors inter tential perceptionof holes.) Thus, color, texture, _1 regions . veneand helpdiscriminatebetweentwo adjacentvisuaJ white tends to emerge. A Color is the first factor. Blackvisuallyrecedes ; black patchon a uniform white field will suggestthe presenceof a hole. But furtherfactorsdeterminewhichpart of the patternis morelikely to be perceivedasa figureandwhichastheground(thefigurebeingthe floating ). The figureis usually part, beyondwhichwe feelthat the groundcontinues convex, smallerthan the ground, brighter; its border is sharp, its texturetight. If brightness , convexity, texture, and so on are co-localized at a certainplacep, thenanythinglocalizedat p (or p itself) will likely be . seenas a figure. As is well known, thesefactorscan act antagonistically An areacanbe big but convex,anotheroneunsharpbut bright. Consider the two examplesin figure 11.1. In A, the circleclearlyemerges againstits white background . This is not the casefor the circle in B. Background , and the white regionin B is similarity winsoverconvexityand smallness visible seenasa part of a bright background , througha holein B. Figure 11.2 illustratesthe conflictbetweenfiguralfactorsat the levelof -of the whitespotpattern. Both A and B havebeenbuilt by superposition a white squareon a black one, but a slight inclinationof the superposed squaredramaticallyaltersthe perceptualeffect. In A we havethe impression of seeinga whitesquareovera blackone, whereasin B the patternis . Moveasgently that of a blackframeovera continuouswhitebackground asyou can the white squarein A to the positionit hasin B and you will " seethat a holeis born. (Onecould, of course , ask Whenexactlyis the hole born?" We do not know the answer , but onceagainthis is an instanceof a much more generalproblem. Whereexactlydo the outskirts of a city begin? WheredoesMont Blancstart?)

8

A

11.1 ~ A floatingpatchanda hole.

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A y~ 11.1. of a white Superposition

square 011a black one. (Adapted from Bozzi 1975.)

Perception of tunnels is therefore a.caseof non-standard perception of a figure over a ground- non-standard becauseof the non- cooperative interplay of Gestalt factors, which pushesback to the ground a portion of the visible scenethat could reasonably be treated as a figure. When you perceivea hollow , it is as if you were dealing with its perfect filler in such a way that it reconstitutesthe unity of the surface(and this agreeswith our perception of a discontinuity). Often the caseis the samefor the perception of a tunnel, apart from .the fact that here you needalso to see(a hint of ) a portion of the background at the other end of the tunnel. Figure and ground: On the importance of ha\ inK a border. We have seen a first characterization of the distinction betweenfigure and ground. We can refine the picture by introducing borders. Somediscontinuities in the visual field are seenas limiting portions of planes and objects. Consider a two -dimensional sketch. The objects thus limited are figures; the border is seenas belonging to the figure and not to the adjoining area of the ground. Following a suggestionby L. Shiman, we can expressbelongingnessor orientation of the border by drawing an arrow directed toward the region of the visual field that is the owner of the border. Take a simple set of four lines disposed so as to form a square (figure 11.3, left). Its favored interpretation is as in a; an interpretation as in b (that is, as of a sort of in the page) is not privileged. Of coursethe distinction is local hole square throughout , as is indicated by the suggestedinterpretation of the pattern on the right (a diamond over a square). Now , one basic law of segmentationof the visual field seemsto be that no oriented border (or portion of a border) can belong to more than one object. Ray JackendofTcalls this the well-formednessconstraint: " The

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D

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Pattern interpretation using oriented boundarie&

visual field must be exhaustively partitioned into regions, each with its own closed directed boundary, plus a background that lack boundaries." (Consciousnessand the Computational Mind , appendix B) JackendofT defends this view against putative counterexamples. For instance, two adjacent figures will produce a boundary with a double orientation , but this is not particularly problematic as long as the two figures themselves have other parts of their boundaries demarcating them from the ground. Though the well-formednessconstraint captures a rather pervasivetendency in the segmentation of the visual field, we feel that it does not constitute the end of the story. In the following section we introduce a counterexample that seemsto be intractable in terms of the standard " equation figures = portions of the visual field provided with oriented " borders. If we are correct, some old wisdom about the distinction between figure and ground deservesrethinking . The strange case of the rotating hole and the shifting border. U sing a computer with a program for generating slide shows, one can assemblea short motion picture featuring a rotating portion of a sceneagainst a still frame such as the pattern of figure 11.2. A few steps(slides) would suffice. Figure 11.4 shows the four first stepsof four such movies, each composed of sevenslides. (Three more stepslead back to the original configuration .) With only seven slides in 2 seconds, the effect is slightly unsmoothed; however, it is informative enough. As one can seealready by looking at the still snapshotsin figure 11.4, b will be preferably interpreted as a rotating patch and c as a rotating hole. (Of course, holes do not rotate, as we know from chapter 8. But our concern here is with perceptual phenomena , be they veridical or not ; our concern is with hole - perceptions , not with perceptions of holes.) What matters more , c and d present a weird effect: the internal border is seen as shifting along with the movement of the square area (which is, of course ,

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Fiawe 11.4 with framesand the pen:eptionof holesrotating. Experiments

unproblematic in b). This is particularly striking in d, where the border in question has no unity at all. But both c and d are troublemakers for the well-formednessconstraint, for the following reasons: ( 1) The border must belong to the frame, since the frame is surely a figure (we seethe lines in c and the white areasin d as distant ground). (2) Yet the border of the frame cannot be rotating or sliding along with the rotation of the hole; at most it undergoesa slight elastic deformation. (3) The hole seems,then, to -have a border of its own, which shifts restlessly. (4) Yet the hole cannot be figure, for one seesclearly the ground at the region occupied by the hole. We have here a caseof something that possess es an oriented border and is not a . yet figure This little experiment makes no big revolutionary claim. But if its conclusions are accepted, it seemsthat we must introduce further complexity that could replace the figure/ground dychotomy. One idea could be that in addition to figural boundaries there are topical boundaries, which confer a figural role on some portion of the visual field (the rotating hole in

Hole Detection

Figure 11. 5 What is the pe~ ptua1 dif Teren ~ betw~ upside down.)

163

concave and convex patterns? ( Turn the page

our movie) without at the sametime suggesting : that sucha role is played in the old . sense by figures Concavevs. convex. Frames are objects with one kind of hole: tunnels (which are essentialfor them). Holes come in other kinds, however. Our next concern is with hollows. In a sense, recognizing hollows is recognizing concave superficial discontinuities. But what factors take part in our interpretation of such gestalts? We are here confronted with a completely new variety of problems, some of which may be quite tricky . (After all , morphological features can hardly make up for the lack of topological singularities.) We shall just hint at a couple of them, starting from a rather simple case: hollows whose superficial gestalt is like the mark left by a ping-pong ball half-pressedinto plasticine. Figure 11.5 reproduces(slightly modified) a curious pair of patterns that were presentedin 1938by Kai von Fieandt. There is an apparent ambiguity in our perception of these patterns, owing to the fact that the illuminant direction is not uniquely determined by the shadow information . In the left pattern we have the impression of seeinga hollow in a bump (a crater on the top of a hill ), whereas in the right pattern we have the impression of seeinga bump (or a hill ) inside a hollow. However, these impressionsare reversedif we imagine that the light comes from below rather than from above. In fact, a moment' s reflection shows that there is only one pattern here: the one pattern is just the other pattern turned upside down. We thus seeat once that as far as concavity or convexity perception is concerned, orientation matters. The samepattern sustains competing visual interpretation as, respectively, concaveand convex. And part of the explanation is that shadow patterns like thesedeliver gradient

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Figure 11.6 Hot Tman's cosine surfa~ . The 3D interpretationof crests and troughsis reversedas you turn the figure upside down.

- --- ~~-.----~~~~---.-~-~ ~ ~~~~~~~ ;~~~~~~~ Fiaure11.7 A simplecaseof convexo - concaveambiguity.

information, that is, information about the metric structure and relative distancefrom the observerof the object' s surface. Similar structures are also present in other configurations, such as Donald Hoffman's " cosine surfaces" (figure 11.6). Here the texture information is displayed by a pattern of cosine waves rotated about a vertical axis. Of course, the pattern is unnecessarilycomplicated as a casein point : the same basic effect can be obtained with much simpler configurations (figure 11.7). Thesepatterns reveal that in the perception of concavitiesand convexities- as in many other perceptual phenomena- a major source of information is provided by surfacetexture. Sometimesthere is more to it. For instance, the cosine surfacewas originally devisedto show that the visual system organizes shapes into parts. Note that the singularities of this surface- the maxima and minima - are organized in co-axial circles. (Singularities of this kind are evident also in simple patterns, particularly patterns with tips sticking out of the ground.) However, the explanation of such phenomenaneednot concern us here. And we neednot enter into the controversy about the way in which textural information is processed.We

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=-

Fiaure11.8 An irregularityin a regularpatterncanbe perceived asa hole. ourselves with the general

observationthat the basicpercep -

tual information in question has to do with surfacetexture. Blanks and boles. It is barely necessaryto record that texture plays other important roles in perceptual stories concerning holes, and in some cases there is no additional actor responsiblefor the feeling that a hole is present . For instance, an irregularity in a regular pattern, such as a blank in a printed pieceof fabric, can be perceivedas a hole (figure 11.8). It may be so perceivedbecauseit marks a discontinuity in the texture, and we know that holes go hand in hand with discontinuities. Sometimes such irregularities are construed as holes in aggregates. Think of a missing book in a stuffed bookshelf, or imagine a narrow free region in a square crowded with people, or a small green meadow in a large, thick mountain forest. Thesepatterns bring us back to the exploded torus that we discussedat the end of chapter 8. There is no unitary host, henceno hole proper. But the gathering power of the context provides the necessaryglue to separatematter from immateriality . " Depicting boles. Supposeone were to ask somebody Could you please " draw a hole for me? This is an ambiguous request. Some pictures are hole-pictures (in describing their content one would make use of the concept of a hole), and some pictures are pictures of holes(their referent is a hole, even though one neednot be able to tell so by just looking at them; think of an extremely blurred photograph of a tunnel' s mouth ). A holepicture can be a picture of a hole, but it need not: one could draw holepictures even if there were no holes around. And a picture of a hole can,

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but neednot , be a hole-picture: one can try to portray a hole and yet come out with , say, a beetle-picture, or a shadow-picture. Now , under what conditions is something a hole-picture (be it or be it not a picture of a hole)? Think of a teachermaking a drawing in order to explain the meaning of the word ' hole' to her pupils. How would she do that? It seemsthat she cannot draw a hole without also drawing something else: the hole' s host. Well, hole depiction is not such a simple matter. Supposethe teacher draws a circle on the board. The circle definesa boundary which is not by itself oriented, and the picture inducesan ambiguous interpretation (a hole or a disk?). Or supposethe teacher draws a black spot on a white sheet. Shedid not depict the object at all - shedepicted the hole directly. This is an embarassingsituation , though; shedrew what is not there, and shedid not draw what is there. Thus, the problems with hole perception have an echo in hole depiction. The content of a picture is in some way linked to perceptual content: we could gain perceptual information about an object by looking at a picture. But we need not have the same content in both cases. Supposeyou are looking at the picture of a typewriter . It looks to you as if the typewriter is red on all sides- this is the content of your perception. But the picture representsonly one side of the typewriter . Although your eyestell you that the typewriter is red all around, the picture saysonly that the visible side of the typewriter is red. It says less than there is to see, though it shows what is visible; but it makes you seemore than it says. Supposenow that you depict a hole. In a sense,you can depict lessthan there is to see(the holed object, i.e., its visible parts)- provided that you depict them in such a way as to make the hole appear. The picture tells you that there is an object whoseparts are arranged in a certain way, and your eyestell you that there is also a hole. But in another senseyou could also depict more. Consider figure 11.9, which representsthe mouth of a tunnel in the background wall , partially covered by a tunneled sheet of white paper. One could argue that here we representsomething that is not there by using a positive mark: a black disk. The representational system usesmore entities than are in fact visible: it saysmore than there is to see. This caseresemblesone in which we use a symbol, say ' A' , to representa relation such as the difference between two objects. A representational ' ' systemin which each object has a unique name could avoid the useof A ,

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F"1I8' e 11.9 A holein a sh~ t of paperin front of theentran~ of a tunnel.

.

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Fipre11.10 A Maaritte -bole . for the fact that there is a difference between (say) objects a and b is ' ' ' ' representedby the differencebetweenthe symbols a and b employed to refer to them- and there is no need to state explicitly that 4 (a,b). Would black spots in the same fashion represent a hole in aconventional , intrinsically non- depicting manner? This is odd; figure 11.9 certainly does not fall into the samepictorial category as figure 11.10. Severalproposals could be put forward at this point . One should start ' by remarking that the fonn of the hole s mouth, as well as its surface, is representedin the same sensein which the fonn and the surface of the white sheetare. Now , one theory is that the black spot is also a kind of " " negativeentity (light is not reflected from it ), even though it is made of black ink : what actually exists is the white spot, which has an internal closedborder. A negativeentity (the tunnel) is representedhere by another

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negative entity (the black spot), and this makes the picture not say more than there is to say. Another theory has it that the black spot is a positive entity , but that representationsare primarily relations not between objects but between facts- that is, object representation is derivative of fact representation. One object (the picture) represents another (a certain portion of the world ); one object (the black spot) representsanother (the tunnel). But the former can representthe latter only becauseof somerepresentationalfacts involving them: the fact that a black spot is in the middle of a white spot representsthe fact that a certain entity is in the middle of another (that a tunnel opens in the middle of a white sheet). Dominating gestalts. We said that the thesis that there exist immaterial bodies is granted support through its theoretical contribution to the explanation - among other things- of certain perceptual phenomena and contents (also of causalinteractions and shapedescription, but let us leave " these aspectsaside). Here one could object: Even though in somecases holes are dominating, and impose themselvesupon us, in most casesimmaterial bodies are not. Consider for instance the unoccupied space in your kitchen. Can you represent it perceptually at all? Do you see this empty space? What impinges upon your mind are, rather, certain middle " sizedry goods piecesof furniture , cutlery, food, walls. Theseare dominating , and pour cause, for they are the basic units for our action, and perception is primarily oriented to action-relevant units. ( This problem was taken seriously by Husserl in Ding und Raum, particularly appendix VII .) Note also that every time theseunits move, the empty spacein your room undergoesa rather complicated modification , of which you cannot keep track without difficulty ; by contrast, the units themselvesdo not changeat all. However, all this could be just a matter of perfectly contingent fact. We can accept that the action-relevant units are the favored candidates for ) normally are perception and, further , that material objects (substances this is not hold that one could such candidates. Nevertheless, necessarily so. First , when one goes into a room , one can measurethe empty space one will cross; one can perceiveand judge that a certain degreeof freedom is left to pne. Moreover, that perception is primarily oriented to material objects can really be contingent. Let us elaborate on this.

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Think negative. This is a typical exercisefor painters: Keep an eye on the hole and not on the doughnut. In Rubin's vase-profile reversible figure (reproducedhereas figure 11.11), this exerciseis dramatized and simplified so that the hole could just be filled by another material object (think of two persons whose profiles perfectly fit the profile of the vase, and who hold up the vase with their faces). This suggeststhat the fact that substances are dominating could be just a matter of accident. In the " Seaof " Holes describedin YellowSubmarine(figure 11.12), holes are dominating and the host object- the plane- is no longer salient. Were we on an infinite plane, we would no longer be perceptually " interested" in that " plane. We would say The world is composedof n holes," and n would be the total number of perceptual units (not n + 1: the plane would not be an

.11 Figure 's11 -profilereversible Rubin vase . figure

Figure11.12 TheSeaof Holes .

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Figure11.13 A hole, an island.

additional unit ). Thus, one could accept the issue of dominating gestalts and yet deny that substancesare the sole dominating units. The reason underlying this denial should be clear by now: Domination dependsheavily on a context. Gestalt theory teachesus so. We will say that domination is a holistic property, like every gestalt property . A red patch imposesitself in a green context but disappears in a red one. Sounds and voices can easily be masked by structured auditory patterns. Domination can also act antagonistically though not destructively, as when two elementsof the same context, which should interact in such a way as to make one of them emerge, have an equal strength of domination . This is clear from the case of reversible pictures (such as Rubin's vase-profile pattern), and there is an analogy in the caseof holes (figure 11.13). We are not propounding, as we were in the context of figure 11.1, the thesis that a single pattern can sometimesbe seenas a hole in a sheet and sometimesas a spot on a sheet. Here we are talking simply of domination . Look at figure 11.13, and grant that you seeboth the hole and the island. Which one is dominant? The answer is that they both impose themselves.We pay attention to the hole, and we head for the island. Complementaryreasoning. That sometimesour reasoninggoesback and forth between the two extremes of positive and negative form is a fact everybody is acquainted with . Take a lake (a dead lake, with no tributary or emissary). We seemto talk above all about a filler - a huge mass of water. But now think of the lake's bottom . You inadvertently shift to what is outside this massof water, for the lake's bottom is not made of water; it is ground and stones. It remains the lake's bottom , though. So what is a lake? Is it just a big hollow , a big geographicaldepression? (Are there empty lakes?) Is it a certain localized massof water? Surely it is

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not both . ' Lake ' is used with certain principal descriptive purposes in mind , which vary accordin~ to our changes of interest .

Hole inversion. How far can we go with complementary reasoning and negative thinking ? Consider this. We can conceive of a being who perceives colors in an inverted way relative to us (Locke, Essay, II -xxxii - 15). For instance, he seesgrassas red and ripe tomatoes as green. The classical problem is whether color inversion would be detectable, provided that an individual undergoing it would have learned predicateslike ' red' and ' ' green on being exposedto red and green objects respectively, though he would have perceivedred in the way we perceivegreen and green in the way we perceivered. There are a number of argumentsfor and against detectability in inversion contexts, but no argument to the effect that an inversion of primary qualities such as shapeor sizemight be undetectable. In fact, the behavior of our unlucky fellow who inverts colors could be quite similar to ours, while the behavior of he who underwent primary quality inversion will surely look different. Jonathan Bennett, in Locke, Berkeley, Hume, describes" the case of S who, going by what he seesand feels, judges a certain jug to have the same sizeas a certain glasswhich is in fact shorter and narrower than the jug . . . . In this case, we can place the glassinside the jug : or fill the jug with water, and then fill the glass from it and throwaway the remaining water. . . . What are we to supposehappenswhen S is confronted by thesemanipulations of the two objects? . . . Each time we contrive a happening with the and the jug, S misperceivesit so that what he seesand feelsstill fits glass in smoothly with his original judgment about their sizes. . . . This requires us to creditS with such inabilities as following . He cannot seeor feel that the glass is inside the jug . . . he cannot seeor feel that the glass is full of water (or that water remains in the jug after the glasshas beenfilled from it ) . . . to keep S in ignorance of his initial sensoryfailure we have had to surround it with ever-widening circles of further abnormalities." (pp. 96100) Good. But Bennett does not consider the possibility that our S could have learned inverted meanings to associatewith ' inside' or 'out side' . The circles of abnormalities get wider and wider. Yet it would just be a matter of degreewhetherS would be able to invert someprimary quality conceptswithout inverting many others. Would a being who undergoesan inversion of primary qualities be in need of interchanging the concept of a hole with any other (named or

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nameable) concept in order to save his conceptual schemefrom general collapse? Can one systematically take insidesfor outsides and vice versa? Granted that inversion is a matter of degree, our thesisis that the concept of inside and that of outside are hardly invertible. Consider Aesop's fable of the fox and the grapes. The fox eventually ends up captive, but in its cage it never stops its bad habit to reason as it did in front of the grapes (when it knew that they were unreachableand consideredthem sour and unworthy). Now the object of its thought is the outside space. The fox thinks that there is no need to be on the other side of the bars of the cage after all , for everything beyond the bars is really inside the cage- and that the fox is the only free thing in the whole universe . Where does the fox go wrong? It cannot be a matter of dimension . only The fox cannot reasonably believeitself to be outside the cage- but not for the reason that the spacewhere it lives is smaller than the remainder of the space. A journey to Flatland will help us. Imagine a round plane world in which a circle has been drawn which corresponds to a cage. Even if the circle were such that almost all of the world were included in it , still there would be reasonsto think that those who live closer to the borders of this plane world are not inside the circle, and not the other way around. A short stay in Flatland will help us further. Recall what was said in chapter 5 about the differencebetweenedgesand holes in two -edgedflatlands . Can a person who walks along the external edge of such a flatland (of the approximate shape of a floppy disk) possibly confuse it with its internal edge? The intuition is that we should always be able, when looking from the inside edge toward empty space, to wave our hands to a friend situated on some other point of such an edge, himself looking into empty space. Were we on the external edge, such convivialities would no longer be obviously possible. To use a more geometricaljargon , consider a straight path connecting two points on the ideal plane of which our flatland is a part . If it is always possiblefor an edgeto be crossedtwice by such a path, then such an edgeis an internal edge. This does not hold if ' straight' and 'curve' have inverted meanings. Thus, our fox can chooseamong a few strategiesfor sour-grapesattitudes toward its situation. For instance, it can believethat it is outside the cage, but then think that straight lines are in fact bent, and that at least some bent lines are straight. Or the fox can believethat it is in a vast open space, but then renounce the possiblity of agreeing with us on the meanings of

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'

' ' ' big and small and cognates. As we suggested , it is difficult to invert the ' ' ' ' conceptsof inside and outside . (We find ingenious attempts at a description of large-scale hole inversions in those cosmologiesaccording to which the Earth 's surfaceis concave and enclosesthe sky. But the hypothesis of a hollow Earth, fantastic as it may be, does not concern a basic feature of our conceptual schemata , and can easily be accomodated within it - though we should leave it no place in proper astronomical or geological theory.) Bubblesand cavities. So far we have been concerned with hollows and perforating tunnels. To unmask the hidden side of things requires further elaboration. Take, again, the caseof the perception of a cavity (a graceful imperfection in a Murano glass) or of a (possibly static) bubble in mineral water. Convexity, brightness, relative dimension, and other factors are all co-localized at the cavity' s place. There is no antagonism betweenperceptual factors, and actually the cavity is seen as an object- a small ball floating in a more or less transparent medium. There is no perceptual reconstruction ofa background as in figure II .IB or figure 11.2B. How do you know that what you seeis a cavity? Where does the hole-character come in? Figural conditions alone do not suffice. Recall what is the distinctive mark of a cavity: it requires a surfaceof its own, disjoint from the object's external surface. When we seea cavity, we seeat least a part of this surface: the part that , from our point of view, looks concave. If we saw only the convex part of an imperfection in a Murano glass, it would be for us as if we saw an opaque marble in it . (This has the consequencethat if we put a transparent marble in a Murano glass it will be seen as a cavity or as a bubble in it , for we can see, from our point of view, the back of its surface, and thereforea part of its surfacethat looks concave- which duly happens.) Last, there is the simple but not negligible fact that, although it could be the casethat you are not able to seecavities in an opaque object from the outside, you perceivesuch cavities correctly from the inside: you are surrounded by a closed surface. Getting closer: sight and touch. The concept of a hole displays acomplexity that can be grasped only by creatures who can make senseof certain operations, such as filling, digging, enlarging, and going or seeingthrough .

