The Structure of Shadows

0.Abstract

This paper deals with two main problems. First, it discusses an application of the descriptive tools of the theory of space representation developed together with Achille Varzi (1994, 1996) to shadows, here conceived of as holes in light. Some adjustments of the theory seem necessary because shadows are dependent not only upon material objects (as holes are) but also upon processes. Second, it discusses the viability of the shadow metaphor as a means for modelling the dynamics of socio-economic units.

1.Models of common sense

Modelling common sense, or finding good theories about it, is a piecemeal project, and it has long been recognised as such (Hayes, in Boden 1990:187). When a conceptual cluster is rather circumscribed and shows clear-cut, interesting and robust interconnections with neighbouring clusters we should be able to represent its structure by a small theory - a theorita (borrowing here a term from the philosopher H.N. Castañeda). Theoritas maximise focus (modulo their presuppositions, they tend to stay within the limits of their subject-matter) and are thus open to massive revision once they get integrated into theories proper. They constitute mid-term examinations of an object matter; however they do claim to be as exhaustive as possible. In the conceptual cluster of intuitive space, whose overall theory is far from being completed, analysis of various sub-clusters and their associated theoritas abound and cover core concepts, such as those of regions of space, parts of objects, objects in space, space-time objects, and more peripheral concepts, such as those of events and of holes. An interesting though little investigated notion is that of shadows; its interest comes from the fact that shadows are eminently spatial but are also linked to some generating process, thereby bordering on the causality cluster (though in an odd way, for they seem to be acausal). In what follows I shall discuss some of the issues raised by shadows, and show their place in an integrated theory of naive space and causality.

2.A theorita of shadows, and its problems

Constructing structured databases is our target, and an unstructured database such as a current dictionary of English might very well be our starting point, incorporating as it does a fragment of lexical (encyclopaedic) competence. The OED considers a shadow as a 'patch of shade', a 'dark shape projected by a body intercepting rays of light'. The patchy nature of shadows is evoked, and so is their relation to light. There is obviously more structure around. Todes and Daniels (1975) and Bas Van Fraassen, in his Laws and Symmetry (p. 217), discussed a very simple theorita of shadows (henceforth the TDV theorita) that summarises in a more articulate way our common sense ontological intuitions with regard to these objects. Let X be any physical (opaque) object. Then

I. If X casts any shadow, then some light is falling directly on X

(This is pretty intuitive: light is necessary for casting shadows).

II. X cannot cast a shadow through an opaque object

(To realise this, consider the following case:

 

receiving light from a source a, b casts a shadow on a body c, and c casts a shadow on d; but b does not seem to cast a shadow through c onto d).

Finally,

III. All shadow is shadow of something.

These three principles are not logically incompatible, but the theorita they jointly constitute runs against some empirical facts and is thus inadequate. For take the following case:

 

 

Object b has now been moved in the shade projected by c onto d. Consider the imaginary lines linking the profile of b to the light source, and project them onto d, so as to delineate shadow s’, which is a portion of the shadow s (of c onto d). What is s’ a shadow of? Not of b (by principle I). Not of c (by II). But then, as b and c and the only candidates around, III is false.

The case is less academic that it might appear at a first sight. There are many examples of causal pre-emption of this sort that are a source of conflict in the legal arena. Suppose that your land parcel a is in the shade of a skyscraper built on parcel b. Suppose now that on parcel c, situated between a and b, another house is being built, which would cast an illegal shadow on a were the skyscraper not there. On which ground would you stop the building? For the house on c does not cast a shadow on your parcel.

Should we give up any of I-III? We’ll see the matter more closely in a moment. For the time being, the puzzle shows that we ought to handle shadows with care. As with many other common sense concepts, if we push analysis too far we discover hidden inadequacies in them. Common sense principles can be very local, for they are developed for the handling of specific cases, and resist integration in grander conceptual schemes. Should we then get rid of the notion of a shadow altogether?