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And this further presupposespossessionof tactual discrimination and kinesthetic sensitivity. Paradigmatically, we tactually notice holes by feeling how they surround parts of our body and limit our movements. Put your hand inside a glass and you will feel the glass all around the hand, which cannot open. We also realize that we are digging a hole by feeling how a surfacesagsunevenlyunder our hand's pressure.And another simple caseis that of the perception of a closedsharp border (edge) in asurfaceone you can follow with your finger. The latter case resemblesgestalt discrimination of a hole in a plane. We seeand feel how the plane could be reconstituded where the hole lies, and we hypothesize the presenceof a background region which exists beyond the plane even when unperceived. The seenregion corresponding to the hole was indifferently black or white (though a small preference is accorded to black, the receding color ); the tactually felt region has the particular quality of emptiness. Emptiness is here correlated to the possibility of free movement in a place where one would have expected hindrance. But in the caseof holes (in contrast to comers or edges) the conquestof free spacefor new movement becomeslimited at once, for you are soon inside the hole. The kinesthetic story we have to tell about hole detection is thus very simple. Holes always imply a detour; that is, a path connecting two points on a surfaceon opposite sidesof the hole's edgeis always longer than the straight path connecting them as the crow flies. Holes mark discontinuities in our possibilities of movement, as comers or wavesdo; but, unlike corners and waves, they deliver lessthan they promise: once you get in a hole, the increasein the quantity of reachable space will be paid for by a decrease in the possibilities of movement. In chapter 2 we suggestedthat some gestaltsare organized around singularities. Sometimesone neglects the overall convexity of a pattern becauseof the strength and peculiarity of a singularity . Perception- visual or tactual- starts with singularities. The eye resemblesthe hand in that each looks for some dramatic salience in its scenesin order to organize the perceivedworld . Singularities provide some. According to philosophical wisdom, primary qualities, like shape or size, are intertwined with our mastering of spatial operations and (bodily) actions. Theseconceptsdevelop as abstractions from classesof functions that objects can perform when considered from the point of view of an

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agent who can only move, locate things, visit them, push them, cut them, and the like. Such functions are typically described in the dispositional mode as mobility or pushability. They require some relational specification , that is, the mention of some other object in relation to which they could exhibit a distinctive pattern of behavior. This is true, for instance, of suchdispositions as fillability , graspability, geometric suitability , and aptitude to. (Not surprisingly, relational specificationsplaya major role here, for what matters most is interaction.) What we said about hole detection in the tactual and kinesthetic caseseemsto squareperfectly with the common picture of primary qualities. Holes and shadows . Holes are made of space, and this space could be . variously qualified What is it to perceivespaceas qualified? Supposeyou seea house whose windows are open in the bright light of the middle of the day. You seethe darknessfill the interior of the house. The holes outlined by the windows appear as if they were filled. But what are they filled with? It is not air , for you do not have the impression of seeing any airy or gaseoussubstance there. What you see is shadows. Shadowsfill the hole, permeatethe spaceit is made of. Consider how strange a property shadowinessis- and we mean here three-dimensional shadowiness. It is as immaterial as the hole itself is. It doesnot get out of the hole. It is confined to the volume of the perfectideal filler , evenif sometimesyou are uncertain in determining whether a certain ' point (more or lessat the filler s free surface) is in the shadow or not - and this reflectsour uneasinesswith the notion of the filler 's (of the hole' s) free surface. (We usually test whether the point is in the shadow by putting objects at the point and looking at how they appear, but this is an indirect test.) Shadowsare in betweenempty spaceand properly qualified space. That the perceptual content could be underdeterminedin relation to the difference betweenshadowsand holes is indirectly shown by the choice of this particular example in recent discussionsconcerning externalism. (Its first use was probably in Tyler Burge's 1986 paper " Individualism and Psychology ." ) An interesting thesis has been put forward by Gabriel Segal, who proposesto qualify the contents that are ambiguous betweenpresentation of a shadow and presentation of a crack by saying that they are contents as of crackdows: " thin , dark marks that could be either shadows or cracks" (" SeeingWhat Isn' t There," p. 208).

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Holes and shadows are similar with respect to another feature, too. It has to do with modality . You have a book and a lamp in the daylight . The lamp is otToDoes the book cast a possibleshadow (considering that the lamp could be turned on)? Now take the samesetting by night . The lamp is still otToDoes the book cast a possibleshadow? Whatever shadow it may cast is a part of the actual shadow surrounding the book (for it is night; everything is dark ). On the other hand, take a cup. It is empty. Isn' t there here a possible filler of the cup (considering that the cup could be filled )? Now take a region of space, on the right of the cup, of the samesize and ' shape as the cup s interior . It is empty too. Were there a cup there, in a sensethere would also be the possible filler (a potential filler of the latter sort requires you to travel to a possibleworld which is one more removed ; recall what we said at the end of chapter 9 on counterfactual reasoning about holes). Hearing boles, and other epistemie exeavatio. . The fact that you are inside something (a cave, a mine, a large flat , a small dig) makes itself audible, thanks to echoes. The sound wave does not find its way out , and comes back to you. But the phenomenology of sound is still a primitive science, and we will not try to develop it here. A final word may be devoted to the inferential processes of getting knowledge about holes. Supposethat you hold in your hands two pieces of the same cheese , of the same size, consistency, and superficial appearance , but the one in your left hand weighshalf as much as the one in your right hand. It would be reasonableto infer that the differencebetweenthe two piecesis due to the circumstancethat some stuff is missing from the piece in your left hand- that that chunk of cheesehas an internal cavity in it .

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TheFieldof

. At the end of our exploration many problems still Back to language remain open, and from these we selecta small group that has to do with language. What does talking about holes or through them amount to? There is, first, a general question concerning the way holes are represented in language- that is, how they enter our way of describing the world and our perception of reality . It is a question that we encountered at the very beginning and that we cannot leave unanswered. Do we need to talk about holes? Or could we adequately and fully paraphrase holecommiting sentencesby meansof sentencesthat remain neutral as to the existenceof holes? Even though paraphrasability does not per se imply elimination (just as the assertability of sentencesabout certain entities doesnot by itself introduce such entities into the world ), it is clear that our realist attitude toward holes would come out weakenedwere that a real possibility . It is therefore our businessto addressit in somedetail. To this end, our enriched hole-theoretic machinery will now prove useful. Second, we shall consider whether an adequatetheory of holes can help solve somebasic problems in the philosophy of language. In particular , we shall focus on one such classic problem: the problem of the nature of inscriptions. Third , we cannot but have at least a quick look at the variety of guises - syntactic, semantic, and metaphoric- in which holes show up in language as a matter of fact. After all , that is how we get a picture of what, with some caution, we may think as the field of emptiness. . The main property of the skinning function introduced A generalDOtiOn in chapter 5 is that it supplies us with a general tool for cataloguing holes. ' That is, any hole s skin is a sphereor a torus or , more generally, a sphere with a certain number of handles and edges. Thus, although tunnels are, from a certain point of view, genetically dependent upon hollows and cavities (every perforation starts with a hollow , and every internal tunnel is a cavity), a look at the skins allows us to also seehollows and internal

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( j , k ) - hole

rank: numberof closededges of the skin

: degree numberof non-separatingcuts of the skin

.1 Fiaure 1% A general notion of a hole.

cavities as degeneratecasesof tunnels (with only one edgeor with no edge at all , respectively). That is the natural history of discontinuities, and that is what gives us a general notion of a hole in spite of the fact that holes come in a variety of sorts and shapes. To be more precise, let the rank of a hole be the number of edgesof its skin, and let the degreeof a hole be its topological genus, i.e., the maximum number of simultaneous cuts (edge-to- edge paths or closed rings) that can be made on the hole's skin without separating it into two unconnected pieces. Then the general notion we are talking about is that of a hole with a certain rank and a certain degree. That is, we can seeevery hole as an instanceof the generalconcept ofaj -edgedhole of degreekin short, (j , k)-hole- wherej , k ~ o. (Seefigure 12. 1.) Thus, aspherical internal cavity would be a (O, O)-hole, a hollow a ( I , O)-hole, a straight tunnel a (2, I )-hole, a Y-shaped tunnel a (3, 2)-hole, an internal toroidal cavitytunnel a (0, I )-hole, and so on. Paraphrasesrevisited. With this general notion in hand, we can seein a new light the claim that holes do not exist. According to such a position (briefly discussedby the Lewisesin their article and defendedby Jackson), holes are a mere faron de parler, for to describe the world (indeed, any world morphologically compatible with ours) you need not quantify over - or refer to - entities of that sort. That is, every sentenceinvolving quantification over or referenceto holes can be paraphrasedby a sentencethat is onto logically neutral with respectto such entities: the latter , unlike the former, would not entail the existenceof holes. Instead of saying that there is a hole in a brick one would say " the brick is holed." As this amounts to interpreting our pairs of integersj and k as a sort of adverb, let us call this position " hole-adverbialism." Is any such position tenable? The availability of a generalconcept covering all sorts of holes-

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determinations of which would describeany kind of hole- might suggest " so. For instance, the sentence"There is a hollow in that brick is tantamount " to " Thereis a ( l , O)-hole in that brick ; hence, it can be paraphrased " as That brick is ( l , O)-holed." More generally, any sentenceschemaof the form ( I ) there is a hole so-and-so in x , insofar as it is tantamount to a sentenceof the form (2) there is a (j , k)-hole in x (for a suitable choice of j and k), can be paraphrasedby a sentenceof the form (3) x is (j , k)-holed. In certain cases, then, existential quantification over holes can be dispensed with in favor of a more neutral idiom. Of course, that requires a " vocabulary with an infinite number of predicatesof the form . . . is (j , k)holed" - one for eachordered pair of integersj , k ~ O. We shall ignore this problem, for we are doing our best to facilitate agreementwith the holeadverbialist. But how far can we go with this? Things get slightly more complicated when we consider sentencesassertingthe existenceof several holesin one sameobject or entity. How can we paraphrasea sentencelike " There are a hollow and three Vshapedtunnels in that brick"? Here our 't won help. infinitely many predicates In spite of thesedifficulties, one could envision various ways to take care of such casesas well. For instance, one could further expand the vocabulary " " so as to include a predicate . . . is (n,j , k)-holed for each ordered triple of integers n, j , k ~ O. The idea would be to take the first integer, n, " " as indicating how many times . .. is (j , k)-holed is being predicated of the object in question. With this vocabulary, our sentencecould easily be " renderedas " That brick is ( I , I , O)-holed and (3, 3, 2)-holed. And by generalizing along theselines, we could paraphraseevery sentenceof the form (4) there are nl holes so-and-so and n2 holes so-and-so and . . . n". holes so-and-so in x , " that is, any sentencewhose " canonical expansion has the form (5) there are ni (jl , k1)-holes and n2 (j2 , k2)-holes . . . and n". (j ". ,k".)holes in x ,

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by a sentenceof the fonn (6) x is (nl ,jl , k1)-holed and (n2,j2 , k2)-holed . . . and (nijik

. )-holed.

If quantification over numbers is allowed, one can account in a similar way for such problematic casesas these: (7) There are as many (j , k)-holes in x as there are in y . (8) There are more (jl , k1)-holes in x than there are (j2 , k2)-holes in y . (9) There are as many (j , k)-holes in x as there are ys such that Py ' ' (where P stands for any property). ( 10) There are more (j , k)-holes in x than there are ys such that Py. Here, the corresponding paraphraseswould be the following : ( 11) For every n ~ 0, x is (n,j , k)-holed iffy is (n,j , k)-holed. ( 12) For every m ~ 0, y is (m, j2 , k2)-holed only ifx is (n,jl , k1)-holed for somen > m. ( 13) For every n ~ 0, x is (n,j , k)-holed iff there are n distinct ys such that P. ( 14) For every m ~ 0, there are m distinct ys such that P only ifx is (n,j , k)-holed for some n ~ m. There may be some awkwardness to such paraphrases, but linguistic awkwardnessis no criterion in onto logical disputes (though it may be a rather indicative factor). But now consider the following pair of sentences , which refer to objects a and b in figure 12.2: ( 15) There is an internal toroidal cavity-tunnel in that brick , i.e., a (0, 1)hole (casea). ( 16) There is a trefoil -knotted toroidal cavity-tunnel in the brick (b). And consider the following pair of sentences , refemng to the objects c and d: ( 17) There are two toroidal cavity-tunnels in the brick (c). ( 18) There are two interlocking toroidal cavity-tunnels in the brick (d).

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Fiaurel U A toroidal cavity-tunnel(a), a trefoil-knottedone(b), two cavity-tunnels(c), and two interlocking ones(d).

Is it possible to provide an adequate paraphrase for each of these sentences ? In a sensethe first pair - ( IS) and ( 16), corresponding to casesa and b are not so hard to deal with . The main difficulty seemsto be that of accounting for the difference between a normal cavity-tunnel and a trefoil -knotted one. Knots are known to cause troubles in the extrinsic topology of three-dimensionalobjects, but presumably we could just solve the present problem by further enlarging our vocabulary as required by . (Such ad hoc solutions adding a predicate modifier for trefoil -knott .edness are starting to look dangerous, but once again we may close an eye.) Can we also account for the difference between c and d? Can we find suitable adverbial paraphrasesof a sentencelike ( 18) without directly referring to the two toroidal tunnels? Here the difficulty starts to look serious . In fact our view is that the difficulty is rather compelling, as we have " " here an equivalent of what is known as the many-property problem : the adverbialist wants to represent hole- complexity by means of adverbial ' ' complexity (i.e., by adverbially modifying the predicate is holed ), but it is known that this generalstrategy doesnot cope well with multiple property descriptions. An examplecomesfrom adverbialist theories of perception (after Frank " ' " Jacksons On the Adverbial Analysis of Visual Experience ). According to these theories, having a certain visual experienceis not a matter of sensinga certain object (i.e., it does not amount to standing in a relation - datum), but rather a matter to a specificperceptual entity , such as a Sense of sensingin a certain way. Supposeyou want to describewhat Suehas the impression of seeingwithout making any referenceto putative perceptual objects. Adverbs help you. If Suehas the impression of seeinga red patch, then you can say that Sue seesredly. If she has the impression of seeinga

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chimera, then you can say that sheseeschimeraly. So far, so good; you can succeedin differentiating between different psychological states without being committed to the existenceof the color red or of chimeras. But now Sue has the impression of seeinga red square and a blue circle close to each other. The adverbialist rendering is ( 19) Sueseessquarely and redly and circularly and bluely. Here the problems start, for ( 19) would also qualify as a description of a situation in which Suehas the impressionof seeinga blue squareand a red circle. One way to cope with this is to use compound adverbs such as ' ' ' ' square-bluely and square-redly (supplied with adequate logical principles to take care of the relations betweensuch compounds and their constituents ). That would allow one to discriminate between the following two possibilities: (20) Sue seessquare-redly and circle-bluely. (21) Sue seessquare-bluely and circle-redly. But this would be a very small step forward. Is there any difference between seeing square-redly and seeing red-squarely? How can we also account for the fact that , say, the red square is inside the blue circle? How can we account for the fact that there is also a green triangle inside the blue circle? And how can we say whether this triangle does or does not overlap the red square? Things get very complicated, and not clearly solvable , even if we stick to such simple examples. Spatial relations present seriousdifficulties for an adverbial strategy. Now , the casewith the holed objects c and d of figure 12.2 is somewhat similar. How can we account for the fact that in d, but not in c, the two " " cavity-tunnels are interlocking (that is, eachgoesthrough the hole in the other)? How can we paraphrase ( 18) in our hole-free language by just ' modifying the basic predicate is holed' ? One possibility was suggestedto us by Michael Tye: (22) There are two interlocking torus-shapedempty spatial regions x and y internal to that brick , and the brick is ( 1, 0, I )-holed at x and ( 1, 0, I )-holed at y. Here, ' at x ' and 'at y' are standard predicate operators, obtained by ' ' ' ' ' applying the preposition at to the individual variables x and 'y (com-

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' pare Tye s Metaphysics of Mind , chapter 4). Since these variables range over spatial regions, (22) commits us to empty spatial regions; but since theseneednot themselvesbe holes, there is here no commitment to holes. This is fine. But notice that the binary predicate " . . . is (n,j , k)-holed at . . . " has now appearedin our language. Correspondingly, new questions arise as to the constraints underlying the use of such a compound relational term. Being holed turns out to be a relation betweenan object and certain spatial regions which are not (occupied by) parts of the object itself. But what are thesespatial regions? How can we account for the fact that holed objects can be holed only at the peculiar empty regions that , on the realist perspectivethat we advocate, correspond to true holes? Theseconcernsarise with other eliminative strategiesthat favor regions of spaceover holes. Take the idea mentioned in chapters 1 and 2 that holes are nothing but regions of space. Or consider the view that holes are just relations betweenobjects and regions of space. Or again, take the stronger variant that maintains that statementsabout holes are to be paraphrased by statementsabout the relation of hosting that holds betweenobjects and certain regions of space. Which regions of spacewould enter this relation? What makes them different from other regions? Perhaps we can expresseverything we say in a substantive-hole language by paraphrasing our sentencesinto an adjective-hole language or in a purely relational idiom , plus mereology. Perhaps, in order to solve problems of the many-property sort, it sufficesto predicate hole-patterns of parts of objects, and to trace back spatial relations among holes in one and the same object to relations among the object' s parts. But does mereology suffice? The caseof figure 12.2 d suggeststhat it does not. The relations among some of the object' s parts (those that are relevant to the spatial relations among the concernedholes) are such that some parts are the fillers of others, which are therefore to be consideredas holed. Our conjecture is that , in the end, the adverbialist and relational strategies are likely to fail. This does not by itself imply that we are bound to admit referenceto or quantification over holes in our philosophically regimented language. One eliminative strategy is still available to the nonrealist : He can try to exercise his onto logical parsimony by providing purely geometric paraphrasesresting on a point -by-point description of the relevant region (eventually combined with an account of the properties exemplified at each point). But the hole-realist has a good reply to this: A point -by-point paraphraseis far too powerful a tool . It does not allow one

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to consider discourse about material objects as being onto logically primary with respect to holes - in fact , it does not allow one to save material objects at all . If immaterial objects can be bit - mapped away , so can material " " things . The geometric eliminative strategy eliminates everything just in order to eliminate nothings .

Inscriptions. Let us now turn to a different aspect of the linguistic relevance of holes. We have seen that it is hard to expunge holes from language . Our next point will be that , in somecases,holes can play an important role in language, in the sensethat an adequate theory of holes can solve some basic problems in philosophy of language. One of theseis help the classicproblem of the nature of certain inscriptions. Inscriptionalism is the view, put forward by philosophers such as Quine or Schemer, according to which intentional idiom can be dispensedwith in favor of reports that relate people to such linguistic items as ink marks and engravings. According to this view, when we say that Suebelievesthat snow is white we should instead say that Sue has a certain relation to the ' ' inscription snow is white . The big problem that the inscriptionalist must faceis: What exactly is that relation? Can it be specifiedin non-intentional terms? But there are other problems. For instance: What is an inscription ?" According to inscriptionalism, words are spatio-temporal concreteentities . This does not seemtroublesome in the caseof strings of sounds or in the caseof ink marks on paper. Yet some difficulty arises when we come to scratches on a surface(e.g., engravings on a gravestone). The latter are also spatio-temporal, but they lack materiality . If holes do not exist, how could inscriptions? By contrast, if one believesthat inscriptions are (or are made of ) holes, and that holes exist, the problem dissolves: lapidary inscriptions are just one more variety of immaterial bodies. The matter getscompounded when one tries to solve the problem of the unity of words in the casein which a word is made up of separateletters (and is not, as in most Western and Arabic handwriting , a continuous mark ). Recently David Kaplan (in " Words" ) hinted at what we could call the hole-lining theory of words. Think negative: words are not strings of letters, but the paper surrounding them. Accordingly , the word in figure 11.10 (the " Magritte -hole" ) would be the piece of paper minus the letters. There is no question of a scatteredink word ; ink is used only to mark the borders of the proper word , a pieceof paper with a certain shape. Accord-

The Field of Emptiness

ing to this theory too the aforementioned stony inscription would simply be the stone (for if you subtract the hole, you get what you already had). On this account, however, the problem of the unity of words is not solved at all. Take again the stony inscription , or , in order to fix ideas, the ordinary ink word. If the word is actually a holed piece of paper, you will encounter all the difficulties we had with the materialist account of holes in terms of hole-linings. 'Tim ' and 'Tom ' are different words but , they are also alike in some . If their difference is a difference betweenforms, say between just respects 'Tim ' -holed and ' Tom'-holed of then the question arisesas to pieces paper, their obvious similarity . One similarity is that each word has 'T as its first letter; but you cannot say that the two piecesof paper (which for you are the words) are both 'Tholed , for you assumedthat they are either 'Tim 'or 'Tom ' -holed, and an object cannot have different forms at the same time. Mereology won't do, for if you accept that words (piecesof paper, ' you said) have parts, and that these parts are sometimes T -shaped or 'Tholed , you are simply saying that there are letters that make up words; but once you try to spell out the details of this make-up the whole thing becomesembroiled. It is not enough to possessholed parts of such and such shapein order to be a 'Tim '-holed pieceof paper; it is also necessary that these parts stand in certain definite spatial relationships to one another , and what these spatial relationships are depends on the relative positions and shapesof the holed parts corresponding to the single letters ' ( miT will not do). On a realist theory of holes no such difficulty arises: the relationships at stake hold between holes- letters are just holes. (This is the view that Kaplan actually endorses.) But certainly the sum of these holes does not make a unity of the samekind (three holes, like the three letters of a stony ' ' inscription of Tim , do not constitute a bigger hole). Hence, the problem of the unity of engraved words is not solved by this theory of words. Unlessone is ready to countenancescatteredholes, one has to accept that words are mereological sums of holes. . Holes are part of the world , and language has to cope with Emptiness them if it is to describethe world . The descriptions in question turn out to be extremely detailed and variegated, and are- like every spatial concept - a never-ending sourceof metaphors. In the remainder of this chapter we will examine three main relationships between holes and language. First

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there is the metaphoric field (which will only be hinted at, owing to its vast ' ' ramifications). Secondcomesthe semanticfield, the position of hole and cognate terms in conceptual space. Last, we will make a more ambitious claim concerning the syntactic field of emptiness. The metaphoric field of emptiness. There are obvious metaphors one can build out of some concepts that suit holes and cognates , and to put some order in the field would be an interesting task . One can think of emptiness , void , nil , and blank ; of lack and absence; of omission , exclusion , and privation ; of need, want , and poverty ; and (less dramatically ) of deficiency , scarcity , shortage , and vacancy . One can then move on to more philo sophical items such as nothingness , non - being, and non -existence; consider events and processes like vanish , disappear , and fade ; pause and take into account intrinsically temporal beings like interval , break , cessation , and suspension ; then focus on thoughts and discover lapses, oversights , and neglects; then rise to the more abstract realm of fault , flaw , and error ; and end up with incompleteness and unsaturatedness , with truth - value gaps and denotationlessness . This would be, however , a purely taxonomical task . Then come the philosophically more engaging questions . Do the arguments in support of hole realism apply also to the concepts listed above? Is one entitled to be a realist about pauses? Are there gaps and flaws? It seems that there is no general rule as to the extended applicability of the realist strategy we envisioned in the case of holes. Holes have spatio temporal (primary ) qualities . Almost none of the items in the list above satisfy this requirement , and this could justify our suspicions as to their affinity with holes and as to their reality . But a few other items have temporal properties - they stretch over time . These are silences and pauses in speech and music patterns . There is a basic notion of a pause: a whistle stops , and then it starts again . ThiS; description will probably be accepted by everybody , but it implies something odd - namely that one and the same whistle was there before and after the pause. This is philosophically suspect. The sound philosopher will say that there are two whistles , and a pause between them . We see here an analogue of the problems we encountered in dealing with ordinary holes. We propose the equation holes : space = pauses : time .

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. Another direction of semantic exploration The semanticfield of emptiness concernsthe thesaurusstructure of holes- i.e., the place occupied by the concept of a hole in a more general semantic field. The obvious suggestion is to look at other superficialities. This move has actually been ' undertaken- even though the result is somewhatdisordered- by Rogets International Thesaurus, which classifiesholes in its class 2 (Space), III (Structure, Form ), C (Superficial Form ), 257 (Concavity), 3 (Hollow ) or 5 (Hole). In class2.III .C Roget' s offers a variety of conceptsemployed in natural languagein order to cope with superficialities of various sorts. We find geometrical properties or accidents like Convexity, Protuberance, Sharpness, Bluntness, Notch , Furrow , Fold , Opening, and Closure, along . with other more qualitative propertiessuchas Smoothnessand Roughness But, as we said, no orientation in the field is given to the reader. The fact that the main purpose of a thesaurusis to list all that comesto mind when one thinks of a certain thing does not exclude that someorder has to be made. Common sense- the source of all thesauri- makes clear a distinction between some geometrically relevant properties (like being notch-shaped) and diffuse properties (like smoothness); between corners and edgeson the one hand, and holes, tunnels, and cavities on the other hand. A thesauruswould serveits purpose much better if distinctions were made among morphological, topological, metric, and other factors. The syntactic field of emptiness . Last but not least we can study how holes leave traces in syntax. We shall follow two main paths here, and introduce them by referenceto one conception of the encoding of ontol oglcal information by linguistic forms. There is an interesting thesis in cognitive linguistics (most prominently " defendedby Len Talmy in " How Language Structures Space ) according to which natural languageinvolves two main distinct systemsof items and forms in natural language, one of which- the lexical system- is open (in the elementary sensethat elementscan be added to it rather easily) and the other of which is relatively closed(addition is much more difficult ). In the latter system fall grammatically relevant items, mostly unsaturated ones (affixes, conjunctions, prepositions), and also grammatical features like the distinction betweensubject and predicate. The cognitive explanation of this phenomenon is that thesetwo classes- the open one and the closed one- have distinct functions; Talmy shows that (at least) spacestructuring featuresare encodedin the closed class, mostly in the form of

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prepositions; relative to spatial properties, spatial prepositions 66represent a skeletal conceptual microcosm" (p. 228). Looking beyond space, one could go one step further and say that grammar (the closedclass) encodesonto logical information (i.e., the categories to which the objects referred to belong) whereaslexicon (the open class) encodeslessgeneralor lessformal classifications. Thus, for instance, " " The apple is red can be parsedat the grammatical level as an onto logical indication that there are here(at least) an object and a quality , and can be also parsedat the lexical level, which informs us as to the speciesand the determinate to which the object and the quality belong. The two systems act on different planes or reflect different levels of generality. There is no grammatical encoding for fine-grained differences (like, say, the differencebetweenred and yellow), and somedistinction gets the honor of grammatical representationonly if it is onto logically salient. ' (We are anxious to stressthat Talmy s philosophical approach construes spatial (and onto logical in general) distinctions and featuresas dependent on the presenceof certain linguistic distinctions and features, and is therefore a variety of linguistic idealism. Our realist view, on the contrary , is that space(or the world ) is not structured by language, but that the structure of space(and the world ' s structure) is reflected or mirrored by language . We simply cannot imagine any plausible way in which the alleged structuring could have taken place at all ; anyway, we are not going to propound here an account of mirroring . It is enough, for our purposes, to observethat neither the idealist nor the realist account is implied by the descriptions and models given by Talmy .) An important case is that of those prepositions (studied by Talmy) whose mastering draws upon the competenceof a subject capable of orientation in space. To consider an example related to our topic, a correct ' mastering of 6through seemsto presuppose, in somecontexts of useof the word, the possessionof the concept of a tunnel or of an operation which is performed following a tunnel-like path. Talmy suggeststhat abstractive processes, relative to spatial properties, underlie the mastering of spatial ' ' prepositions like 6through or 6in. Thus metric and shapeproperties can be disregarded(and are usually registeredin the lexicon and not in grammar: no element of the closed class expresses facts relative to squarenessor being one foot long), whereastopological properties are of the outermost importance. 66Forexample, the use of in requires that a ReferenceObject [ i.e., an object determining the orientation or the position of the speaker]

The Field of Emptiness

189

be idealizableas a surfaceso curved as to define a volume. But that surface can be squared off as in a box, spheroidal as in a bowl , or irregular as a piano-shaped swimming pool . . . . None of these variations of physical manif~station affect the useof in." (p. 262) Topology is relevant in the case of ' through' when this preposition is used to describe a path inside a material object along which a vehicle movesor lies without physical interaction with the object (penetration is therefore excluded): the object, or a relevant part of it , must be topologically equivalent to a torus. There remains one thing to be said about the syntax of emptiness in ' ' ' ' ' general. Emptinessis marked by the suffix -less (as in formless and waterless'), which usually takes a noun or a predicate and yields a predicate. The resulting predicate is in turn suitable for standard nominalization (as ' . Now ' -less' suffixes are submitted to some in 'formlessness , ) elementary rules. One thing can be sensibly said to be ; -lessonly if it can be ; . If the above account of the relevanceof the closed, grammatical classof linguistic items and forms is correct, the fact that we have a ' -less' suffix is an index of its cognitive significance. Again, we should like to say, it is one more manifestation, this time in the linguistic realm, of the basic principle of thought about holes: Think negative.

: Outlineof a Theory Appendix

In the chaptersabove we presentedour views in a rather informal setting, trying to show the philosophical importance and consequencesof a realist attitude toward holes rather than spelling out a full -fledged theory of holes. In this appendix we attempt to addressthis task more directly by summarizing somebasic tenetsof our account in a rather systematic- though by no meanscomplete- fashion. For convenience, we divide the presentation into four main sections: ( 1) a preliminary ontological part , which introduces the basic binary relation " " is a hole in or ( through) along with some relevant facts; (2) a mereological part , which systematizessome fundamental principles governing the interplay between the host-hole and the part -whole relations; (3) a topological part , which summarizessome basic facts concerning surfaces and the taxonomy of holes; and (4) a morphologicalpart , focusing on the fact that objects with holes constitute - as we have put it - the morphological manifold offillable things. This organization does not exactly parallel the order we followed in the text, where mereological facts were examined only after- and somehow on the basis of - a topo -morphological characterization of holes. Indeed, the interplay among thesedifferent domains (as well as betweentheseand other domains, such as kinematics or causality, consideredin the text but not included here) is an interesting issuein itself, but it would take us too far afield to addressit here. For the modest purposes of this Appendix, suffice it to note that the order followed here permits a rather nice and intuitively simple step- by-step construction. The exposition proceeds more geometrico. Each section begins with basic definitions, followed by some basic principles, or axioms, followed by a list of a few noteworthy consequencesor theorems. ( Numbers in

192

Appendix

bracesindicate the axiom or definition from which a theorem is derived.) It is understood that there is no pretense of completenessor logical elegance. In fact we have been quite relaxed in our choice of axioms and theorems, as our aim is first and foremost perspicuity. An informal spelling of every formula, sometimeswith a brief comment, is also provided (the reader may want to skip the formulas and just focus on these informal renderings). In the formalization , we assumebasic logical notions and notation. We use-' , A , V , -+, and +-+as connectivesfor negation, conjunction, disjunction , material implication , and material equivalencerespectively; V and 3 for the universal and existential quantifiers; and l for the definite descriptor . To simplify readability , we dispensewith quotation devicesas far as is practicable and rely on standard conventions to minimize the use of parenthese . (In particular, we assumeour connectives to bind their arguments in decreasingorder of strength as listed, so that negation binds the strongestand equivalencethe weakest.) The underlying logic is deliberately left vague, as after all we think holes are utterly neutral in this respect. A preferred alternative is some sort of a free logic, where improper descriptions and other possibly empty expressions can be admitted bonafide ; however, everything that follows could in principle be dealt wit " within the framework of a standard first -order logic, with I treated as an improper symbol. In addition , we assumefamiliarity with some basic principles of extensional mereology and topology .

1 Ontology The main thesisis that a hole is an immaterial body located at the surface (or at some surface) of a material object. Since the notion of a surface is essentiallya topological one, and sincethe property of being immaterial is reflected in the morphological property of being tillable, the onto logical basisis concernedfirst and foremost with the generaldependenceof a hole on its host. Notation Nt .t

Hxy = x is a hole in (or throughy

.

This is our primitive relation. We need a binary relation to expressthe basic intuition that holes are dependent entities. A hole is always in (or through) some object.

Outline

of a Theory

193

Definition

DI .I

Hx = df 3yHxy . We write ' Hx ' for " x is a hole" . Sinceevery hole is onto logically dependent on its host, being a hole is defined as being a hole in (or throughsome thing .

Basic Axiom

AI .I

Hxy -+-, Hy. The host of a hole is not a hole.

Some~Theorems

Tl.l

Hxy - + -, Hyx . Being a hole in (or through) is an asymmetric relation: a hole cannot host its own host.

Tl .2 -, Hxx. ) is an irreftexive Beinga holein (or through itself.

relation : a hole cannot host

TI .3 Hx -+., 3yHyx. Holes do not have holes: they cannot host one another (though they can

haveholesasproperparts). Tl .4

3xHx -+ 3x -, Hx . Holescannotbethe only thingsaround.

2 Mereology As immaterial bodies, holes have parts and can bear part -whole relations to one another (though not to their hosts). The main principles concerning theserelations can be formulated within the framework of classicalextensional mereology supplementedwith somespecificaxioms on the behavior of the onto logical relation H.

194

Appendix

Notation

N2. 1 x Sy = x is a part ofy .

This is one of many possible mereological primitives: a reflexive, antisymmetric , transitive relation (i.e., a partial ordering), according to classical mereology. Definitio . . 02 .1 x < Y = dfX SY A IX = y . ' X < ' means" x is a " y proper part of y ; i.e., x is a part of y other than y itself. This is a transitive and asymmetric (henceirreflexive) relation. 02 . 2

X 0 Y = df 3z(z S X A Z Sy ). 'x 0 ' means" x " y overlapsy ; i.e.. x and y have someparts in common. This is a reflexive and symmetric (but not transitive) relation.

02 . 3

1:xAx = df lZ'v'y (y 0 Z +-.. 3w (Aw A yo w . 'I.xAx ' stands for " the fusion of all x such that Ax." The existenceof such an object is always assumedin classicalmereology, provided there is some x such that Ax . We do not assumeit unlessotherwise specified.

02 . 4

x uy = df1: W(W S x v W Sy ). 'x u ' stands for " the sum of x and y," i.e.. the smallest thing whose parts y are either parts of x or parts of y. This is an idempotent. commutative, distributive operation.

02 . 5

x ny = df1: W( W S x A W Sy ). 'x n ' standsfor " the product of x and y," i.e.. the largest thing whoseparts y are both parts of x and parts of y . This too is an idempotent. commutative, distributive operation. though one that is defined only if x 0)I.

02.6 x - y = dr1:W(W~ x A ., W0 y). 'x - ' stands for " the difference of x and y," i.e., the largest thing contained y in x that has no part in common with y .

02.7 x ~ Y = dfHx A x . sY. 'x

' ::sy means" x is a hole-part of y," i.e., a holethat is a part of y. This is a partialordering , like .s ; it appliesonly wheny is itselfa (part of a) hole.

Outline of a Theory

02 .8

195

x -< Y = df X ~ Y A -, X = y . " ' " 'X 0( y means x is a proper hole-part of y, i.e., a hole that is a proper part of y . This is a transitive, asymmetric, irreOexiverelation, like < .

Basic Axioms

A2.1

Hxy -+ IX 0 y. No holeoverlapsits own host(thoughthe sumof a holeand its hostmay bea legitimatehostfor differentholes:e.g. the sumof adoughnuty andits holex- if sucha sumexists- will not be a hostof x, but it will be a host of, say, a cavitythat maybehiddeninsidey).

A2.2 Hxy 1\ Hxz -+ 3w(w < y n z 1\ Hxw ). Any two hostsof a holehavea commonproperpart that entirelyhoststhe hole. (Of course , intuitivelya holehasonehost; but if weallow for mereosums or splittings, then everyhole hasa virtually infinite classof logical hosts, partiallyorderedby < . See03.2 below.) A2. 3 Hxy v Hzy -+ Vw(w ~ x uz -+ Hwy). Any hostof a holeentirelyhostsall commonhole-partsof anysum involv ing that hole. A2.4

Hxy 1\ y ~ Z - + X 0 Z v Hxz . Any object that includes the host of a hole is a host of that hole, unlessits parts also include parts of that very hole.

A2.5

Hxy A Hzw A X 0 Z - + Y 0 w. Overlapping holes have overlapping hosts. (However, two holes may well occupy the same region, or part of the same region, without sharing any common parts. Holes are immaterial, and can penetrate one another; mereological overlapping is not implied by spatial co-localization.)

A2.6 Hx - + 3z(z < x ). No hole is atomic (though holes need not have proper hole-parts; otherwise every hole would correspond to a pile of infinitely many, gradually smaller holes).

196

Appendix

SomeTheorems T2. 1

Hxy -+ -, X Sy . Holesarenot partsof their hosts(althoughthe hostsof a holemayhave differentholesasparts) {A2. I } .

T2.2

Hxy -+ -, X ~ y . Holesarenot hole-partsof their hosts(althoughtheycanbehole-partsof partsof holes) {A2. I }.

T2.3

Hxy -+ -, y S x. The hostof a holeis not part of it { A2. I }.

T2.4

Hxy " y S Z -+ -, Hz. The host ofa hole is not part of any hole {AI .I + A2.4- A2.S} . This is a of theontologicalaxiom(AI .I ) andof thethesisthat a hole generalization a cannothostanything(T1.3).

T2.S

Hxy " Hxz -+ y 0 z. Any two hostsof the sameholeare overlapping ; ie., a holecannothave two discretehosts{A2.2}.

T2.6

Hxy -+ 3z(z < y " -, Hxz ). Not everypart of a hole's hostis a hostof the hole(thoughit couldhosta differenthole) {A2.2}.

T2.7

Hxy -+ 3z(z < y " Hxz ). Hostinga holeis havingsomeproperpart that entirelyhoststhehole; ie., thereis no minimalhostfor a givenhole {A2.2}.

T2.8

Hxy -+ Vz(z -< X -+ Hzy). Hostinga holeis hostinganyproperpartsof theholethat areholesthemselves {A2. 3}. (Note that the samedoesnot hold relativeto < .)

T2.9

Hxy " Hxz -+ Hx (y uz ). Thesumof anytwo hostsof a holeisitselfa hostof thathole{A2. 1 + A2.4}.

T2.10 Hxy " Hxz -+ Hx (y nz ). Theproductof anytwo hostsof a holeis itselfa hostof that hole{A2. 1 + A2.2 + A2.4}.

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197

T2. 11 Hxy A Hxz -+ 1 Hx (y - z). of two hostsof a holeis not itselfa hostof that hole{A2.2}. Thedifference

T2. 12 Hxy -+ "9"Z(, x 0 z -+ Hx (y u z . A holein an objectis alsoa holein any mereologicalsumincludingthat object, providedthe sum doesnot includeany parts of the hole itself {A2.4}.

T2.13 Hxy -+ 3z(, x 0 Z A Hx (y n z . A holein an objectis alsoa holein somemereological productinvolving that object, providedthe productdoesnot includeany partsof the hole itself{A2.2}.

T2.14 Hxy -+Vz-, Hx (z - y). The differencebetweenan objectand the host of a hole is not a host of that hole{A2.2}.

T2.15

-, H (1:z.z = z). If the universalindividualexists,it surelyisn't a hole {AI .I + A2. I }.

T2.16 -, H (1:z.z ~ z). The null individual(if it exists)cannotbea hole {A2.6}.

T2. 17 -, 3z.z < y -+ -, 3xHxy . Atomsareholeless{A2.2}.

T2.18 Hx -+ Hx (1:zHxz) Thefusionof all hostsof a holeis a hostof that hole{A2. 1 + A2.4}.

3 Topology Topology constitutes in many ways a natural next step after mereology, although various mereological notions could formally be defined in terms of topological ones. Particularly in a theory of holes, topological notions are important to account for the fact that every hole is located at some surfaceof its host as well as for taxonomic purposes.

198

Appendix

Notation N3 .1 xc y = x is connected with y . This is a reflexive and symmetric relation capturing the intuitive notion of touching or being co-localized at some point . We take it to satisfy the clausethat what is connectedwith a part is also connectedwith the whole, so that x : Sy implies Vz(z c x -+z C y). (We do not, however, assumethe converse, which would have the effectof reducing mereologyto topology .)

N3.2

gx = the genus of x . Intuitively , the genusof an object is the maximum number of simultaneous C\(ts that can be made without separatingthe object into two unconnected \ pieces(0 if it is a sphere, 1 if it is a torus, etc.). This notion could be defined in terms of C, but that would lead us too far afield.

Definitions

03.1

Cx = dfVyVZ(X = Y uZ -+ Y c Z). 'Cx' means" x is self- connected " i.e. x doesnot consistof two or more ( ) ; , disconnected parts.

D3.2 hx = df1: y(Hxy A Cy). 'hx' standsfor " the " ' principalhostof x, i.e., x s maximallyconnectedhost (a notion that we take to be definedonly when x is a hole). We may intuitivelyregardthis asthehostof the hole, everyotherhostbeingeither a topologicallyscatteredmereologicalaggregateincludingthe principal hostor a potentialpart of this latter (seeabovead A2.2).

D3.3 X)( y = dfX C Y A IX 0 y . 'X)( ' means" x is " y externallyconnectedwith y ; i.e., x is connectedwith y but doesnot overlapit. This is an irreflexiveandsymmetricrelation.

D3.4 X <JY = dfX . sy A Vz(Z)( X -+ I Z)( y). 'x ~ ' means"x is an interior " y part of y ; i.e., x is a part of y that is externally connectedonly with thingsthat are not so connectedwith y itself. Thisis a transitiverelationincludedin oSandclosedunderbothu andn.

D3.5 x 'x

Y = dfX SY A -, 3z.z~x.

' " " y means x is a superficial part of y ; i.e., x is a part of y that has no interior parts of its own (or , intuitively , that only overlaps those parts of y

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199

that are externallyconnectedwith the geometriccomplementof y). This too is a transitiverelationclosedunderuandn. y 1\ Cx A Vz(z y 1\ Cz -+ IX < z). ' ' means" x is a surfaceof " ; i.e., x is a superficial maximallyconnected y Sxy part of y.

D3.6 Sxy = dfX

D3.7 Hcavxy= dfHxy 1\ 3z(Szy 1\ Vw(w S Z+-+X >( w . " ' H X ' means" x is a cavity in y ; i.e., x is an internal hole enveloped C8vY . A cavity is a topologicallynon-erasablediscontinuity by an entire host surface .

D3.8 Htunxy = dfHxy 1\ Vz(z SY 1\ Cz 1\ Hxz -+ gz ~ 0). " ' " Htuaxymeans x is a tunnel(or a perforation) throughy . This is alsoa hole, characterized by the factthat its hosthas topologicallynon-erasable no connectedpart of genus0 entirelyhostingthe hole. ( Notethat a hole mayat oncebea tunnelanda cavity: it maybea cavity-tunnel.)

'

03.9 HbolXY= dfHxy 1\ I Hcavxy1\ I Htunxy. " XY' means" x is a hollow (or a depression ) in y ; i.e., x is a holein y Hhoa whichis neithera tunnelin y nor a cavityin y. This is alwaysan external, , characterized by the fact that the relevant topologiCallyerasabledisturbance hostmusthavea part of genus0 entirelyhostingthe hole.

'

: Axioms Basic

A3.1 Hx -toCx. Holes are self- connected; i.e., there is no scatteredhole.

A3.2 Hxy -+x C y. Holesareconnectedwith their hosts.

A3.3 Hx -+ 3y(Hxy " Cy). host. Everyholehassomeself- connected

A3.4 Hx " y -<X -+ 3z(z )(x " , z >
200

Appendix

SomeTheorems T3.1 Hxy -. x )( y. Holes are only externally connectedwith their hosts; i.e., a hole and its host touch each other, but have no parts in common { A2.1 + A3.2} .

T3.2 Hxy -+ 3z(z y " xC z). Every hole is connected with some superficial part of its hosts { A2. 1 + A3.2} . (Holes are superficial entities; they go hand in hand with surfaces.)

T3.3 Hxy --toVz(z
T3.4 Hxy 1\ Z~ X -+z>
T3.S

Hxy A Z ::Sx A W oSY - + (Z)( W - + X )( w). A hole is externally connected with every part of its host that is so connected with some hole-part of that hole { A2. 1 + A3.2} .

T3.6

Hx A Z ::Sx A Hzw - + x c w. A hole is sure to be connected(externally or not ) with the hosts of its own

hole-parts{A2. 1+ A3.2}.

T3.7 Hx A Hy -. (Hx uy -. XCy). to fonn a hole Only holes that are connected with each other can join -

{A3.1}.

T3.8 Hxy -+ 3z(z < y 1\ Hxz 1\ Cz). Hosting a hole is having someproper self- connectedpart that entirely hoststhe hole{A2.2 + A3.3}.

T3.9 Hxy 1\ Hxz -+ y C z. Any two hosts ora hole are connectedwith each other { A2.2 + A3.3} .

.

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201

T3.10 Hxy A Hxz -to.., y )( z. Two hostsofa holecannotbeexternallyconnected{A2.2 + A3.3}.

T3.11 Hxy A Hxz -to3w(w oSy n z A Hxz A Cz). part that entirely Any two hostsof a holehavea commonself- connected hoststhe hole{ A2.2 + A3.3}.

T3.12

Hx -toChx . Principal hosts are alwayssure to be self- connected{A2. 1 + A2.2 + A2.4 + A3.3}.

T3.13

Hx -toHxhx . Everyholeis a holein its principalhost { A2. 1 + A2.2 + A2.4 + A3.3}.

T3.14 Hx A z ::Sx -tohx = hz. The principalhostof a holeis alsothe principalhostof any hole-partsof that hole{A2.3 + A3.3}. T3.15 HcavxyA Hzy -+ IX >
A Hzy -+ IX < Z. T3.16 HC8vXY Cavities are maximal holes; i.e., no hole can include a cavity as a proper

part{A2.1+ A3.3 + A3.4}. A Hzy A X0Z-+Z oSX. T3.17 Hcavxy

Cavities are maximally connected holes; i.e., a cavity includes every hole with which it is connected { A2.1 + A3.3 + A3.4} .

-+IX ~ Z. T3.18 HtunxyA HbolZY A hole that qualifies as a tunnel with respectto a hollow ' s host cannot be

partof thathollow{A2.3}.

-+-, Z Sy . T3.19 HtunxyA HbolXZ A hole cannot qualify as a hollow with respect to any part of a host relative to which it qualifies as a tunnel { A2.3} .

Appendix

4 Morphology Topological conceptsmake it possible to distinguish and classify objects with different types of holes, but we needmorphology in order to account for the feeling that blind hollows, perforating tunnels, and internal cavities are all part of a single family . The relevant notion operating here is that of filling : objects with holes- or , better, holes in objects- constitute the morphological manifold of fillable entities. Notation N4.1 Fxy = x is (perfectly) filled by y. Holescan be filled, and we meanhereperfectlyfilled. Theycan be filled (without losingtheir statusof holes) insofarastheydeterminea (partially) concavediscontinuityin the surfaceof their host. Definitions

04.1 Fx = dr03zFxz . 'Fx' means" x is tillable" i.e. x can be ; , . Here perfectlytilled by something for possibility and in the followingwe use<>and c as modalconnectives . and necessity , respectively

04.2 Fcomxz= dr3w(w ~ Z A Fxw). " ' means" x is Fcomxz completelyfilled by z ; i.e., thereis somepart of z that 1\ perfectlyfills x. This is a monotonicrelation, in the sensethat FcomXY . Y oSz - Fcomxz '

04.3 Fparxz= dr3w(w ~ x A Fcomwz ). " ' 'F " parxzmeans x is partiallyfilled by z ; i.e., thereis somepart of x that is completelyfilled by z. This too is a monotonicrelation, in the sensethat Fparxy1\ Y oSz - Fparxz . Note that a partial filler neednot be wholly inside a hole(it may" stickout" ). whichmeansthat everycompletefiller also qualifiesas(a limit casesof) a partial one.

D4.4 Fproxz= dr3w(w ~ x A Fwz). " ' 'F " ) filled by z ; proxzmeans x is properly(thoughperhapsincompletely and i.e., somepart of x is perfectlyfilled by z. This is the dual of FCorn ++Vw(w oSZ- Fparxw is relatedto Fparby the equivalenceFproxz ). (Thus, .) everyperfectfiller is both completeandproperin this sense

Outline of a Theory

203

04.5

ax = df 1:z (z hx 1\ X >( z). 'O'x ' stands for " the skin of x " i.e. the fusion of those , , superficial parts of x ' s principal host with which x is externally connected (a notion that is meant to apply only when x is a hole). This is a slight departure from the ' text, where the skin is defined as the part of the filler s surfa(:e that is in ' contact with the host, or as that part of the host s surfa(:e that is in contact with the filler. However, these definitions are essentially equivalent and serve the same purpose: the topology of the skin reflects the morpho logical complexity of the hole.

04.6

; wzx = df W Z 1\ -, W C ax . ' wzx' means " w is a free " ; superficial part of z relative to x ; i.e., w is a 's of z that is not connected with x s . superficial part host( ) ( This notion is meant to apply only when x is a hole and z a corresponding perfect filler.)

Axioms A4 .1

Fx +-+3y(Hy A X ~ y). Somethingis tillablejust in caseit is part of a hole; i.e., fillability is an exclusivepropertyof holesandtheir parts.

A4.2 Fxy A Fz -+ -, y 0 z. Perfect tillers and tillable entities have no parts in common (rather. they may occupy the samespatial region).

A4.3 FcomXY A Z C X -+Z C y. A complete filler of (a part of ) a hole is connectedwith every part of (that part of ) that hole.

A4.4 FproXY A Z C Y -+Z C x. Every part of a proper filler of (a part of ) a hole is connectedwith (that part of ) that hole.

A4.5 Fxy A Z < X -+ FcomZY . A perfect filler of (a part of ) a hole completely fills every proper part of (that part of ) that hole.

A4.6 Fxy 1\ Z < Y -+ Fproxz. Everyproperpart of a perfectfiller of (a part of ) a hole properly fills (that part of) that hole.

204

Appendix

SomeTheorems T4.1 Ox -.. Fx . Everyholecanbetilled (and continueto bea hole) {A4.1} .

T4.2 Hxy -. -, Fy . Onecannottill the host of a hole(exceptin a looseway of speaking : we can, e.g., saythat somethingmaterialis tillablemeaningthat it is the host of sometillableentity proper) {AI .I + A2.4- A2.S + A4.1}.

T4.3 Fxy -. -, Hy . No tiller canbea hole(thoughit canbeholed, like the tiller of an internal tunnel- cavity) { A4.1 + A4.2} . T4 .4

Fxy - + -, Hxy . A filler cannot host the thing it fills { A2. 1 + A4.3} .

T4.5

-, Fxx . Being filled is an irreflexive relation: holes or parts of holes cannot fill

themselves {A4.2}.

T4.6 Fxz-+-, Fzx. Being filled is an asymmetric relation: holes or parts of holes cannot fill their own fillers { A4.2} . T4 .7

Fxz - + -, Z 0 x .

Fillers do not haveany partsin commonwith the holesthey fill (though someof their parts are spatia-temporallyco-localizedwith parts of the holes, holesbeingimmaterial) {A4.2}.

T4.8 Fxy A y S Z-+-, Fz. A tiller is not part of any tillable entity (just as a host is not part of any hostedentity: compareT2.4) {A4.2}.

T4.9 Fxy /\ z ~ y -+ -, Fz. A perfecttiller cannothaveany tillable part (thougha completetiller, or evena partial tiller, can) {A4.2}.

205

Outline of a Theory

T4 .IO

Fxz A Hxy A W SY A xC W - + Z)( W. A hole' s perfect filler is externally connectedwith every part of the hole' s hosts with which the hole itself is connected (i.e.. it perfectly "fits " the hole) { A4.2 + A4.3} .

T4 .11

Fxz /\ w oShx - + Z )( W+-+X )( w. A hole' s perfect filler is externally connected with exactly those parts of the principal host with which the hole itself is connected { A4.2 + A4.3} .

T4 .12

Fxy A Z S ax - + Z)( y . A hole' s perfect filler is externally connectedwith every part of the hole' s skin { A4.2 + A4.3} .

T4.13 Hxy -+ax

y.

The skin of a hole is a superficial part of each and every host of that hole

{A3.1 + A4.2 + A4.3}.

T4.14 -, HxO'x. Nothingis a holein its own skin; i.e., the skin of a holeenvelopsthe hole but doesnot qualifyasa hostof it {A2.2 + A3.1 + A4.2 + A4.3}.

T4.15 Fcomxy A w :s: ax -+ WCy. A complete filler of a hole is connected with every part of the hole ' s skin

{A4.2 + A4.3}. T4.16 Fxy A Z < Y-+Fparxz .

fillerof(a partof) a holepartiallyfills(that Everyproperpartof a perfect partof) thathole{A4.S}.

T4.17 Fcomxy A Z S x -.. Fcomzy .

T4.18 Fproxy A Z SY -toFproxz . Any part of a properfiller is alsoa properfiller (of thesameentity) {A4.6}.

T4.19 Hcavxy" Fcomxz" Cz -. Fxz. An internalcavity admitsof no imperfectcompleteself- connected fillers: it canonly becompletelyfilled in a perfectway {A4.3} .

206

Appendix

A FparxZA Cz -+ Fproxz. T4.20 HC8YXY fillers: it An internalcavity admitsof no improperpartial self-c:onnected canonly bepartially filled in a properway {A4.3 + A4.4}. A Fxz -+ -, 3w.; wzx. T4.21 HC8YXY An internalcavity admitsof no perfectfillers with freesuperficialparts {A4.3 + A4.4}.

T4.22 HhoaxyA Fxz -+ 3w(; wzx -+ Vv(; vzx -+ V S w . Thefiller of a hollow is sureto haveexactlyonemaximalfreesuperficial part {A4.3}.

T4.23 Hcayxy" " Hxy A cVz(Fxz -+ -, 3w.; wzx). A cavity is a hole whose fillers cannot have any free superficial parts { A4.3} .

T4.24

HholXY+-+Hxy A g( O'X) = 0 A cVz (Fxz - + 3w.; wzx ). A hollow is a hole whose skin has genus0 and cannot be connectedwith every superficial part of a perfectfiller; i.e., the filler of a hollow must have

freesuperficial parts{A4.3}.

T4.25 Hxy -+ 03z (Fparxzv Fproxzv Fcompxz ). Everyholecanbefilled partially, properlyor completely{A4.1}.

Puzzlesand Exercises

The hole in tlae baD (figure A ). Take a ball and make a hole in it by removing a small round disk. What type of hole is that: a hollow , or a tunnel?

FlaweA.

The double handle (figure B). Does an object with two hollow handles define a tunnel- perhaps a tunnel with two scattered ushaped proper parts?

FiaweB.

. TIle . wdenmarble (figure C). Consider a cavity with a marble floating inside it . Is it a partially filled ordinary cavity? Is it an empty convexoconcavecavity betweentwo solid objects (i.e., a cavity in a scatteredhost)?

208

Puzzlesand Exercises

Fi88ec .

Imaginea hole shaped like a Klein bottle . Is it a hollow. a tunnel or what?

FIaweD.

The Sierpiilski-Meager sponge(figure E). Think of the fractal sponge described in chapter 2. One can conceive of liquid flowing through the holescut out at eachstep of the construction. The holes surely are tunnels, as it is impossible to reducea ring on the surfaceto a point by continuous deformation. But what about the limit sponge? In that caserings are single points! Or can we say that the tunnels touch one another asymptotically?

Fil8eE .

Puzzlesand Exercises

209

The Peanofneture (figure F). Is it possible to imagine an object with a " total " fracture , say, a fracture constructed like a Peano curve? Such a fracture would be totally connected. How many faces would the object have: one, two, or infinitely many?

Fiawe F.

Zeno' s hoUow-tu DDel(fIgUre G ). Take a block of continuous matter and make a hollow on the top, so that the hollow ' s height is 1/ 2 of the block's total height. Make a second hollow inside the first one, reduced in the ratio 1/ 2; the result is a bigger hollow with a small hollow as a proper part . Now supposeyou keep repeating this processover and over. At each step, the result is always a hollow with hollows as proper parts. Supposeyou go on ad infinitum. What do you end up with : a hollow? a tunnel?

Fipre G.

Thesplittingbole(figureH). How would you describethe situationdepicted here? Doesthe hole get biggerand bigger, and then split into two distinct holes?

210

',""-'-~" ' !I,/ \I \, , " -....-...'/J (AirBall )

Puzzlesand Exercises

i\"/

I,'!~ .;, ~

Fia8eH.

The hole-rissure-bole hole (fia - e I). Suppose two internal cavities are connectedby an internal crack, e.g. , a filamentous crack. Does that make them a single (dumbbell) cavity? .

fissure

Ji' ia8eL

The bottle in t- book (figure J). Supposeyou cut the pagesof a book so as to hide a bottle of whisky inside it . Each pagehas a tunnel in it; but the entire book - when closed- can be said to have a cavity . Can we say that the cavity has tunnels as proper parts? Can we say that it is made of tunnels? Moreover, can we say that the cavity in the bottle is part of the cavity in the book (and overlaps with the tunnels)? The whisky is a (partial ) filler of the cavity in the bottle ; is it also a partial filler of the cavity in the book?

Fta- e J.

211

Puzzlesand Exercises

The tube in the blocks (figure K ). Imagine two blocks with a tube that goesthrough them. Imagine that you enter the tube from the left, go down, walk , and then go up and exit from the right . You walk through the tunnel in tube. But you also go through the two tunnels in the blocks, through which the tube has been inserted. How many tunnels, in all, did you go through?

T

F1i8eK.

ne notebook (figure L ). How many holes are there in this notebook (assumeit has 10 pages)? Are they all tunnels? Are there any interesting mereological relations among the holes?

Fi88eL

212

Puzzlesand Exercises

Exercise (figure M ). Supposeyou exercisesome pressurewith a finger on a thin layer of plasticine until the finger' s shape sticks out on the ODDositeside. Have you made a hole, a bump, or both?

"" "" = ==== ===== = ~ FiaureM. Self-reciprocal BatMan (figure N) . We described a BatMan pattern (chapter 5) as involving two reciprocal fillers. Supposethat Bat and Man are two parts of a single connected object. Would that be a caseof selfreciprocal filling ? Can an object be host and guest of the samehole( s)?

FiaureN.

Exercise(figure0 ). Cut theskins(seechapter5) in theright columnwith someedge-to-edgecut, anddescribewhatwouldresultfrom acorresponding cut in the host objecton the left. Do the samewith the skinsof the tunnelsdepictedin figure4.14.

213

Punlac and Exercises

0' ~

ofcut ezample -edge edg &oto

0' ~

0 ~

~

o.

Holesin shadows (figureP). Wesaid(in chapter11) that holesaresomewhat akin to shadows . But we also saw(in chapter8) that therecan be holesin shadows . Now, imaginea threedimensionalshadow.What kind of holescantherebe in it, if any?

.

Fi88e P.

214

Puzzlesand Exercises

The " hole through . hole in . hole" (fiaure Q). That is what topologists call the surfaceof the object depicted here. How manyholes(i.e.t tunnels) does the actual solid objeCthave?

FIaweQ.

FJa- . R.

The tube in the hole (flpre S). Think ofa tube spinning inside a hollow. How would you describe the tunnel inside the tube? Does it also spin? Would you say that a part of the hollow is moving while the rest is still? Or would you deny that the tunnel and the hollow have parts in common? Would you say that the tunnel is still, regarding the tube as a spinning partial filler of the hollow?

.

Puzzles and Exercises

215

Fiawe8.

ne rolUngring (figure T). Imagine a wedding ring rolling on an inclined plane. Surely the hole in the ring is not still, as it moves down along with the ring. But is it also rolling ? Supposethe rolling object is a disk with an elliptical tunnel. How would you describethe corresponding movement of the hole?

~ -----Fia8e T.

Thespringin tile ring (figureU). Explainwhyloop a, whichgoesthrough the twistsof the spring, doesnot properlyidentifya tunnel, whereasloop b does.

(J b Fipre u .

216

Puzzlesand Exercises

The goldenbracelet(figure V) . How many tunnels are there in a bracelet built up from n interlocking rings (here n = 12)1 How many mock tunnels (seechapter 10)1

Figurev. The honoweenhole (figure W) . How many hollows are there in this Hallowe'en pumpkin ? How many tunnels can you count? Think of this object as a folded, closed Flatlandian world (seechapter 5): describeit and classify the holes.

Fiaure W.

The two pretzels(figure X ). Argue that although thesetwo objects can be transformed into each other by elastic deformation (fry that!), they cannot be treated alike by the hole theorist. Now , what about the possibility that theseare skins of cavity-tunnels?

Puzzlesand Exercises

217

Fiawex.

Penr.-e's skin(figureY). Try to perfectly 's fill up thetunnelin Penrose notoriousimpossible tribar. Try to drawtheskinof thetunnel.

FiI - . Y.

Mabi_ nd(figureZ). Consider a Flatlandin theshape of a Moblus strip,withalittlediskremoved . Isthatahole(viz., a tunnel -cavity ), or are theretwodistinct holes (oneforeachwayoflookingatit)?Onecouldhave , in relation to Flatlands imagined , a criterion to theeffectthatif one makea ringthatcannotbeshrunkto a pointthenonehasfounda can hole. Wouldthiscriterion workonMobiusland ?

Fil - . z.

218

Puzzles and Exercises

Hyperboles. We can easily talk about holes in three-dimensional space, and we have an idea of what they could be like in a lower-dimensional world (a Flatland ). What about higher-dimensional worlds? What is a hole in , say, a four- dimensional object? Is there any kind of hyperfiller in a four- dimensionalobject? " 11IehoW holy relief. Supposesomebodymade a hole in a marble stone with a relief. Argue that its perfect filler is not the one that reconstructsthe original pieceof work . Exercise. world .

Draw a natural

history of discontinuitiesin a Flatlandian

Exercise [ hard] . Count the number of holes in your kitchen, them into hollows, tunnels, and internal cavities.

Classify

Exercise[veryhard] . Write a programthat a robot could profitablyuse to put certainobjectsinto certainholes.

Further Reading

There is not much literature devoted specifically to holes. Yet over the years many authors have said a few words about them and related issues. The following is a small selection . It includes our favorites together with other readings that we found relevant or thought to be interesting , puzzling , curious , intriguing , or otherwise remarkable . All entries are annotated and are usually integrated by quotations and additional , nested biblio graphic references. Where a published translation of a quotation is not credited , the translation is our own . Edwin A. Abbott , Flatland. A Romanceof Many Dimensions(London: Seeleyand Co., 1882; reprinted by Penguin Books, 1952). The memoirs of A Square's extraordinary journey through a perfectly bidimensional world (and more). We talk about this in chapter 5 and elsewhere. Some problems concerning holes on Flatland are discussedby Charles Howard Hinton in " A Plane World " (in Scientific Romances , London: Swann Sonnenschein , 1884; reprinted in Speculationson the Fourth Dimension. Selected Writings of Charles Hinton, ed. Rudolf v. B. Rucker, New York: Dover, 1980, pp. 23- 40): it is explained, for instance, that a two -dimensional house wall cannot have two or more openings , for otherwise it would not have any support. Thesedisquisitions deviate from Abbott 's in that Hinton ' s creatures are constrained to living on the edge of their (disk-shaped) planet Astria, while Abbott ' s Flatlanders can move freely about their two -dimensional land. Hinton also went back to flat worlds in 1907 with An Episodeon Flatland: Or How a Plain Folk Discoveredthe Third Dimension(London : Swann Sonnenschein ; partly reprinted in the 1980 selection, pp. 163- 204). Other views include Dionys Burger's Sphereland (English translation by Cornelie J. Rheinboldt, New York : Crowell , 1965; reprinted by Barnes& Noble, 1983) and Alexander Keewatin Dewdney's The Planiverse. Computer Contact with a TwoDimensional World (London: Pan Books, 1980). A more technical discussion of what it meansto live in Flatland (and of how Flatlanders could succedin establishing the shape of their world ) can be found in the first chapters of Jeffrey R. Weeks' book The Shapeof Space. How to Visualize Surfacesand Three-Dimensional Manifolds (New York and Basel: Marcel Dekker, 1985).

220

Further Readin~

David M . Arm strong, A Materialist Theory of the Mind (London: Routledge and Kegan Paul, 1968). The remark that empty spacehas spatio-temporal properties is usedto support the thesis that it would be impossible to distinguish a material object from a nonmaterial " object in virtue of its primary qualities only . Thus, secondaryqualities, conceived of as irreducible properties, are thrown into the breach to provide the " stuffing for the matter (p. 282; Arm strong regardsimpenetrability as a secondary the Physical World (London: Routledge and quality). Seealso his Perceptionand " . . . we can : . 187 185 Paul 1961 , speak of the shape, size and duration ), pp Kegan of an empty spaceor vacuum just as much as that of a physical object, so there is " no differentiating mark there (an argument to be found in embryonic form in Berkeley and worked out by Hume in the Treatise, book I , part IV , section 4). In ' " Jacksons words, Arm strong suggeststhat if we attempt a definition [ of material object] in terms of primary qualities alone we find that each primary quality either " fails to distinguish a material thing from empty spaceor involves us in circularity of the rotating homogeneoussphere (Perception, p. 130). For Arm strongs problem " " his see 8 above in discussed Identity through Time, in Peter van ), ( chapter Reidel, 1980), pp. 67 78. Compare also Inwagen (ed.), Time and Cause(Dordrecht: " D. Robinson, " Re- Identifying Matter , Philosophical Review 91 ( 1982), pp. 317341. " Michel Aurnague and Laure Vieu, A Three- Level Approach to the Semanticsof " in Cornelia Zelinski Wibbelt ed. The Semantics of Prepositions: From ( ), Space, Mental Processingto Natural LanguageProcessing(Berlin: Mouton de Gruyter , forthcoming). " Here it is argued that in a sentencelike " Thereis a hole in this pieceof cheese the ' in ' links a " localized" object completely included in the interior of the preposition " referenceobject. The hole is a part of the interior of . . . the pieceof cheese. . . , the ' ' ' meronomy being a piece-whole case. It is not just an inclusion, for the hole is at" ' tached to the object just as the interior is, i.e. both are determined by the object. Seealso Laure Vieu, Semantiquedes relations spatiales et inferencesspatio-tempo' ' relies: Une contribution d I etude des structures formell es de I espaceen Langage Naturel (Ph.D . dissertation, Universite Paul Sabatier de Toulouse, 1991), where " holes are characterized as portions of space individuated by an object, be it by comparison with a typical form" (or a previous form ) or be it on account of the topological nature of the object. (p . 220) Each of theseworks includes a discussion of such arguments as " Thereis a hole in the sheet. The sheetis in the drawer. Ergo - there is a hole in the drawer," which are rated invalid on grounds of involving " different typesof part-whole relations. (On this seeD. A. Cruse, On the Transitivity " of the Part-Whole Relation, Journal of Unguistics 15( 1979), pp. 29- 38, and Morton " Winston, Roger Chaffin, and Douglas Herrmann, A Taxonomy of Part-Whole " Relations, Cognitive Science II ( 1987), pp. 417- 444). (We address this issue in chapter 7.)

FurtherReading

221

Thomas Bachler, Horst Ehrler, Peter Formell a, Christoph Irrgang, and Ottomar Kiefer, Das Loch (Kassel: GesamthochschuleKassel Fotoforum , 1985). A photographic tribute to holes, and to their role in photography: "Through a hole we come to see the light of day. . .. Hole means breakthrough.. . . Through the hole' s opening light floods into the darkness." (p. 3) More ethnographic curiosities can be found in Le trou. Mode d 'emploi, Neuchitel : Musee d' Ethnographie, 1990. Christer Backstrom, " Logical Modelling of Simplified Geometrical Objects and Mechanical Assembly Processes," in Su-shing Chen (ed.), Advancesin Spatial Reasoning , Volume1 (Norwood : Ablex, 1990), pp. 35- 61. Describesa logical model for geometric reasoning in a simplified world , called a 2D + world : " a two -dimensional world with the restriction that all bodies are axis parallel rectangles, but with holes allowed in them. Holes are also axis parallel rectangles, but ... through holes are allowed without splitting a body into two, thus preserving most of the topologically interesting problems of the real world " . (p. 37) The formalization is basedon axioms constraining the geometrical relations betweenobjects to the effect that: " A hole must be totally inside a body. . .. Two holes in the samebody must not overlap.. .. Bodies must not overlap, but a body can be (totally or partly ) located in a hole in another body.. . . A body is not allowed to consist of holes only ." (p. 43) Jonathan Bennett, Eventsand Their Names(Oxford: Clarendon Press, 1988). Outlines a theory where matter is regardedas qualified space: " . . . we can be helped to understand the notion of a thing in spaceif we analyze it in terms of qualitative variation of space. The basic idea is that for there to be an atom in a given region of spaceis for that region to be thus rather than so." (p. 117) We use this idea in ' chapter 3. In chapter 11 we also refer to Bennett s Locke, Berkeley, Hume: Central Themes(Oxford: Clarendon Press, 1971), where the caseof a subject is discussed " who , going by what he seesand feels, judges a certain jug to have the samesizeas a certain glasswhich is in fact shorter and narrower than the jug ." (p. 96) Seealso W. D. Joske, Material Objects(London: Macmillan , 1967), where it is argued that he who inverts primary qualities " would be a creature who found small pictures cover large patches, who claimed to be in contact with objects when he seemedto us to be some distance from them, who could walk through what we took to be solid walls, who took a surprising long time to walk around some things and a " surprisingly short time to walk round others. (p. 48) George Berkeley, A Treatise concerning the Principles of Human Knowledge (Dublin , 1710), in The Worksof GeorgeBerkeley Bishopof Cloyne, edited by A. A. Luce and TE . Jessop(London and New York : Thomas Nelson and Sons, 1949). Part I , 10, contains a remark to the effect that color - and secondaryqualities in " general- are inseparablefrom extension: But I desire anyone to reflect and try, whether he can by any abstraction of thought , conceivethe extension and motion of a body, without all other sensiblequalities. For my own part, I seeevidently that

222

Further Reading

it is not in my powerto framean ideaof a bodyextendedand moved, but I must to exist withal giveit somecolouror othersensiblequalitywhichis acknowledged other all from abstracted motion and extension In , mind. in the , , short , figure only ." (We discussthis at the beginningof chapter8.) qualities,areinconceivable , hrsg. von F. Pfihonsky(Leipzig, BernardBolzano, ParadoxiendesUnendlichen es of the : Paradox 1851 ). Englishedition and translationby Donald A. Steele . 1950 ) Infinite(London: Routledgeand KeganPaul, " See 59 on the " porosity of the world. Comparealso 66 on the distinction : " By the boundaryof a bodyI understand betweenboundedand unboundedsegments " atoms which still belongto it. ethereal extreme the collectionof those Thus, two bodiescan only be in contactwhereone of themis boundedand the otheris not (i.e., hasno lastor extremeatom). CharlesVernonBoys, SoapBubblesandthe ForceswhichMould Them(London: ; NewYork: E. andJ. B. Young, Societyfor the Promotionof ChristianKnowledge Their Coloursand. the Forceswhich Bubbles : edition , . New 1890 Soap enlarged ) Mould Them( NewYork: Dover, 1959 ). on soapbubbles(and weredisappointed know to wanted Everythingyou always not to find here). A classictextof modernscientificliterature, beautifullyillustrated . The otherfundamentalreading and full of tips on performingsoapyexperiments Plateau Ferdinand Antoine , Statistiqueexplrimentaleet thloriquedes is Joseph moleculaires ), the seules aux soumis es (Paris: Gauthier-Villars, 1873 forces liquid to geometry , first text to explainthegeometryof soapbubblesand their relevance . (Remarkably, the existenceof minimal notablyto the theoryof minimalsurfaces surfacesof genusn ~ 1, not obtainablewith soapfilms, wasonly provedin recent " , The Computer-Aided Discoveryof New Embedded years: seeDavid Hoffman " 9 (1987 ), no. 3, pp. 8- 21.) Compare Minimal Surfaces , Mathematicallntelligencer " 1866 in " der ), Popularwissenschaftliche alsoErnstMach, Die Gestalten Fliissigkeit( ), pp. 1- 16. For a morerecentoverviewseeFred (Leipzig : Barth, 1896 Vorlesungen J. Almgren, Jr. and Jean E. Taylor, " The Geometryof Soap Films and Soap " , The Bubbles , ScientificAmerican235(1976), no. 1, pp. 82- 93, and Cyril Isenberg . 1978 : Tieto Cleveden Bubbles and , ) Science ( Soap of SoapFilms " fenomenicarealizzabili Paolo Bozzi, Osservazionisu alcuni casidi trasparenza ' " . ed. Arcais ( ), Studiesin Perception con figure a tratto, in Giovanni B. Floresd . 88 1975 Giunti : Martelli Firenze and , ) , Milano pp Metelli ( Festschriftfor Fabio 110. , reportingthe transparency An examinationof somebasicpatternsof phenomenic of a the of 11 .2 in here superposition figure interestingpattern reproduced of theperceptionof a rotatingholein whitesquareon a blackone. Our discussion ' a frame(chapter11) is alsoindebtedto Bozzis seminalwork on the topic.

Further Reading

223

vomempirischen FranzBrentano,Psychologie (2 vols., Leipzig: Meiner, Standpunkt 1924- 1925 ). Englishtranslationby L. L. McAlister et al.: Psychologyfrom an ). (London: Routledgeand KeganPaul, 1973 EmpiricalStandpoint We could not speakof the magnitudeof a point or of a line (in chapter6 above) " without referringto Brentano's notion of " plerosis of a boundary, which is defined in which it is a boundary: " the of directions the number as a function of spatialnatureof a point differsaccordingto whetherit servesasa limit in all or only in somedirections.Thus a point locatedinsidea physicalthing servesas a limit in all directions,but a point on a surfaceor an edgeor a vertexservesas a limit only in somedirections. And the point in a vertexwill differ in accordance distinctions with theedgesthat meetat thevertex.. . . I call thesespecificdifferences ." (p. 157) The conceptis also discussedin many other Brentanian of plerosis zu Raum,Zeit und texts: seee.g. the first essaysin Philosophische Untersuchungen Chisholm Roderick und Korner . von Kontinuum (Hamburg: , hrsg Stephan on Meiner, 1976 Investigations ; Englishtranslationby Barry Smith: Philosophical ). , Time, andtheContinuum , London: CroomHelm, 1988 Space " " ), , PhilosophicalReview95 (1986 Tyler Burge, Individualismand Psychology . 3 45. pp Containsinterestingmaterialon shadowsandcracks(pp. 42- 43). SeealsoGabriel " ' " ), pp. 189- 214, , SeeingWhat Isn t There, PhilosophicalReview98 (1989 Segal " aredefinedas thin, dark marksthat couldbeeithershadowsor wherecrackdows " havebeenRobertMatthews cracks." (p. 208) Otherdiscussants , BurgeonShadows " " Self Authoritative and Cracks, and Tyler Burgeagain, Knowledgeand Perceptual Individualism," in R. H. Grimm andD. D. Merrill (eds.), Contentsof Thought ; Martin , 1988 ), pp. 77- 86and" pp. 62- 76 respectively (Universityof ArizonaPress " " , Defence Davies, Individualismand PerceptualContent, and Gabriel Segal Individualism," Mind 100(1991), pp. 461- 484 and pp. 485- 494 of a Reasonable . respectively Keith Campbell,AbstractParticulars(Oxford: BasilBlackwell, 1990 ). " If therecannotbe a hole in , therecannotbe a true part filling the place space " , David wherethe holecannotbe." (p. 145) Seealsofootnote5: In correspondence Lewishas urgedthat there can be indeededgesfor space , and so, on suitable ). If this is correct, it would provide support for , holes(closededges topologies , and Lewis an atomisticapproachthat viewsspaceas a compoundconstruction thinks this is a viable and perhapsthe bestoption. You cannot get a hole by abolishinga volume, and it is this fact that showsthat partsof spacearenot true parts." ' Lewis Carroll (CharlesLutwidge Dodgson), Alices Adventuresin Wonderland ). (London: Macmillan, 1865 " Down, down, down. Would the fall nevercometo an end? . . . 'I wonderif I shall fall right throughthe earth! How funny it 'll seemto comeout amongthe people

224

Further Reading

that walk with their heads downwards! The Antipathies, I think - ' . .. Down , down, down" (chapter 1: Down the Rabbit- Hole): Alice' s speculation about a classic problem raised by Plutarch and addressedby people like Oresme, Bacon, and Galileo. SeeMartin Gardner' s discussionin The AnnotatedAlice (Harmondsworth: Penguin Books, 1965, p. 27, note 4) and do not ignore the invitation to consult Camille Flammarion 'g " A Hole through the Earth" (Strand Magazine 38 ( 1909), " " pp. 349- 355), if only to look at the lurid illustrations. Gardner also reminds us " that the fall into the earth as a devicefor entering a wonderland has beenusedby ' many other writers of children s fantasy, notably by L. Frank Baum in Dorothy and the Wizard of Oz, and Ruth Plumly Thompson in The Royal Book of Oz. Baum also usedthe tube through the earth as an effectiveplot gimmick in Tik - Tok " ' of Oz. Carroll s interest in the matter also comesout in the 7th chapter of Sylvie and Bruno Concluded(London: Macmillan , 1893), where a method is describedfor " running trains with gravity as the sole power source. How? Easily .. . Each railway is in a long tunnel, perfectly straight: so of course the middle of it is nearer the centre of the globe than the two ends: so every train runs half-way down-hill , and that gives it force enough to run the other half up- hill ." Ignoring friction and air resistance, the time neededfor the train to travel from one end to the other would ' always take exactly the same time as Alice s fall - just over forthy -two minutes ' regardlessof the length of the tunnel. Carroll also anticipated Abbot s speculations on Flatland in the introduction to his Dynamicsof a Particle (a 1865pamphlet of ' political satire); and in Alice s Adventures, the playing-card characters provide another example of flat creatures, incapable of finding the gardeners that Alice hides in the interior of a flower-pot (chapter 8: The Queen's Croquet-Ground ). ' Finally , a mention of Alice s encounter with the Cheshire Cat and its persisting a of a smile that is not rigidly dependenton its face. " Well! grin , unique example ' I ve often seen a cat without a grin," thought Alice; " but a grin without a cat! It ' s the most curious thing I ever saw in all my life." (end of chapter 6, Pig and Pepper) Adelbert von Chamisso (Louis Charles Adelaide de C. de Boncourt), Peter Schlemihls wundersameGeschichtemitgetheilt von Adalbert von Chamisso und von Friederich Baron de la Motte Fouque( Niimberg: Schrag, 1814; herausgegeben reprint Stuttgart : Reclam, 1980). English translation by Sir John Bowring : Peter Schlemihl(London: Whittaker , 1824). A classic story featuring object-independent, reified shadows that can be cut and taken away: " He shook my hand, kneeled down in front of me, and I saw him quitely detach my shadow from the grass, from my head to my toes, with admirable " dexterity. He then took it , rolled it up and folded it , and lastly put it away. (p. 23 of the German reprint ; our translation) A century later, JamesMatthew Barrie will tell the story of Peter Pan having his shadow sawn back to his feet by Wendy (Peter Pan and Wendy, London: Hodder and Stoughton, 1911). Good examplesof the sort of reification that includes cartoon-holes. (We mention it at the end of chapter 9.)

Further Reading

225

" RoderickM. Chisholm , " PartsasEssentiaJ, to Their Wholes, Review of M eta26(1973 ), pp. 581 603. physics

" The first of a seriesof works that Chisholm devoted to a defenseof mereological " i.e. the view that a true individual can neither essentialism, gain nor lose any , parts; seealso the discussionwith Alvin Plantinga in the Reviewof Metaphysics27 Allen and Unwin , 1976), ( 1975), pp. 468- 484, and Person and Object (London: " Essentialism: David see a rebuttal B for , Mereological Wiggins ( Appendix " Asymmetrical Essential Dependence and the Nature of Continuants, Grazer PhilosophischeStudien7 ( 1979), pp. 297 315; for a critical overview Peter Simons, Parts. A Studyin Ontology(Oxford: Clarendon Press, 1987), chapter 7). As Chisholm e.g. to himself emphasizes , this view has a long pedigreeand can be traced back " Abelard (" no thing has more or lessparts at one time than at another : compareD . P. Henry, Medieval Logic and Metaphysics, London: Hutchinson University Library , 1972, p. 120), Leibniz (Nouveaux Essais, " xxvii II ) and Hume ( Treatise, I -iv -6). We discussit in chapter 8 in connection with the question of whether holes ' are parts of their hosts. In that context we also refer to Saul Kripke s thesis that " constitution has the modal strength of identity : seethe third lecture of Naming " in Donald Davidson and Gilbert Harman eds.), Semanticsof and Necessity, ( Natural Language(Dordrecht and Boston: Reidel, 1972), pp. 253- 355, addenda Harvard University pp. 763- 769; reprinted as Naming and Necessity(Cambridge, Mass.: " Press, 1980). For a recent exchange, seeMarc Johnston, Constitution Is " " Not Identity , Mind 101 ( 1992), pp. 89- 105, and Harold W. Noonan, Constitution " Is Identity , Mind 102( 1993), pp. 133 146. " Ernest Davis, " A Framework for Qualitative ReasoningAbout Solid Objects, in , G. Rodriguez (ed.), Proceedingsof the Workshopon SpaceTelerobotics(Pasadena Ca.: NASA and JPL , 1987), pp. 369- 375. Reasoningabout a die dropped inside a funnel: will it fall out the bottom? The account usesvarious geometric and topological notions and relies on an ontology ' " " " supplied inter alia with a sort of pseudoobjects, i.e., point sets that move ' around with objects, like the hole of a doughnut, the opening of a bottle, or the " center of massof any object. (p. 373) Unfortunately , it turns out that an adequate formalization of the analysis requires about 90 non-logical terms (plus standard arithmetic operators) governedby about 140axioms. (The notion of pseudoobject" '" appearsalready in Davis Shapeand Function of Solid Objects: SomeExamples Science, New York University (Technical Report No. 137, Department of Computer " , 1984), where a pseudoobjectis said to be analogous to an object, except for three differences: its shapecan be an arbitrary point set, its position is dependent on some real object, and it has no physical reality, and is therefore not affectedby " physical laws. (p. 3) This paper also contains definitions of various morpho topological notions including tunnels.)

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" PaulAdrienMauriceDirac, " QuantumSingularities in the Electromagnetic Field, . 60 72 . Proceedings of theRoyalSocietyAI33 (1931 ), pp " A hole if therewere one, wouldbea newkind of particle" : theofficialformulation , of Dirac's predictionof theexistence of antiparticlesin relativisticquantumtheory. -energyparticlesarenot protons(asDirac himself The" holes" in a seaof negative assumed two yearsearlierin " A Theoryof Electronsand Protons," Proceedings of theRoyalSocietyA 126(1929 ), pp. 360- 365) but a " a newkind of particleunknown " to experimental , behavinglike positiveenergyparticlesof positivecharge. physics The " positron" wasactuallydiscoveredby Carl David Andersona few months " later, in thesummer1932(see" Energiesof Cosmic-RayParticles , PhysicsReview 41( 1932 . 405 421 . The reader interestedin this importantchapterof modern ), pp ) ' physics(Diracs holetheoryis generallyconsidered the first steptowardsa correct theory of the quantizedelectronfield) may consult the papersin BehramN. Kursunogluand EugeneP. Wigner(eds.), Reminiscences Abouta GreatPhysicist : Paul AdrienMauriceDirac (CambridgeUniversityPress 1987 . Seealso Simon , ) " " -EnergySea Saunders , The Negative and HarveyR. Brown, , in SimonSaunders ThePhilosophy of Vacuum (Oxford: ClarendonPress , 1991 ), pp. 65- 109. PierreDuhem, Le mixteet la combinasion . Essaisur I'evolutiond'uneidee chimique (Paris: C. Naud, 1902 ; reprintedby Fayard, 1985 ). historical Mainly investigationof the notion of a mixture. Includes useful references to classicexplanationsof solidity in termsof fittings of shapesat the microscopiclevel. (Usethis to try and solvethe Eift'el puzzlewe introduceat the endof chapter8). By Duhemcheckalsothe 8th volumeof Le systeme du monde (Paris: Hermann, 1958 ), particularlythefirst two chapterson emptiness andho" or vacui. George Dunning, YellowSubmarine , from an original story by Lee Minoft' - Subafilm, 1968 (KingFeatures ). ' Remember the Beatles journeyto the Seaof Holes? Explicit usageis alsomadeof the ideaof a cartoon-hole, especiallyby RingoStarr(a holein his pocketis used 's Band. Theideahas to freeSergeantPepper beenused,amongothers, by Robert ) Zemeckisin WhoFramedRogerRabbit? (TouchstonePictures , 1988 ), basedon ' Gary K. Wolf s novelWhoCensored RogerRabbit?(New York: Ballantine , 1982 ). Wementionit at theendof chapter9. FriederichDiirrenmatt, " Der Tunnel," in Die Stadt. ProsaI -IV (Ziirich: Arche, 1952 ). New versionin FriederichDu" enmattLesebuch (Ziirich: Arche, 1978 ). time we travel train and enter a tunnel by , we hopeit is not that tunnel: "Every How we got into this tunnel, I don't know, I haveno explanationfor that.... Thereis no evidencethat somethingmight be wrong with the tunnel, apart of coursefrom the fact that it doesnot end." Onecanstart a shortstory with somebody enteringa hole. Diirrenmatt madean entirestory out of this entrance .A tunnel- a deep,neverending hollow? neverending

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" ? The Hole Story," John Earmanand John Norton, What PriceSubstantivalism 38(1987 BritishJournalfor the Philosophy ), pp. 515- 525. of Science -Time. Absolute The story is taken up in John Earman, WorldEnoughandSpace , 1989): chapter8, 4 " The versusRelationalTheoriesof SpaceandTime(MIT Press " . The of various First Hole Construction , describes ways makingholesin space " ism: A entirechapter9 elaboratesuponthis: GeneralRelativityand Substantival " " " , BritishJournal Very HoleyTruth . Discussionin J. Butterfield, The Hole Story " Substance Teller , Relations , for the Philosophyof Science40 (1989), pp. 1 28; Paul " 100 Review Time of Nature the , Philosophical Space , and Argumentsabout " : and Substantivalism Holes Robert and . 397 , 363 , 1991 Rings ; Rynasiewicz ), pp ( " , Philosophyof Science59 (1992), pp. 572On the Programof LeibnizAlgebras 589. : von A bis Z (Stuttgart: Thienemanns , Geschichte Michael Ende, Die unendliche Story (Garden 1979 ). Englishtranslationby Ralph Manheim: The Neverending ; reprintedby PuffinBooks, 1985). City, NY: Doubleday& Co., 1983 " 'A hole?' the rock chewergrunted. 'No, not a hole,' said the will-o'-the-wisp "' . 'A hole, afterall, is something . This is nothingat all . (p. 24) despairingly M. C. Escher(Zwolle: Till , 1959 ; revised Maurits C. Escher , Grafiekentekeningen andenlargededition, 1966). Englishtranslationby JohnE. Brigham: TheGraphic Workof M. C. Escher(New York: Meredith, 1967 ). " wood his 1952 on , Dragon: Howevermuchthis dragon engraving Commenting triesto bespatial, heremainscompletelyflat. Two incisionsaremadein the paper on which it is printed. Then it is folded in sucha way as to leavetwo square . But this dragonis an obstinatebeast,andin spiteof his two dimensions openings hepersistsin assumingthat he hasthree; so he stickshis headthroughoneof the " holesand his tail throughthe other. (p. 22) beiwechselnder Beleuchtungsrich Kai von Fieandt, 0 berSehenvonTiefengebilden . 1938 Helsinki of Institute , ) from the , University Psychological tung(Reports A studyon theperceptionof concavityandconvexityin two-dimensionalpatterns herein figure11.5 (seealso" Dasphanomenologi includingthepatternreproduced " a 6 (1949), pp. 337- 357; scheProblemvon Licht und Schatten , Acta Psychologic " , Local Shading for a recent, computer-aided perspective , seeAlex P. Pentland " IEEE Transactions Machine and on Pattern Analysis Intelligence6 , Analysis ' hole detection of discussion . In our 2 t miss don 187 ) ; (1984 ), pp. 170 'figure suchas curious other with s Fieandt 11 we patterns patterns compare (chapter' ) " , ScientificAmerican Hoffmanscosinesurface(" TheInterpretationof VisualIllusions 's accountof a similar 289(1983 ), no. 6, pp. 154- 162). SeealsoC. Peacocke " , casein which orientationmatters: the perceptionof a c and a 0, Scenarios " Cambridge . The Contents ed T. Crane in ( and Experience of ( ), , , Perception Concepts , 1992 ), pp. 105- 135(the exampleis due to Ernst Mach, UniversityPress ; Englishtranslation , Jena: Fischer, 1886 Beitraegezur Analyseder Empfindungen Sensations to the : , Chicago: Open Contributions M. Williams C. of Analysis by Court, 1897 , p. 106). , reprinted1914

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FurtherReadin ~

Robert B. Fisher, " Representation , Extraction and Recognitionwith Second " Order TopographicSurfaceFeatures , Imageand VisionComputing10 (1992 , ) 156- 169. A significantexampleof theroleholesandothersuperficialparticularsmayplayin artificial imageprocessing : " Representation for many three dimensionalobject -order recognitionsystemslack highly salientvisual details. Here, eight second volumetricprimitivesaredefinedto extendtherangeof descriptivefeaturesusable " . . 156 The for object representation and thus recognition (p ) primitives(called 'second " first -order' because , they are smallerfeaturesthat add detail to the surface , ratherthan specifythe overallshape , and second , theydenotemorespecific , " higher-leveldescriptiveshapes) are split into four positive(extruding) features , , spike, ridge, fin , and four corresponding bump negative(intruding) features , dent, hole, groove , slot. Moreover, they are classifiedaccordingto " the number of " . 157: dimensionswherefeaturesizesare significant (p ) bumpsand dents are O-dimensional , fins and slotsare 2 dimensional . By ; the othersare I -dimensional the sameauthor, seealso From Surfacesto Objects . ComputerVisionand Three Dimensional SceneAnalysis(Chichester : Wiley, 1989 ). CompareE. R. Davies,Machine Vision: Theory, Algorithms , Practicalities(London: AcademicPress , 1990 ): . chapter13is entirelyon holedetection Richard M. Gale, Negationand Non-Being, AmericanPhilosophicalQuarterly , No. 10(Oxford: BasilBlackwell, 1976 MonographSeries ). ' " Two long footnotes in the Author s own words, to Plato's theory of non-being , in the Sophist , ... and to his accountin the Timaeusof anotherkind of nonbeing - the Receptacle or emptyspace ." (p. 63) The Receptacle is defendedas a " basisfor the explanationof the ultimate groundsof individuationof spatio" temporal(concrete ) individuals, and as an entity necessary to " an adequateaccount ... of theobjectivityof concreteindividuals." Unfortunately" it is mysterious andelusive to it is by reference ; our only access to thosemetaphysical problemsto whichit givesan intellectuallysatisfyingresolution." (pp. 63- 64) (Wementionthis in our discussionof matterand spacein chapter3.) Martin Gardner,KnottedDoughnuts andOtherMathematical Entertainments ( New York: Freemanand Co., 1980 ). " Chapter5, Doughnuts:Linked & Knotted," pp. 55- 66, Motto: " As you ramble on through life, brother / Whateverbe your goal, / Keep your eye upon the doughnut/ And not uponthe hole!" (signedAnon). Peter Geach, " What Actually Exists," Proceedings of the AristotelianSociety , Volume42 (1968 Supplementary ), pp. 7- 16. Holesas accidentalindividuals. A discussionof the principle"x is actualif and " only if x eitheracts, or undergoes , or both (p. 7) in relationto someentia change nongrata, like numbers , events , properties , surfaces , andholes.Thoughsomepara-

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" phrastic strategiescould be envisioned in some cases, [ t] here are, however, some is much harder to get rid that individuals accidental phrasesostensiblydesignating ' . The initial noun' The black box is this of surface . phrase in a of top plausible way ' is not easy to regard as a nominalization of ' The box is top- surfaced or the like: ' ' nor it is e~sy to seehow this sentenceand black can be fitted together in some tantamount to the original sentence(and there is the syntactically coherent string ' ' . further difficulty that black is clearly a first order predicate' of actual objects) ' The hole in the tooth was smaller than the dentist s finest probe' ; can for Similarly " ' we rephrasethis, using The tooth was holed' ? (p. 12) " : Early Understandings Susan A. Gelman and Henry M . Wellman, Insides and Essences " of the Non Obvious, Cognition 38 ( 1991), pp. 213- 244. An experimental study on the understanding of the inside-outside distinction in that early childhood. 3- to 5-year-old children were tested on their recognition ' " insides are typically more important than outer surfacesfor an object s identity " and functioning : the reported results seemto testify against a more traditional view held eminently by researchersof the Piaget school (compare Jean Piaget, Plays. Dreams. and Imitation in Childhood, New York : Norton , 1951). JamesJ. Gibson, The EcologicalApproachto VisualPerception(Boston: Houghton Mimin , 1979; reprinted by Erlbaum, 1986). " geometry An attempt to formulate a theory of surface layout; a sort of applied " the for that is appropriate study of perception and behavior. It is based on such concepts as: open environment, enclosure, detached object, attached object, denumerability, partial enclosure, hollow object, place, sheet, fissure, stitch, diheral (convex: edge; concave: comer), curved concavity, curved convexity. We read that " An enclosureis a layout of surfacesthat surrounds the medium in somedegree. . . . A partial enclosureis a layout of surfacesthat only partly enclosesthe medium. It is may be only a concavity. But a cave or a hole is often a shelter. A hollow object an enclosure but outside the from It is an . an object that is also an enclosure object from the inside, part of the total surfacelayout facing outward and the other part inward. . . . A fissure is a layout consisting of two parallel surfacesenclosing the medium that are very close relative to their size. The surfacesof rigid solids often have fissures(cracks). . . . A convexdiheral is one that tends to enclosea substance the medium and and to make an edge; a concavediheral is one that tends to enclose ' " to make a corner. (pp. 34 ff ) The readerwill find in Gibson s book severalfascinating ideas, but little organization. Nelson Goodman, Languagesof Art . An Approachto a Theory of Symbols(Oxford University Press, 1969). ' ' Distinguishesbetweenpicturesof a manand man pictures, where man picture and " unbreakable one-place predicates, or class terms, like cognates are considered 'desk' and ' table' " . 21 . We use the distinction in chapter 11). (p ) (

Further Reading

P. L. Heath, " Nothing," in Paul Edwards(ed.), The Encyclopedia of Philosophy (Londonand New York: CrowellCollier and Macmillan, 1967 ), Vol. 5, pp. 524525. Arguingthat nothingis givenonly in relationto what is, that absenceis a mere privation of something , the point is madethat ..A hole is alwaysa hole in something : takeawaythe thing, and the holegoestoo; moreprecisely , it is replacedby a biggerif not betterhole, itself relativeto its surroundings and so tributary to , " somethingelse. (p. 524) AnnetteHerskovits andSpatialCognition , Language . An Interdisciplinary Studyof thePrepositions in English(CambridgeUniversityPress , 1986 ). Treatsholesasa typeof " spatialentity," alongwith ordinarysolidobjects,partsof , and a few more. " A hole is definedonly in relation to anotherobjectin space whichit occurs.. . . A holemaybeentirelyboundedby its associated object, or part of its boundarymay bevirtual, definedby extrapolatingthe object's surfacespast the hole. Thus the hole in a shirt is boundedby planesdefinedby extendingthe two sidesof the fabric over the hole, and by the edgesof the fabric aroundthe hole." (p. 63) Explicit useis also madeof the Gestaltlaw of " good form" (Kurt Koftka, Principlesof GestaltPsychology , NewYork: Harcourt, Brace& Co., 1935 ): " As ' ' to a complete speakingof a holeimpliesreference , normal shape , thegeomet ric descriptionapplicableto thereference . . . is the of objects region spaceoccupied ." (p. 69) by that completenormalshape David Hilbert and S. Cohn-Vossen Geometrie(Berlin: Springer , Anschauliche Verlag , 1932 ); Englishtranslationby P. Nemenyi: Geometryand the Imagination (NewYork: Chelsea , 1952 ; reprinted19903 ). " . iii A classicpresentationof geometry" in its visual, intuitiveaspects (p ), including " " an account in largebrushstrokes of somebasicfactsconcerningthecurvatureof surfaces( 28) and the topologicaldifferencebetweenholed spheresand toruses ( 45). Basedon a courseof lecturesgivenby Hilbert in the winterof 1920- 1921at Gottingen. " " DonaldD. HoffmanandWhitmanA. Richards , Partsof Recognition , Cognition 18(1985 ), pp. 65- 96. Includesa discussionof holesas " negativeparts" : whereasa " positive," material part can be thoughtof as an object" stuckinto" the whole, segmented off at the concavitiesin the 2D image,a holeis seenasa placewherean objectis " scooped out" of the whole, leavingbehindconvexitiesin the host's surface . Comparealso " Richardsand Hoffman's " Codon Constraintson Closed2D Shapes , Computer Vision,GraphicsandImageProcessing 31(1985 ), pp. 265 281. DouglasR. HofstadterandDanielC. Dennett, TheMind's I . Fantasies andReflections on SelfandSoul(New York: BasicBooks, 1981 ). On the ontological problem: " Our world is filled with things that are neither mysteriousandghostlynor simplyconstructedout of the buildingblocksof phys-

231

ics. Do you believe in voices? How about haircuts? Are there such things? What are they? What , in the language of the physicist, is a hole- not an exotic black hole, but just a hole in a piece of cheese , for instance? Is it a physical thing ? .. . These things are not physical objects with mass, or a chemical composition, but they are not purely abstract objects either- objects like the number 1[, which is immutable and cannot be located in spaceand time. Thesethings have birthplaces " and histories. They can change, and things can happen to them (Introduction, authored by Dennett, pr . 6- 7). In correspondenceHofstadter stresses, against what he calls traditional , " concrete" holes, the importance of more general, abstract " " " , pattemy disturbances. Such are the ones described in his Commentary '" ' ' About Gebstadter s Essay Cheltenham Aachen Garamond Eurostyle , in Stefan Schadler and Walter Zimmermann (eds.), John Cage- Anarchic Harmony. Ein ' Buchder Frankfurt Feste 92/ Alte Oper Frankfurt (Mainz: Schott, 1992), pp. 57- 66. Edmund Husserl, Ding und Raum, Vorlesungen1907, hrsg. von Ulrich Claesges (HusserlianaXVI , The Hague: Nijhoff , 1973). Appendix VII deals with the perception of empty space(pp. 361- 362). In footnote 2 the suggestionis made that someempty spaceis necessarybetweengiven things " or phantasms: " If nothing spatial is given, no spaceis given either. Husserl considers and seemingly rejects the hypothesis that we imagine figures between the " objects, and that we attribute spatiality to these figures: Now , what about the of ] configurations into the space in imaginative projection [ Hineinphantasieren between[ Zwischenraum]? Can imaginatively projected colours be bearersof perceived " depth?" He further analysesperception through seriesof objects ( If I look at the street through a window , let us say through a tube (or through a tube I look at a window together with somehouse-wall , through this window again at another housein which again there is a window , etc.) then I have simultaneously seen(and " " have in my present " visual field ) severallayers, i.e., patchesof the visual space ), and the casesin which we seeair betweenthings. By Husserl we should also quote a passagefrom the Ideas: "The most perfect geometry and its most perfect practical control cannot help the descriptive students of nature to express precisely (in exact geometrical concepts) that which is so plain, so understanding, and so , umentirely suitable a way to expressin the words: notched, indented, lens-shaped belliform , and the like - simple conceptswhich are essentiallyand not accidentally " " inexact, and thereforealso unmathematical ( Ideenzu einer reineDPhanomenologie und phanomenologischen Philosophie. Erstes Buch. Allgemeine Einftihrung in die reine Phanomenologie," 74, JahrbuchfUr Philosophieund phanomenologische Forschung1 ( 1913), pp. 1- 323; English translation by W. R. Boyce Gibson: Ideas: GeneralIntroduction to Pure Phenomenology , London: Allen and Unwin , 1931). " Don Ihde, " On Hearing Shapes, Surfaces, and Interiors , in R. Bruzina and . Dialoguesand Bridges(Albany : State University B. Wilshire (eds.), Phenomenology of New York Press, 1982), pp. 239- 251. A phenomenologicalsketch of the possibility offered by hearing in relation to the detection of primary qualities like shapeand dimension. Seethe last section of our chapter 11.

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Further Reading

Peter van Inwagen, Material Beings(Cornell University Press, 1991). " " Chapter 13( Artifacts, beginningwith a picture showing four walls of sand heaped up with a bulldozer) indirectly suggestsa way of treating what we call ready-made holes (in our chapter 10). and the ComputationalMind (MIT Press, 1987). Ray JackendotT, Consciousness ' Appendix B is devoted to an enrichment of Marr s 2 ! dimensional sketch, a level of visual representation somewhat half-way between Oat image and threedimensional representation(seeDavid Marr , Vision, San Francisco: Freeman and Co., 1982, especiallychapter 4). Following a suggestiondue to L. Shiman (Grammar for Vision, PhiD. dissertation, MassachusettsInstitute of Technology, 1975), Jackendoft' introduces the notion of a 2 ! structural description among whose " primitives are boundary and region, the former being treated as obligato rily " directedtoward the regions to which they belong. (p . 331) This is exemplified with respectto a pattern obtained by drawing a circle inside a square: depending on whether we take the direction of the circle' s boundary as pointing toward the interior or toward the exterior of the circle, the reading of the figure shifts from (i ) a disk superimposedon a square to (ii ) a square with a circular hole in it . (We discussthis account in chapter 11, figure 11.3.) Frank Jackson, Perception. A RepresentativeTheory (Cambridge University Press, 1977). " etc., is to Representativeof the view that to say that there are holes, empty spaces ." (p. 131) The take paradigm examplesof nothingsand make them into somethings Ludovician account is pushed one step further , suggestingthat one can " translate statements putatively about holes in terms of statements about hole-surrounds. ' 'There are many holes in that pieceof cheesejust saysthat it contains many hole' surrounds; There are the samenumber of holes in A as in B' just saysthat A and B " have the same number of hole-surrounds; and so on and so forth . (p. 132) We mention and briefly criticise this view in chapter 3. We also refer to Jackson in chapter 12, when we consider the many-property objection to adverbialism: see his " On The Adverbial Analysis of Visual Experience," Metaphilosophy 6 ( 1975), pp. 127- 135. Jerome Klapka Jerome, Three Men in a Boat ( To Say Nothing of the Dog! ) (Bristol: Arrowsmith , 1889; reprinted by Penguin Books, 1957). The forms known to geometry do not include holed objects: " . .. I took the tin ofT myself, and hammered at it with the mast till I was worn out and sick at heart, whereupon Harris took it in hand. We beat it out flat ; we beat it back square; we battered it into every form known to geometry- but we could not make a hole in it ." (p. 117)

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Immanuel Kant , Critik der reinen Vernunft (Riga: Johann Friedrich Hartknoch , 1781). English edition and translation by Norman Kemp Smith: Critique of Pure Reason(London: Macmillan , 1929). " . . . If I lay the ball on the cushion, a hollow follows upon the previous flat smooth shape; but if (for any reason) there previously exists a hollow in the cushion, a leaden ball does not follow upon it ." (A203/ 8248- 8249) Although Kant seemsto be confusing the terms of the causal relation, we can take him as saying that the responsibility for the shaping of objects dependson what other objects are (or are made of ), while the shaping itself bearsno responsibility for what other objects are (made of ). (We mention this in chapter 9.) David Kaplan , " Words," Proceedingsof the Aristotelian Society, Supplementary Volume 64 ( 1990), pp. 93- 119. Footnote 8 on p . 97 hints at what could be labeled the " hole-theory" of words: " Vou are creating a physical object when you inscribe a word in stone, but the token of the word is not the great big heavy physical object, the physical object which is the token of the word is the light -weight space." (We discuss this in chapter 12.) " Lajos Kardos, Ding und Schatten. Eine experimentelle Untersuchung fiber die " Grundlagen des Farbensystems, Zeitschrift fUr Psychologie, Erganzungsband23 (Leipzig: Barth, 1934). First thorough investigation of shadow perception. An interesting experiment is : draw a line following the profile of a shadow; it will suddenly look like suggested a grey patch. This suggeststhat in ordinary cases, a shadow is seenwhen one is able to attribute the standard colour of a surfaceto the area in the shadow (therefore , shadowsdo not darken the surfacesonto which they are cast). Toomas Kanno , " Disturbances," Analysis 37 ( 1977), pp. 147- 148. Holes as disturbances, i.e., entities found in someother objects " not in the sensein which a letter may be found in an envelope, or a biscuit in a tin , but in the sensein 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 of telling whether an object x is ' in ' an objecty in the sensepeculiar to disturbancesis to enquire whether x can migrate throughy .... That which a disturbance is in is its medium" (p. 147; compare T. J. M . Bench-Capon' s " A Note on Mr . Kanno 's Disturbances," pp. 148- 149). Fred Dretske, " Moving Backward in Time," PhilosophicalReview71 ( 1962), pp. 94- 98, considersdisturbances" which spreadoutward at a certain rate, and a disturbance, once again, is an occurrence or happening." See also Peter Simons' remarks in Parts, where moving disturbances such as waves are defined as " a special and interesting kind of continuant: moments which continuously change their fundaments." (p. 308) More recently, Kanno has proposed to replace the tenn " distur bances" with the neologism " derivities," i.e., derived entities (" Note Toward a Fonnal Metaphysics of Knots , Holes, Glows, Sneezes , Artifacts, Living Organisms "' ' , Rivers, and Other Derivities , manuscript, York University, 1990).

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Wolfgang Kohler , Gestalt Psychology(New York : Liveright, 1947). " " Chapter 6, Organized Entities, provides an account of the gestaltic dimension of our talk about holes and related entities. " The . .. referenceto larger wholes is implied in many terms which we continually use as banal words. We do not generally realize that the meaning of such words points beyond the local facts to which the words may seemto be attached. From a large list I will give only the : The German 'Rand' (in English ' brink ' or 'edge') is such a following examples ' ' word ; again Anfang (' beginning' ), 'Ende' and 'Schluss' ('end' and 'close'), 'Stuck' ' and ' Teil (' piece' and ' part ' ), 'Rest' (' rest' or ' remnant' ); also 'Loch' (' hole' ) and ' ' ' ' ' ' Storung ( disturbance ). It will at once be seenthat a place can appear as a hole only inasmuch as it constitues an interruption of a larger entity, the other parts of which have figure character. Mutatis mutandisthe same holds for the meaning of 'disturbance' . It is by no meansnecessaryto restrict the list to instancesin which the words apply to sensory facts. In the case of thought processes an event is a ' disturbance' only with regard to a larger and otherwise unitary whole which it . . .. Similar casescan easily be found among the adjectives and verbs. interrupts 'H ohl' ' hollow ' ' ' ' ' ' ' ' ' ( ) and offen ( open ), complete and incomplete belong in this class, in that their meaningsrefer to specific experiencedunits in which theseadjectives are alone applicable. . .. Essentially, the meaningsof such words remain the same in all provinces of experience; for the principal phasesof organization are not restricted to any specificfields." (pp. 203- 205) " " Serge Lang, Great Problems of Geometry and Space, in The Beauty of Doing Mathematics. Three Public Dialogues(New York : Springer-Verlag, 1985). A most lively introduction to rubber geometry showing the enormous importance of holes for the characterization of compact, unbounded surfaces. The discussion includes questions about holes in two- and three-dimensional manifolds: How many holes does a knotted torus have? (p. 95), or: Can one be left with only holes? " (p. 87), or again: Are we living on something which is the analogue of a threedimensional sphere? What happensif we look far out in space, do we find a hole? One can also ask the question in dimension two , but for us dimension three is more relevant. We seethree-dimensional space, and we have telescopeswhich are more and more powerful. If we can seesufficiently far, what are we going to find? Are we living on an object equivalent to a sphere? or are we going to find holes? This is getting serious. You can really raise this question about the nature of the universe." (p. 93) Gottfried Wilhelm Leibniz, Nouveaux essaissur l 'entendementhumain, in Oeuvres Philosophiqueslatines et franroises de feu Mr . de Leibnitz, ed. R. Eric Raspe (Amsterdam and Leipzig, 1765). English edition and translation by Peter Remnant and Jonathan Bennett: New Essayson Human Understanding,(Cambridge University Press, 1981).

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" We quote Leibniz on the Ship of Theseus, which the Athenians were constantly " II - xxvii-4 and which ) posesa philosophical puzzle for a theory of identity repairing ( . The puzzle (mentioned in chapter 7 above and again in chapter 8) is already described in Plutarch' s Lives, 22- 23: " The vesselin which Theseussailed and returned safe ... was preserved by the Athenians up to the times of Demetrius Palerus . .. ; being so refitted and newly fashioned with strong plank , that it afforded an example to the philosophers in their disputations concerning tbe identity of things that are changed by addition , some contending that it was the same, and others that it was not " (see David Wiggins, Samenessand Substance , " Oxford: Basil Blackwell, 1980, p. 92). A fuller statement is given by Hobbes: For if , for example, the ship of Theseus. . . were, after all the planks were changed, the same numerical ship it was at the beginning: and if some man had kept the old planks as they were taken out , and by putting them afterwards together in the sameorder, had again made a ship of them, this, without doubt , had also beenthe same numerical ship with that which was at the beginning; and so there would " have been two ships numerically the same, which is absurd (ElementorumPhilosophiae, Sectio Primade Corpore, 1655; English translation by Sir William Molesworth : Elementsof Philosophy, The first Section ConcerningBody, in The English Worksof ThomasHobbesof Malmesbury, London: John Bohn, 1839; reprinted by Scientia Verlag, 1966, pp. 136- 137). Leonardo da Vinci , I manoscritti e i disegnidiLeonardo da Vinci. II CodiceArundel 263 nel Museo Britannico ( 1478- 1518), pubblicati dalia Reale Commissione Vinciana (4 vols., Roma: Danesi, 1923- 1930). Selected English edition and translation by Edward MacCurdy in: The Notebooks of Leonardo da Vinci (London: Reynal and Hitchock , 1938; reprinted by G. Brazilier, 1958). Leonardo put forward the view that surfacesare somesort of abstraction, with no divisible physical bulk. " What is it . . . that divides the atmospherefrom the water? It is necessarythat there should be a common boundary which is neither air nor water but is without substance, becausea body interposed between two bodies prevents their contact, and this does not happen in water with air. . .. Therefore a surfaceis the common boundary of two bodieswhich are not continuous, and does not form part of either one or the other, for if the surfaceformed part of it , it would have divisible bulk , whereas, however, it is not divisible and nothingnessdivides " thesebodies the one from the other. (pp. 75- 76) This view, which comesclose to " " ' Euclid s conception of a surface as that which has length and breadth only (Elements, Book I , Definition 5), has been thoroughly scrutinized by A vrum Stroll conception of (in Surfacesand related works) and contrasted with the physicalistic " " surfacesas the topmost layers of atoms (G. A. Somorjai, SurfaceScience, Science 201 ( 1978), pp. 489 497). (We briefly mention thesetwo views at the beginning of chapter 2.)

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Stanislaw Lesniewski, Podstawy ogolnej teoryi mnogosci. I (Moskow: Prace Polskiego Kola Naukowego w Moskwie, Sekcya matematyczno-przyrodnicza, 1916). English translation by D. I . Barnett: " Foundations of the General Theory of Sets. I " , in Stanislaw Lesniewski, Collected Works, eds. Stanislaw J. Surma, Jan Srzednicki, D. I . Barnett, and Frederick V. Rickey (Dordrecht , Boston, and London: Kluwerj NijhotT, 1992), Vol. I , pp. 129- 173. Mereology- the theory of parts and wholes- plays an important role in our inquiry . This work by Lesniewski is generally considered the first rigorous treatment of the part -whole relation. The other standard referenceis Henry S. Leonard and Nelson Goodman, " The Calculus of Individuals and Its Uses," Journal of Symbolic Logic 5 ( 1940), pp. 45- 55. Although the underlying logics dit Ter, both treatmentsagreeon the basic principles and are often referred to jointly as " classical extensional mereology." As for the historical background, one can go as far back as to Aristotle (seeMetaphysics, ~ 26- 27), though one should perhaps work one' s way backwards starting from Husserl' s third Logical Investigation. For a general overview, including later developments, see Rolf A. Eberle, Nominalistic Systems(Dordrecht: Reidel, 1970), and Peter Simons, Parts. A Study in Ontology (Oxford: Clarendon Press, 1987). David K . Lewis and StephanieR. Lewis, " Holes," AustralasianJournal of Philosophy 48 ( 1970), pp. 206- 212 (reprinted in David K . Lewis, Philosophical Papers. Volume1, Oxford University Press, 1983, pp. 3- 9). A most penetrating philosophical dialogue which was, in many ways, the inspiration for our own work. The Lewisesaddressand try to solve the major difficulties surrounding a nominalist cum materialist account of holes: as we seem unable satisfactorily to paraphrase talk about holes into talk about perforated material objects, the suggestionis made that holes are (parts of ) material objects, viz. hole" linings. The lining of a hole, you agree, is a material object. For every hole there is a hole-lining; for every hole-lining there is a hole. I say the hole-lining is the hole" (p. 5). (Discussedhere at some length in chapter 3 and elsewhere.) Otto Lipmann and Hellmuth Bogen, Naive Physik (Leipzig: Ambrosius Barth, 1923). " " Presumablythe first official appearanceof the term naive physics referring to the capacity for intelligent action in relation to everyday tasks and objects. The book provides an application to causality and natural law of the ideas concerning the " " Physik des naiven Menschen set forth by the Gestalt psychologist Wolfgang " Kohler (see Intelligenzpriifungen and Anthropoiden . I ," Abhandlungen der Koeniglich PreussischenAkademieder Wissenschaften (Berlin), phys.-math. Klasse, no. I , 1917; English edition by E. Winter: The Mentality of Apes, New York : Harcourt , Brace & Co., 1925; in fact it is in Kohler 's correspondencethat there " " appearswhat is perhaps the first occurrenceof the term naive physics : compare Siegfried Jaeger (ed.), Briefe von Wolfgang Kohler and Hans Geitel 1907- 1920, Passavia Universititsverlag , 1988, p. 156). Some of the very first experimental

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work on naive physics- partly inspired by a study of the naive conceptionsof the ' Aristotelian spokesman Simplicio in Galileo s Dialogue- was perfonned by the ' Italian Gestaltist Paolo Bozzi in the late SOs (seeFisica Ingenua, Milan : Garzanti, " " 1991). More recently, the idea of providing an adequatetheory of this naive or " has been taken "" seriously in certain areasof Artificial Intelligence qualitative physics " "" and robotics as representingthe core knowledge that an intelligent agent must have to make its way in the real world . The seminal paper is Patrick J. Hayes, " ""The Naive Physics Manifesto, in Donald Michie (ed.), Expert Systemsin the Micro Electronic Age (Edinburgh University Press, 1979), pp. 242- 270; a second ""Manifesto" and other influential contributions may be found in Jerry R. Hobbs senseWorld( Norwood: and Robert C. Moore (eds.), Formal Theoriesof the Common "" Ablex, 1985). (Interestingly, in his pioneering treatment of liquids, Naive Physics " I: Ontology for Liquids , pp. 71 107of samevolume, Hayes pays no attention to liquid disturbancessuch as wavesand- more importantly for us whirlpools and bubbles, although they constitute important casesof naive physical phenomena.) Seealso the two special volumes of Artificial Intelligence 24 ( 1984) and 51 ( 1991), " both on " Qualitative Reasoningabout Physical Systems. John Locke, An EssayConcerningHuman Understanding, London , 1690(reprinted by Clarendon Press, 1985). " Holes and absencesposea problem to causal theories of knowledge: . . . one may truly be said to seedarkness. For supposinga hole perfectly dark , from whenceno light is reflected, it is certain one may seethe figure of it , or it may be painted. . . . The privative causesI have here assignedof positive ideas are according to the common opinion ; but in truth it will be hard to detennine whether there be really cause, till it be detennined whether rest be any more a any ideasfrom a privative " " privation than motion. (lI -viii -6) I will not here detennine but appeal to every ' of it consists man one s own experiencewhether the shadow of a , though nothing but the absenceof light (and the more the absenceof light is, the more discernible is the shadow) does not , when a man looks at it , causeas clear and positive an idea in his mind as a man himself, though covered over with clear sunshine? And the " thing . (lI -viii -5) Compare Jenny Teichmann, picture of a shadow is a positive " "" Perception and Causation, Proceedingsof the Aristotelian Society 71 ( 1971), pp. 29- 41: " Things seen include, colours, edges, distances, beams of light, shadows, reflections, glints, gloss, holes, movement, contrast, the Aurora Borealis, lightning, "" " differenceand similarity . (p. 38) (If one objects that differencesand similarities, holes, darknessand shadows, surfacesand colours, have each of them an entirely " different onto logical status from that of a material object, and that , accordingly, ""the senseor sensesin which we seethese things is quite different from the senseof " then the answer is that "' in see a material as it occurs see seeinga surface, object, each other that more like are all much a material and darkness object seeing seeing " anyone of them is' like seeingan unbrokendouble-yolked egg. (p. 38)) In the text we also cite Locke s argument on the inverted color spectrum (lI -xxxii - 15): we use it in chapter 11 to introduce the topic of hole inversion.

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Lucretius (T. Lucreti Carl ), De rerum natura, English translation by W. H. D. Rouse (Cambridge, Mass: Harvard University Press, and London: Heinemann, 1924; revisededition by Martin Ferguson Smith, 1975, 19822). The reader might find some interest in Lucretius' explanation of " the reason for " earthquakes (book VI , 535- 551), which moves from the intriguing remark that " the earth below as above is " [ ] everywherefull of windy caverns (p . 165, our italics). For a more qualified advice on what goes on under earth, see Haroun Tazieff, Les volcans et la derive des continents (Paris: PressesUniversitaires de France, 1972): the explanation of such big hollows as volcanosis traced back to the presenceof great fissures. The compositionality of holes is also taken for granted (look at the magnificent figure 24, representingthe Batur Caldera). Of course, the best way to discover what' s hidden in the depth of our planet is to drill a hole in its crust. Read Willard Bascom's A Hole in the Bottom of the Sea. The Story of the Mohole Project (Garden City , NY : Double day & Co., 1961). Nicolas de Malebranche, Traite de la nature et de la grace( 1680- 1712), in Oeuvres completesde Malebranche, Vol. V (Paris: Vrin , 1958). English translation by Patrick Riley: Treatiseon Nature and Grace(Oxford: Clarendon Press, 1992). The seminal text on momentary entities. We use this idea in chapter 8 in relation to what we call " Male branche-holes." The study of Malebranche-worlds has been undertaken by John Bigelow and Robert Pargetter, " Vectors and Change," British Journal for the Philosophyof Science40 ( 1989), pp. 289- 306. Benoit B. Mandelbrot , Les objectsfractals : forme. hasardet dimension(Paris and Montreal : Flammarion, 1975). English edition: Fractals: Form. Chanceand Dimension (San Francisco: Freeman and Co., 1977). Classic book on fractals, full of holed carpets, sieves, spongesand chunks of Emmenthal cheese . In later editions ( The Fractal Geometryof Nature, San Francisco: Freeman, 1982), the neologism " trema" (Greek Tpr;jJ(X= hole) is introduced to designateportions of spacemodelled after different geometricforms, cut away and removed from an object according to a deterministic or aleatory procedure. On such fractal constructions as the Sierpinski- Menger sponge (mentioned here in chapter 2), seeL. M . Blumenthal and K . Menger, Studiesin Geometry(San Francisco : Freeman, 1970). Alexius Meinong, " Viertes Kolleg fiber Erkenntnistheorie" ( 1917- 1918), in Kol leghefteund Fragmente. Schriften aus DernNachlass, hrsg. yon Reinhard Fabian und Rudolf Haller, Erganzungsband zur Gesamtausgabe (Oraz: AkademischeDruck u. Verlagsanstalt, 1978), pp. 337 401. Holes as a typical example of non-existents: " Ioss of reality : (A ) in the transition from what is simpler to what is more complex: incompatibility , subsumption under ideal Superlora, and of particular importance: lack, hole, boundary, represented, desired, becauseof the implied non-existence." (p. 366) But a hole is, according to Meinong, also an example of something real, becauseit has " an independent

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" " ] (p. 367). Seealso Zweites Kolleg fiber gegenbecoming and passing [ Vergehen " standstheoretischeLogik ( 1913), pp. 239 272, especiallypp. 252- 253. Alan Alexander Milne , Winnie-the- Pooh (New York : Dutton , 1926; reprinted by Puffin Books, 1982). ' ' We all know about Pooh s being a complete but very imperfect filler of Rabbit s ' of ' re stuck.' ' It all comes' said Pooh ' ' " ' hole: The fact is , said Rabbit , you , crossly, '" not having front doors big enough (chapter II , In Which Pooh Goes Visiting and Gets Into a Tight Place, p. 28). Christian Morgenstern, Galgenlieder(Berlin, 1905). English translation by Max ' Knight : Christian Morgensterns Galgenlieder ( Gallows Songs) (University of California Press, 1963). on the idea Witty poetry with severaldivertissementson emptiness for instance " that a certain amount of emptinessis essentialto a good fence: One time there was a picket fence/ with spaceto gaze from henceto thence. / An architect who saw this sight / approached it suddenly one night, / removedthe spacesfrom the fence/ and built of them a residence./ The picket fencestood there dumbfounded / with pickets wholly unsurrounded, / a view so loathsome and obscene, / the Senatehad to intervene. / The architect, however, flew / to Afri - or Americoo." " " ( The Picket Fence, p. 17) " " Charles Sanders Peirce, The Logic of Quantity ( 1893), in Collected Papers of CharlesSandersPeirce, Vol. IV , The SimplestMathematics, ed. Charles Hartshorne and Paul Welss(Cambridge, Mass.: Harvard University Press, 1933). ' We mention Peirces puzzleat the beginning of chapter 2: Which color is the line of " demarcation betweena black spot and a white background: black or white? It is certainly true, First, that every point of the area is either black or white, Second, that no point is both black and white, Third , that the points of the boundary are no more white than black, and no more black than white. .. . This leaves us to reflect that it is only as they are connectedtogether into a continuous surfacethat the points are colored; taken singly, they have no color , and are neither black not white, none of them." ( 7, no. 127, p. 98) Plato, Phaedo, English translation by R. S. Bluck (London: Routledge and Kegan Paul, 1955). " " ' Socrates description of the earth and of the cavities all over its surface does not " " " of the like this: one due to which is omit mentioning a pulsation something chasms of the earth is ... bored right through the earth- the one that Homer ' meant, when he said that it is very far ofT, where is the deepestabyss of all below ' " the earth ( 111c d; compare Illad , VIII , 14, 481). We are then informed that it is " possible to descendin either direction as far as the centre, but not beyond, for the either side begins then to slope upwards in the face of both sets of on ground ' ' streams." ( 112e) SeePlutarch s question and Alice s speculationsas she was falling " Down the Rabbit-Hole."

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Plutarch, De facie quaein orbe lunaeapparet , Englishtranslationby Harold Cherniss : " Concerningthe FaceWhich Appearsin the Orb of the Moon," in Plutarch's Moralia in SixteenVolumes(Cambridge , Mass: Harvard University Press , and London: Heinemann , 1957 ), Vol. XII , pp. 35- 223. A classicreferencefor a classichole-problem: " Not that incandescent massesof forty tonsfallingthroughthedepthof theearthstopwhentheyarriveat thecentre, thoughnothingencounteror supportthem; and, if in their downwardmotionthe impetusshould carry thempastthe centre, they swingbackagainand return of themselves ?" (924a- b) The questioncan be tracedbackto Plato (seeabove) and wasaddressed esde coelo, 11-7; Du ciel, 30a- b) , e.g., by Nicole Oresme(Question and RogerBacon(OpusMalus, part IV, chapterXV). It was finally answered by Galileo Galilei in the secondday of his Dialogosoprai duemassimisistemi del mondo(Firenze: Gio. BatistaLandini, 1632 ; Englishtranslationby Thomas : Dialogueon theGreatWorldSystems Salusburyrevisedby Giorgio de Santillana , " , 1953 Universityof ChicagoPress ): if the terrestrialglobewereboredthroughthe centre, a cannonball descending throughthat well would acquire, by the time it cameto the centre, suchan impulseof velocitythat, havingpassedbeyondthe centre, it would springit upwardsthe other way as great a distanceas it came down. That would beall the waybeyondthecentre,diminishingthe velocitywith decreasements like to the increasements . Thetimespentin acquiredin the descent this secondmotionof ascent ." , I believe , would beequalto the time of descent Willard van OrmanQuine, " Quantifiersand PropositionalAttitudes," Journalof 53(1956 Philosophy ), pp. 177- 187. The inscriptionalisttheoryof intentionalattitudes: sentences relatingspeakersto " " linguisticitemsor inscriptions suchas ink-marks or engravingsare taken to . SeealsoQuine's WordandObject performthe functionsof intentionalsentences ' s TheAnatomyof Inquiry( NewYork: Knopf, , 1960 (MIT Press ) andIsraelSchemer 1963 ). (Seeour chapter12for connectionswith holes.) HansReichenbach der Raum-Zeit-Lehre(Berlinand Leipzig: Walter , Philosophie de Gruyter, 1928 and John Freund: ). Englishtranslationby Maria Reichenbach ThePhilosophy of SpaceandTime(New York: Dover, 1958 ). Remindsusthat " In everydaylanguagewecall the torusa surfacewith a hole. But the holeis a matterof the third dimension ; the surfaceof the torus hasno hole. Whenwe walk on the surfacewe alwaysfind ourselvesin an uninterruptedenvironment . Nevertheless wecalledthe holein the torus manifests , the phenomenon itselfin experiences of the surface ; weformulatetheseexperiences by the existence of curveswhich cannot be contractedto a point and amongwhich obtainsan undeterminedbetweenness relation.... Such considerationsshow that, indeed , ' ' ' perceptswithout conceptsare blind . This striking remark of Kant s is better illustratedby mathematicalanalysisthan by the argumentationof his philosophical system ." (p. 62)

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einerTheoriedesMaschinen Kinematik:Grundzuge FranzReuleaux , Theoretische W. A. B. translation . 1875 : F. Kennedy: wesens by ) English , (Braunschweig Vieweg : London Machines a Macmillan , : Outline ( TheKinematics of Theory of of Machinery 1876 ; reprintedby Dover, 1963). of constrainedmotion, Pioneeringtreatiseon kinematicsunderstoodasthescience for recentwork in an force time and as ideas such from ( inspiration abstracting " " : Two Aspectsof Shape , : seee.g. Yoav Shoham naivephysics , NaiveKinematics : FinalReport,Rep. CSLI-85-35, SRI sense Summer in JerryR. Hobbs(ed.), Common " International, AI Center, 1985, pp. 4.1- 4.25). In chapterVII , KinematicNotation " Reuleaux a symboliclanguagebasedon twelveprimitive class, proposed ' ' ' and'tooth' ' which vessel ) alongwith four form-symbolsfor plane, (among symbols ' ' ' .curved' .full and , open. Thesesymbolscan be combinedwith one anotherin order to signifycomplexforms. The languageis completedby somerelational signswhich indicatethe variouspossiblecombinationsof objectsin kinematic , wherepredicates chainsor couplings(asa matterof fact, it is a pre-Fregeanlanguage 52ff : in nouns as common ). It is also are used subjectposition compare , which are worth pointing out the specialrole assignedto the form-symbols " to distinguishbetween... the portion of spaceenclosedby the figure, necessary ' ' " a prism, ' P+' denotesthe andthe portion enclosingit. (p. 253) Thus, if P denotes -' ' , while P will be'the geometriccomplement by the prismaticsurface bodyenveloped ' . of ' P+' (thesurfacebetweenthe two is denotedby po ). Think negative 's InternationalThesaurusIV Edition, Revised Robert , by PeterMark Roget,Roget , 1977,1991 ). L. Chapman(New York: Harper& Row, CollinsPublications Classifiesholesin class2 (Space ), III (Structure, Form), C (SuperficialForm), offersa or 5 257(Concavity), 3 (Hollow) (Hole). In class2.III .C the Thesaurus of with to in natural of superficialities language cope variety conceptsadopted varioussorts. (Wetalk aboutthis in chapter12.) Wefind, amongothers,geometri, , Bluntness , Sharpness cal propertiesor accidentslike Convexity, Protuberance more with Closure Fold Notch, Furrow, , along qualitativeproperties , Opening , . Unfortunately,no orientationin the field is and Roughness suchas Smoothness and idiomaticentries, the readermayconsult . For reader to the metaphorical given of , MetaphorsWe Live By (University GeorgeLakoff and Mark Johnson " in a as such container . 1980 clearing Press , metaphors , ) Interestingly Chicago " " wood" or "falling into a depressionare classifiedas speciesof the genus onto." logicalmetaphors Criterion of Substance and JoshuaHoffman, " The Independence Gary Rosenkrantz " 51(1991 Research andPhenomenological ), pp. 835- 848. , Philosophy as ordinarily understood An attempt" to analyzethe conceptof individualsubstance " absences " of them." (p. 835) but substances not are holes that , arguing , includingthe level C of Severallevelsof ontological generalityare introduced " " " a level C category concreta , and x is a substanceis definedas x instantiates " This . instance an of ( capabilityis givenan whichis capable having independent

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elaborate definition requiring inter alia " that it be impossiblefor an entity of a level C category to have as a part an item which instantiates a nonequivalent level C " " category. ) The point is then made that it is possible for there to be a privation which has as a part an item which instantiates a nonequivalent [ level C] category . .. . For example, consider a privation such as a hole in a bagel, or a silence betweentwo temporally separatednoises. It would seemthat if a hole exists, then it has as a part each one of the extendedplaces inside the hole. For instance, the hole has a certain volume of spaceas its right half, and another volume of spaceas its left half." The example is meant to show " that it is possiblefor some privations to have places. .. as parts. But the categoriesof being a place and being a time are at level C and neither of thesecategoriesis either equivalent to being a proper part or equivalent to being a privation . It follows that ... a privation could not be a substance." (p. 848) Edgar Rubin, Visuell wahrgenommeneFiguren (Copenhagen: Gyldendalske Boghandel, 1921). Keep an eye on the hole, and not on the doughnut: a difficult exercisewhen we come to Rubin' s famous face-globlet reversiblefigure. The pattern was devised in about 1915, but it is this 1921book that made Rubin' s work made-to-order Gestalt material. (We refer to it in our discussionof dominating gestaltsin chapter 11.) David Sanford, " Volume and Solidity ," Australasian Journal of Philosophy 45 ( 1967), pp. 328- 340. Two bodies can collide, penetrate and occupy the same spatio-temporal zone, which shows that solidity and impenetrability are conceptually independent " (compare Does Locke Think Hardnessis a Primary Quality ?," Locke Newsletter 1 ( 1970), pp. 17- 29; seealso Antony Quinton , " Matter and Space," Mind 73 ( 1964), ' pp. 332- 352). Note that Sanford s thesis is stronger than the one according to which two individuals can occupy the same spatio-temporal region: such two individuals could be categorially distinct (say the statue and the piece of marble), whereasin Sanford' s casethey are categorially homogeneous(they are both solid bodies). Sanford also defines the notion of a material plane: a sort of two -dimensional solid which has no volume and is neverthelesscausally interactive. (We mention it in chapter 9.) Jean- Paul Sart re, L '~tre et Ie neant. Essai d 'ontologie phenomenologique(Paris: Gallimard , 1943). English translation by Hazel E. Barnes: Being and Nothingness. An Essayon PhenomenologicalOntology (London: Methuen & Co, 1957). Includes a pictoresque analysis of the human tendency to fill holes (and suggests " the importance of the opposite tendency, to poke through holes, which in itself demandsan existential analysis," p. 613, note 17). Sartre thinks that every hole is an " appel d'etre," an " appeal to being." An example, if one, of picaresquephiloso. Critique de la raison dietetique, phy. Michel Onfray (Le Ventre des philosophes Paris: Grasset & Fasquelle, 1989, pp. 134- 135) also draws attention to a literary

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text by Sartre, Les Carnetsde la dr6le de guerre. Novembre1939- Mars 1940(Paris: Gallimard , 1983; English translation by Quintin Hoare, War Diaries: Notebooks from a Phoney War. November 1939- March 1940, London: Verso, 1984), where Sartre's ""metaphysique du trou " is much more extensively developed. Here are some illustrative excerpts: ""Theworld is a kingdom of holes . .. the hole is bound up with refusal, with negation and with Nothingness. The hole is first and foremost what is not. . . . The vertiginous thrill of a hole comesfrom the fact that it proposes annihilation , it rescuesfrom facticity .. . the hole is often resistance... . But at the same time, in the act of poking into a hole- which is rape, breaking-in , negation - we find the workman ' s act of plugging the hole... . To plug a hole is to transfonn the empty into the full , and thereby, magically, to create material possessingall the featuresof the holed substance.. .. At the root of thesesorceries I rediscover the craftman's idea of fitting -together. ... Fitting together magically entails fusion" (Camet 5, Thursday 21, December 1939, pp. 149- 153). " " Stephen H. Schaunel, What is the Length of a Potato?, in F. W. Lawvere and SH . Schaunel(eds.), Categoriesin Continuum Physics(Lecture Notes in Mathematics 1174, Berlin and Heidelberg: Springer-Verlag, 1986), pp. 118- 126. "" Argues that if we want counting to be finitely additive when an extendedbody (or " " ' pile) is written as a union of parts which are not clopen then we re forced to " " = count a doughnut as zero. In fact the fonnula number (A u B) number (A ) + number (B) - number (A ~ B)" is satisfied by ordinary piles of potatoes, as well as " " by piles in which even a piece of a potato counts as one, but can fail to be " "" satisfied if our pile includes a doughnut. (p. 120) Brian Skynns, ""Supervaluations: Identity , Existence, and Individual Concepts," Journal of Philosophy69 ( 1968), pp . 477- 482. The Fregeanaccount of atomic statementsinvolving non-denoting namesis taken to suggestthat such a statement " lacks a truth value by virtue of a " hole" in its structure of reference." (p. 478) Bas Van Fraassen's method of supervaluations " " . ( Singular Tenns, Truth -Value Gaps, and "Free Logic, Journal of Philosophy 63 1966 . is 481 495 described as an Aristotelian notion of Redemption ( ), pp ) adding to the Fregeannotion of Sin" in that " if the logical structure is suchthat every way " " of filling up the holes makes it true (false) , then the sentenceis true (false) " regardlessof the holes (p. 479, his italics) Likewise, in GOdel, Escher, Bach: An Eternal GoldenBraid (New York : Basic Books, 1979), Douglas Hofstadter otTersan ' " interpretation of Godel s IncompletenessTheorem as revealing that there were ""holes" in the axiomatic " irreparable systemsof Russelland Whitedead (p. 24; see " . 471: also pp 465 then why not simply plug up the hole? . .. after a while, the whole processbegins to seemutterly predictable and routine .. . any such system digs its own hole; . . . the systemhas a hole which is tailor -made for itself; the hole takes the featuresof the system into account and usesthem against the system" ). Moreover, " G Odel's Theorem has a counterpart in the theory of computation , discovered by Alan Turing, which reveals the existenceof ineluctable " holes" in even the most powerful computer imaginable." (p. 26)

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" " Barry Smith, Ontology and the Logistic Analysis of Reality, in Peter Simons and G. Haefliger (eds.), Analytic Phenomenology Dordrecht: Kluwer ,( , forthcoming). " An attempt to show how mereology, taken together with certain topological notions, can . .. serve as a basis for a complete formal ontology of the commonsenseworld ." Smith' s work can be seenas following in the footstepsof Whitehead' s Processand Reality (New York : Macmillan , 1929), although this latter was restricted to the ontology of events. On the interplay betweenontology, mereology, and topology , seealso JE . Tiles, Things That Happen(AberdeenUniversity Press, 1981), Section 6, and Bowman L. Clarke, " A Calculus of Individuals Based on 'Connection"' Notre DameJournal , of Formal Logic 22 ( 1981), pp. 204- 218 (further " " developments in Individuals and Points, Notre Dame Journal of Formal 26 . 1985 61 75 . ( Logic ), pp ) Interesting applications to Artificial Intelligence have beenproposed in a seriesof papersby David A. Randell and Antony G. Cohn (the " original theory is in Modelling Topological and Metrical Properties in Physical " Processes, in R. J. Brachman, H . J. Levesque, and R. Reiter (eds.), Principles of Knowledge Representationand Reasoning. Proceedingsof the First International , Los Altos: Morgan Kaufmann, 1989, pp. 357- 368). Our appendix can Conference be read against the background of thesemore generalframeworks. Avrum Stroll , Surfaces(University of Minnesota Press, 1988). A philosophical perspectiveon what has mostly been regarded as an exclusive property of geometryand materialsscience(with few exceptions: see,e.g., Thompson Clarke, " SeeingSurfacesand Physical Objects," in Max Black (ed.), Philosophyin America, London: Allen and Unwin , 1964, pp. 98- 114). The differencebetweenthe Leonardo and the physicalistic conception of surfaceswas first discussedby Stroll in " Two concepts of surfaces," in P. A. French et al. (eds.), Midwest Studies in Philosophy, Volume6, Studies in Metaphysics (University of Minnesota Press, 1979), pp. 277- 291; see also Ernest W. Adams, " Stroll on Surfaces," Inquiry 31 " " ( 1988), pp. 551- 552; Avrum Stroll , On Surfaces: A Rejoinder, Inquiry 32 ( 1989), " On the . W. 223 231 Ernest Adams of Surfaces, Solids, and ; , pp Dimensionality " " Spaces, Erkenntnis24 ( 1986), pp . 137- 201; Peter Simons, Faces, Boundaries, and Thin Layers," in A. P. Martinich and M . J. White (eds.), Certainty and Surfacein Epistemologyand PhilosophicalMethod. Essaysin Honor of Avrum Stroll (Lewistonf QueenstonfLampeter: Edwin Mellen Press, 1991), pp. 87- 99. Len Talmy , " How Language Structures Space," in Herbert Pick and Linda Acredolo (eds.), Spatial Orientation: Theory, Research,and Application (New York : Plenum Press, 1983), pp. 225- 282. It is argued that space-structuring grammatical features are encoded in natural languagemostly in the form of prepositions. In particular, the suggestionis made that abstractive processes, relative to spatial properties, underlie the mastering of ' ' ' ' prepositions such as through or in . Thus, metric and shape properties are usually registeredin the lexicon, whereastopological properties are of the outermost " importance and are registered in grammar: For example, the use of in requires that a ReferenceObject be idealizable as a surfaceso curved as to define a volume.

FurtherReading

245

But that surfacecan be squaredoff as in a box, spheroidal as in a bowl, or irregular as a piano-shapedswimming pool . . . . None of thesevariations of physical manifestation " affect the use of in. (p. 262) (We discuss this view in chapter 12.) Useful material can be found also in Ewald Lang, Kai -Uwe Carstensen, and Geoffrey Simmons, Modelling Spatial Knowledgeon a Linguistic Basis. Theory- Prototype - Integration (Lecture Notes in Artificial Intelligence 481, Berlin and Heidelberg: Springer-Verlag, 1991). " Kurt Tucholsky, " Zur soziologischen Psychologie der LOCher (signed Kaspar Hauser), Die Weltbuhne , March 17, 1931, p. 389 (now in GesammelteWerke, hrsg. von Mary Gerold-Tucholsky und Fritz J. Raddatz, Reinbekbei Hamburg: Rowohlt " . 152- 153). English translation by Harry Zohn: The Social Verlag, 1960, Vol. 9, pp " Psychology of Holes, in Germany? Germany! The Kurt Tucholsky Reader(Manchester : Carcanet Press, 1990), pp. 100- 101. " " A very first " treatise on holes, written supposedly to fill the hole between " " the previous selection and the next one. Starting from the remark that a hole " ' " is where something isn t and is permanently accompanied by a non hole ( I' m to assert moveson sorry, but there is no such thing as a hole by itself" ), Tucholsky " some basic hole-theretical principles: holes are static ( there are no holes in transit " " and cannot meet without fusing. Some questions are then raised: If you ) knock down the dividing wall betweentwo holes, does the right edgebelong to the left hole? Or the left edgeto the right hole? Or each to each? Or both to both? . . . If a hole is plugged up, where does it go? Does it squeezeto the side and merge with matter? .. . What happens to the plugged-up hole? .. . Where one thing is, there cannot be another. Where there already is one hole, can there be another? " And why are there no semi-holes?" Also by Kurt Tucholsky is the sketch W 0 Kommen die LOCher im Kaese her?" (signed Peter Panter), VossischeZeitung, August 29, 1927 (now in Vol . 6 of the GesammelteWerke, pp. 210 213) a tragically hilarious family discussionon the apparently simple question asked by a child: " Where do holes in cheesecome from?" (We cowardly ignored the issuein our chapter 10.) " " Mark Twain (Samuel Langhorne Clemens), What Stumped the Bluejay, in A Tramp Abroad (London: Chatto & Windus 1880; reprinted in The CompleteShort Stories of Mark Twain, ed. Charles Neider, New York : Bantam Books, 1984, pp. 159- 163). ' " . .. a bluejay lit on that house, with an acorn in his mouth , and says, Hello , I reckon I' ve struck something.' When he spoke, the acorn dropped out of his mouth ' and rolled down the roof , of course, but he didn t care; his mind was all on the thing he had struck. It was a knot hole in the roof. He' cocked his head to one side, shut one eyeand put the other one to the hole, like a possumlooking down a jug; then he glancedup with his bright eyes, gavea wink or two with his wings- which - and says, ' It looks like a hole, it ' s located signifies gratification , you understand '" ' like a hole, blamed if I don t believe it is a hole! A most entertaining short story about a bird trying to fill up a housewith acorns.

246

Further Reading

Michael Tye, The Metaphysicsof Mind (Cambridge University Press, 1989). " " Chapter 4 ( Bodily Sensations; seeespecially pp. 79- 91) suggestsan interesting way of treating spatial relations within an adverbialist theory, hinting at the possibility that such a treatment could be useful for the purpose of paraphrasing hole-committing sentencesby meansof hole-free ones. ( Weconsider it in chapter 12.) Seealso "The Adverbial Approach to Visual Experience," Philosophical Review 93 ( 1984), pp. 195- 225, where solutions to Jackson's Many -Property Problem are examined.

Index

Note: Decimal numbers (e.g., 5.6) representfigures.

Abbott, EdwinA., 72 Absences , I , 186 Abstractobjects , perceptionof, 156 Accidents , 132 Aesop, 172 , 124- 128 Aggregates , 125- 128 causallyconnected Amoeba, 153 Argie(character ), 2 Arguments from causalrelevance , 119- 120 non/ Dc;' sa/tus, 122- 123 from relativityof movement , 121- 122 Armstrong, David M., 118, 124, 131 Artifacts, 148- 149 andfunction, 148 andorigin, 148 Artificial intelligence , 3- 4 Ascription , dispositional . 111 'At ' 182 , Atomicholes, 108 Atoms, 33 , 107 mereological BatMan holes, 61- 62, 5.6 Bennett , Jonathan , 33, 171 , George , 109- 110 Berkeley Blanks, 114 Bolzano , Bernard, 10 Bodies , material of, 36 complements Borden, 13, 160- 163 oriented, I60- 163 Boundaries , 10, II , 75 topical. SeeFigures Bozzi, Paolo, 160 Breakinglines, 79ft". Breakingpoints, 79ft"., 6.1 Brentano , Franz, 79 Bubbles , 109, 114- 115, 146, 173 dynamic, 115 By (relation). 17. 125- 126. 156- 157

Cartoonboles , 35, 139 - 115 - 158 , 5, 7, 110 , 114 , 130 Causality , 157 - 158 andcausal , 157 theoryof perception - 119 - 125 immanent , 118 ~124 , 132 andshapes , II 5ff. Cavities , 6, 36, 39ff., 53ff., 64, 66, 70, 84- 145 86, 103 , 142 , 4.1, 5.10, 5.16 , 52- 53 counting donut-shaped . SeeTunnels dumbbell , 103 , 143 overlapping , 146 Cbamisso von, 140 , Adelbert

, 117. SeealsoShipof Theseus Change of matter. 16. 32 of shape , 123 of size, 16, 144- 145, 152 CheshireCat, 19 Chisholm, RoderickM., 136 Circularholes. SeeRotatingholes Classification of holes. SeeTaxonomy Closure,causal, 116, 125- 128, 149- 150 Coincidence spatio-temporal,90, 96 of two distinctholes, 91 Color, III andextension . SeeBerkeley patchesof, 140 Commonsense , 3, 49, 70, 110, 187 Common-sensetheoryof holes, 3 Concavity, 19- 23, 74, 173 requiredby holes,21 inexact. SeeHusserl , essentially Concepts Constitution, 32, 114, 153. SeealsoMatter; Space andidentity, 136 Contact. 10

248

Countingholes, 26, 30, 87 Cracks, I , 13, 79- 86 closed , 82, 122 ". contactfacesof, 798 "., 86, 114, 146, 6.1 filamentous , 798 asflat holes, 79 "., 84ft'. internal, 798 intersectionof, 80 Klein, 86 Moblus, 81- 82, 85- 86 Creatingholes, 5 , 131- 133, 136, 138 Creatingmachine

Curvature , 21- 23, 76 Cutting , 10, 101

holes, 84 Degenerate Degreeof a hole, 178- 184 , 6, 9, 15- 19, 33, 35, 948, 105Dependence 106, III , 136- 137 dedicto(notional), 19 dere, 19, 116, 136 fillable, 101- 102 , 136 generic rigid, 19, 136 , 19, 35 superficial Depiction, 165- 168 mediated , 166 , 1, 39, 70, 4.1 Depressions , 131- 133, 136, 138 Destroyingmachine Destructionof boles, 5, 151 Detectionof boles, II . SeealsoPerception , 62- 63, 147- 149 Diaphragms on holes, 148 dependent conceptually Discontinuities , 9, II , 13, 15- 16, 19- 20, 39, 60, 101, 104, 165 concave , 163 naturalhistoryof, 132- 133, 152, 178 , 13, 60 superficial , 5- 6, 110- 112 Dispositions groundof, 110- 112 , 110 ungrounded Di~ tivity, 103- 105, 145 Disturbances , 14- 16 Earth cut, 20- 21 hollow, 173 , , andtangency Edgeinclusion,intersection 102, 7.11 , 13, 70, 72, 74- 76, 101- 102 Edges cavitiesand, 145 Ys. boles, 75, 172 internal, 172 surface , 82, 6.5

Index

Eift'el Tower, 128 Eliminationof holes. SeeParaphrases ; Realism Emmenthalch~ , 1, 2, 13, 52, 95 96, 106, 129 , 144- 145 expanding immutable , 153 , 185- 186 Emptiness andsemanticfields, 187 '. andsyntax, 187ft 142 Channel , English ' ' Enlarge, 28- 29 Entities , 120- 121. Seealso momentary Malebranche-holes potential, 14- 15 circular, 118- 119, rotatinghomogeneous 124- 125

, 6, 9ft'., 13, IS, 19, 87- 88, 104 superficial Essential holes . I. 5 Eventcausation , 157 Events , 157 , 225 , 5, 109 , causal Explanation roleof holes , I , 130 Explanatory , 115 , underwater Explosions

Fa~ , 53, 84 counting , 86 of, 53 insulation of, 53 isolation non-f~ , 68, 71 :es,53 vs. surfac ". Fieandt , KaiYon, 163ft - 163 , 158 Figureva. ground - 163 , 160 , andborders Figures , 34, 56, 111 Fillabilityandfillablethings - 133 . SeealsoDispositions 113 , 132 - 147 Filledholes , 151 , 56, 63, 145 Fillen, 5, 7, 25, 35, 57, 76- 77, 85, 110 , 114 116 , 137 actual , 115 canonical , 58 perfect of holes andc1assification , 63ft". , 57ff., 94, 5.1, 5.3, 5.5 complete onholes , 60 dependent conceptually fitting, 57 f~ f~ of, 64 in Flatland , 73 - 146 andhole-making , 145 - 140 andidentityof holes , 129 , 138 andindirectpe~ ptionof holes , 157 partial,49, 57ff., 87, 94- 95, 97, 5.1, , 5, 54, 57ff., 68, 84, 93, 121 122 perfect 175

249

Index

, 56- 57, 59, 121 , 137 , 146 potential , 61- 62, 5.8 reciprocal scattered , 58, 95, 97 Filling,7, 33, 56, 58- 59, 71 , vs. fillen, 57 Fillings Fine,Kit, 91 Fishyhole,27- 28, 3.3 - 128 Fissures , I , 122 , 126 , 145 , 151 in liquids . 146 Flatland . 72- 77. 172 infinite, 74 , 75 pulsating , 32- 33 Floatingexpanses Floatingholes, 135 Fox andgrapes(example ), 171- 172 Fractures , 85 Function, 56, 100, ISO.SeealsoArtifacts andidentity, 132- 133 Fusion, 7, 63, 143, 145- 147, 149, 151 of hosts,96- 97 , 58 mereological Gale, RichardM., 33 Genus , 48 Geometriccomplement , 12 Gestalts ~ concave , 39 cut in, 39- 40, 79, 104, 141, 145 , I68- 170 dominating , 21- 23, 94, 101 superficial Goodman , Nelson, 89, 158 Gradientinfonnation, 163- 165 Grooves , I , 13- 14, 105 vs. holes, 105 and ridges, 13- 14 ), 95 Gruyere(example Guests , of holes, 2, 5, 34- 35, 110. Seeal.fo Fillers Half boles, 105 asimpossibleto dig, 87 . SeeTori Handlebodies Handles , 147 Heraclitus , 90- 91 Histories,spatio-temporal, 15, 18 Hole-adverbialism , 178- 184 ". many-boleobjectionto, 179ft Holeinversion , 170- 173 Hole- linings(surrounds ), 18, 25- 31, 88, 90, 123, 157, 3.1. SeealsoLudovician Theory maximal, 31 , 31 superficial Hole-monism,97- 98

Hole- part relations, 95ft'., 143 Hole- parts, 57, 87- 89, 90- 105, 143, 7.1. Cf. Parts, holed and disstivity , 104 ys. hole- shapedparts, 101 Hole- perception, ys. perception of holes, 158ft'. Hole- pictures, ys. pictures of holes, 165168 Holes. Also seespecificentries beginning with adjectives- e.g ., Atomic holes in aggregates, 125- 128 and dispositions, 110- 113 and edges. SeeEdges in holes, 99. Seealso Hole- parts in liquids . SeeSoft holes Hole-surrounds. SeeHole- linings Hollows , 6, 39ft'., 53, 64, 70, liS , 142, 4.1, 5.10, 5.16

. 42- 43 Homeomorphism of holes . 60

Homotopy, 42 Host, ora hole, 2, 5- 6, 16- 18, 34- 35, 5157, 84, 110, 114, 117, 133- 137 Hosts actual, 130- 137 andmovingholes, 117 , 63 Hourglass Hurricane,eyeof, 113- 114 Husserl , Edmund, 168

Identification . 1, 2, S. See

; Perception Identity Identity, 1- 2, 6, 26, 30- 31, 35, 90, 129137. SeealsoContinuity andcounterfactual C%.See dependen Causality,immanent and function, 132- 133 Immaterialbodies , 6, 33- 37, 79, 87- 88, 91- 92, 123, 155- 158, 184- 185, 3.8- 3.10 andidentityacrosspossibleworlds, 136 minimaltheoryof, 36- 37 ImmaterialTheory, 34, 37, 92- 95, 3.8 . SeePenetration Impenetrability , 56 Imperfections ' In' 188 , transitivityof. SeeLocation In (relation), 94- 99 va. part of, 95 In a hole(relation), 58- 59, 95, 99- 100, 106, 7.9 In a host(relation), 106 In virtueof (relation), 17. Seeabo By Individuals,calculusof, 89

250

Individuation, 33 , 184- 185 Inscriptionalism , hole-lining theoryof, 184- 185 ' 'Inscriptions Inside, 27 Inside/outside, 1, 11, 20, 27, 171- 173 Insulation, causal,53 Interaction causal,35, 117 patternsof, 116, 125- 126, 143, 174- 175 Invisibleman(example ), 109- 111, 117118, 158 , of holes, 60 Isomorphism Jackendoft ', Ray, 160- 161 Jackson , Frank, 30- 31, 178, 181

Joins , 142 off~ , 79 solid, 127 holes , 138 Jumping

Kant, Immanuel , 130 Kant' s cushion, 130 Kaplan, David, 184- 185 Kanno, Toomas, 14 Knots, 14 Kripke, Saul, 136 Lack, 186 Lake, empty, 170- 171 Lava(example ), 114 Leibniz, GottfriedWilhelm, 90 Leonard, HenryS., 89 Leonardo - - - --~- ~ da Vinci. 10- 11 Leskiewski , 89 , Stanislaw -less(suffix), 189 R., 2,. 25Lewis, David K. and Stephanie 31, 88, 123- 124, 178. SeealsoHolelinings; LudovicianTheory Lexicalsystem , 187- 189 Lexicon. 187 Line generation , 82ft". Linguistic system closed, 187- 189 open, 187- 189 Localization , 129 spatio-temporal, 1 Location, 98- 100 practical, 100 Locativeprinciples , 98- 100 Locativestructure,98- 100 Locke, John, 155 , 44- 45 Loop equivalence ~ -- . . 42ft'., 47, 103, 126- 127 for ~._..-.:- c: objects,44- 45

. Loops hanrllina

Index

infinite, 67 va. rings, 4S Lot's wife (example ), 132, 136 Ludovicianholes, 2S LudovicianTheory, 31, 27, 88, 136, 3.7 Madeof (relation). SeeConstitution; Matter Madeof thesamestuffas(relation), 58 ) in, 113, 8.1 Magneticfields, holes(pockets Magritte-holes, 167, 184 Malebranche-holes, 134- 135, 139 Manifolds , 55- 56 morphological . SeeFlatland two- dimensional Matter, 9, 16, 111, 113, 115. Seea/so Constitution annihilationof, 118- 119, 152- 153 anddispositions , III ftux of. SeeShipof Theseus of holes, 25 holein, VI. holein object, 16- 17 andidentityof hole, 129 splittingof, 153 Maximumholes, 18 Media, 109 Memory traces (example), III Mereological difference, 101 Mereological sum, 91, 101 Mereology, 1, 81ff., 92, 106- 18, 124, 183. Seealso Parts and ontology, 105- 108 standard, 89 Metaphorical holes, 186 Migration of ~ dents, 132 of disturbances, 14 of holes, 18, 131- 139, 141- 148. Seealso Jumping holes Minimal holes, 31 ' Miss Anscombe s weddin ring (example), 121- 122 Moblus strip, 81 Mock holes, 149- 151 Monotonicity , 101 Morphology , 6- 1, 55ff., 10- 11, 15- 16 insufficiency of , 105 and skin, 10 va. topology, 6, 55, 15, 163. Seealso Object topology Moving holes, 7, 91- 92, 117, 121, 134, 135, 7.2, 7.3. Seealso Jumping holes; Migration ; Rotating holes and hosts, 117

2S1

Index

Muranoglass(example ), 173 Naive physics, 3- 4 , 47. Seealso Common

sense Naturalholes, 17 Nestedholes, 94- 95. SeealsoHolesin holes spinning,25, 123- 124 Nets, 126 Notches , 105

Objects completelyholed, 17- 18 . SeeShipof continuouslychanging Theseus cut in, 39- 40, 79, 104, 141, 145 holed, I , 5 essentially infinite, 66- 68, 74, 169 material, 9 scattered , 96, 106, 153, 184- 185 , 133 temporallyscattered topologyof, 6- 7, 41, 49- 51, 54- 55, 59, 68, 70- 71, lOS, 181 , 109 transparent 1- 3 7 Ontology ' Outside',27 , ,

, 106 , 124 Overlap , 96 mereological -tempOral spatio . 95- 96

holes, 39ft'., 4.1 Paradigmatic , 2, 3, 6, 25- 31, 177- 184 Paraphrases andcanonicalexpressions '. , 179ft geometrical183- 184 . SeeDependen Parasites ~ Particulars , 1, 14- 15, 33, 126. Seealso Substances , 6, 9ft'., 13, 15, 19, 87- 88, 104 superficial Partlyin (relation), 59 Parts, 6, 11, 14- 15, 59, 87, 105- 106, 116, 136. SeealsoMereology ; Surfaces actual, 30, 125- 126 fillable, 101- 102, 7.12 freesuperficial , 63 holed, 185 of holes, 25- 26, 39, 49- 52, 57. 88- 89. See alsoHole- parts internal. 14- 15 , 12 maximallyconnected moving, 95- 96, 7.7 non-f~ superficial , 93 andpen: eption, 164 potential, 14- 15, 30, 34- 35, 51- 52, 88, 93- 94, 96, 103, 136, 143

proper , 14, 89 superficial , 11- 14, 19, 21 - 13, 52 , 57, 63 , 88 , 93 , 115- 116, 122, 129- 130 of tunnels , 143 Paths , 41 closed . See Rings Pauses, 186 Peirce , Charles Sanders , 10 ' Peirce s puzzle , 10- 11, 110, 2. 1 Penetration , 33 - 36 , 92 , 95 - 96 Perception adverbialist theory of , 181- 182 auditory , 176 cavity , 173 of color , 110 of extension , 110 hole- , I 58ff ., of holes , 158ff . hollow , 163- 165 kinesthetic , 173- 175 of lines , 110 mediated , 156- 158 shadows in , 165 tactual , 173- 175 tunnel , 158- 163 ' ' Perforated , 2 Perzanowski , Jerzy , 26 Peter Schlemihl (example ), 140 Pictures of holes . See Hole - pictures Planes , material , 129- 130 Plasticine brick (example ), 16

Plasticinestatue(example ), 16 Plasticityof holes, 123- 124 Plerosis , 79 Potentialboles , 94, 137, 148- 149 Predicateoperators . SeeHole-adverbialism Prefabricated boles, 146- 150 . Seespecificprepositions Prepositions spatial, 188- 189 , bolesas. SeeRealism Projections Properties holes are not , 14 holes have, 56 Pseudo- cube, 130 Pseudo-solid, 130

Qualities,primary, 116, 174- 175, 186. See alsoShapes inversionof, 171- 173 Quantifying,overholes, 2S Quine, Willard vanOnnan, 184 Rank, ora hole, 178- 184, 12.1 Ready-madeholes, 139, 146- 148

Index

. Seealso Realism , 188 , 2- 3, 10, 118 Interaction ; ; Ontology patterns Paraphrases andperception , I SS , 3- 14, 34, 31, , complementary Reasoning - 113 11, 16- 11, 110

, 33 Receptacle RedSea(example ), 146 Reference Object, 188- 189 Reification , 140 Reism, 31 Relationaltheoryof holes, 183 holes. SeeCartoonholes; Removable holes; Migrationof holes, Prefabricated Ready-madeholes Replacingholes, 152 . SeeDepiction Representation ' Reversible figure, Rubins, 169- 170, 11.11 , 13 14 Ridges 13- 14, 21 andgrooves ' s, Starr ), 139 (example pocket Ringo Rings, 41- 42, 64, 4.4 vs. loops, 45 non-homotopic,45ft'. Robot(example ), 3- 4 Roget, PeterMark, 187 Rotatingholes, 161- 163 Rubin, Edgar, 169

Sanford, David, 129 holes, 185 Scattered Schemer , Israel, 184 Seaof Holes(example ), 169 Self . 12 - ----connectedness

Shadows, 7, 9, 33, 35, 134- 135, 155, 175176 independent, 140 and perception, 165, 175 Shapes, 5, 9, 111- 112, 116 Shiman, L ., 160 Ship of Theseus(example), 90, 130 131, 9.1 Shoemaker, Sidney, 131 Sierpinski-Menger sponge, 17 18 Simons, Peter, 19 Singularities, 22 23 as organizen , 174 punctual, 22 Skinning function , 68ff., 71, 177, 5.16 Skins, 7, SO- 51, 68ff., 75- 77, 101, 104, 177 Soft holes, 113- 115, 146 principle for , 115

Sp8 (:e, 112 absolute , 121 anddispositions , 112- 113 '., 175 empty, 33, 124- 125, 156, 168ft , 39 encapsulated holeso<x: Upy,9 andmatter, 32 asmatterof holes, 6, 32- 33, 95, 112, 123 of, 37 o<x: Upancy qualified, 32- 33, 146 regionsof, 2, 32, 34, 89, 108, 129 130, 137, 183 , 118- 119 Spatio-temporalstages 171 of invenion , , Spectrum Splittingof holes, 7, 92, 145, 4.3 Stroll, Avrum, II . SeeHole-parts Subholes SUMta .n~ . SeealsoParticulars

- 150 mock,149 splittingof, 153 -part-of(relation ), 88 Superficial , 31, 33, 37, 88, 92- 93, Theory Superficial 95, 3.7 Surf~ , 6, 9- 16, 18- 19, 34, 53, 71- 72, 79, 84- 85, 87- 88, 98, 101

asabstractions , 11 asactual , 136 cosine , 165 defined , 12 of, 53ff., 64- 66, 142 , 5.10 disjointness vs. f~ , 53 notionof, 11 ' ordinary sstuff,11 asoutermost layerofobject asoutermost , 12 partsofobjects , 115 permeable vI. actual , 14- 15 potential to, 70 holes reducing ~ ttered , 72 of, 11 sides information Gradient . See ~ texture Surf

',25 'Surrounds -27 also .See 16 -17 Survival Ship ,153 ,91 ,139 , Theseus of -132 ofholes ,130 Talmy, Len, 187- 189 Taxonomyof holes, 5- 6, 39, 63ff., 71 187 Thesaurus '. 188 ' Through , Time, testof, 112 Topology, 41, 48, 68 andholemaking , 141- 145 intrinsicVI. extrinsic,44 object. SeeObjects

Index

Tori, 41 double, 49 , 127, 165 exploded knotted. 65

, 132 Tropes - 143 Tunnels , 6, 39ff., 64, 70, 127 , 141 , 4.1, 5.10, 5.16 accordion , 103

cavity-, 39, 41, 53ff., 63- 64, 85, 182,4.2, 5.16, 12.2 counting , 12, 44- 53, 73 dumbbell, 103 freefacesascriterionfor, 64- 66, 5.10 in Flatland. SeeTunnel-cavities instantaneous , 143 knotted, 43- 44, SO , 181 Mont Blanc(example ), 92, 135 mouthsof, 104- 105 overlapping , 49, 87 partsof, 143 perceptionof, 158- 163 relativeto partsofbost, 51ff. scattered , 47, 151 Simplon(example ), 135 Tunnel-cavities , 72- 73, 5.19 VI. cavity-tunnels, 73 Tunneledboles, 47- 48 Tye, Michael, 182 Undetectable holes, 155 Unfilledholes, 34- 35 Universe , expanding , 144- 145 , 39, 41, 100, 155, 159 Vagueness Valley-hollows, 74 Vorti~ , 14 Waves , 15- 16, 114- 115, 174 asholes, 114- 115 Wholes.SeeParts

253