3.An eliminative strategy for shadows, and its inadequacy

One should first show that "eliminating" shadows is not an easy business - for they are not idly inherent in our common-sense ontology. The basic idea behind elimination is economy: if we can describe a certain portion of the world that prima facie is inhabited by entities of a certain kind without even mentioning such entities and without losing descriptive finesse, we make some savings (that can be useful if our theory is to be implemented in some energy-consuming system). The eliminative strategy is thus an interesting move to perform if one wants, say, to reduce the number of primitives in one's database. Eliminativist strategies abound in the social sciences; one example is methodological individualism, the doctrine that social facts are to be explained not by reference to abstract entities such as the interest of nations or class conflicts, but by reference to the (idealised) psychology of the individual person, usually modelled as a system of preferences. On the other hand, the basic objection to elimination is that, after all, descriptive fine grainedness might get lost in the process (cf. Casati and Varzi 1994 for a similar point about holes). The trade-off between economy and descriptive adequacy governs the choice of the types of entities a theory allows for.

How does the eliminative strategy work in the case of shadows? One can imagine paraphrasing sentences (purportedly) about shadows by means of sentences about shaded objects. For instance,

(1) There is a shadow on the wall

will be replaced by

(2) The wall is shaded.

How the whole thing develops should be clear: in order preserve descriptive adequacy paraphrases are bound to become increasingly clumsy (thus the economy principle is no longer satisfied). This is apparent from some geometrically sophisticated descriptions that are quite straightforward in sentences allowed to refer to shadows and are pretty difficult to translate into sentences that are not:

(3) The shadow cast by the table on the wall overlaps the shadow cast by the chair in three different zones.

This suggest that we set aside the eliminative strategy, at least until we obtain a better understanding of the things we want to eliminate.

 

4.Improving the theorita: self-shadows and cast shadows

The TDV theorita could be considered inadequate on another account: it leaves out some interesting facts about shadows that will now be our concern (this is not meant to be a strong criticism, for that very theorita was not produced with the claim to cover completely the phenomenon of shadows in general).

Consider first a neglected side of shadows. It is not only objects that cast shadows: objects also have shadowy parts. In some cases one is unable to make perceptually a clear distinction; still, in most cases the distinction is clear. Consider for instance the north face of a building and the shadow it casts on the street and onto the south face of another building. The question now is: Is the first building shading itself?, i.e., Is a self-shadow after all a case of a cast shadow? Common sense seems to answer this question in the negative. Though proper and cast shadows may be phenomenologically indistinguishable, they are experienced as two types of phenomena, asymmetrically linked to the processual structure of shadows. Thus let us assume this distinction for the time being, and discuss the role of processual asymmetries later.

5.Improving the theorita: Mereotopological facts and problems

Where are shadows after all? Let us distinguish three senses of location.

(1) Some things are in other things because they share parts (thus my arm is in me because all of its parts are parts of myself).

(2) Some objects are located relatively to other objects because they are topologically connected to these (without having any part/whole relation to these; thus my arm is on the table because - among other things - it is in contact with it, even though it does not share any of its parts with the table).

(3) Finally, some objects are located 'at' other objects without having any clear part/whole or topological relation to the latter (this is the case of spoons in cups, of fillers in holes, of objects at their places; see Casati and Varzi 1996; Vandeloise, 1994).

To which locative category do shadows belong?

If shadows are just darkened parts of bodies, then they are parts of bodies, and are located in the first sense. What if shadows are outside bodies? There are cases in which shadows are clearly located in the third sense (see below, the shadow that fills a hole: they do not share parts, and are not connected to one another, although their regions coincide - they are simply co-localised). But can we say that shadows are in contact with the bodies they are cast over? It does not seem that common sense has a definite answer to this questions. But I assume that we can conveniently represent common sense as putting shadows in contact with the bodies they are cast over.

6.A morphology of shadows:

We shall now abstract from the physical, and even from some of the metaphysical, facts about shadows, and concentrate on their geometry, but in a very peculiar sense. We grant that shadows are produced by some projection method, and we study their morphology. A morphology is here a sort of very qualitative, very abstract characterisation of some facts concerning shadows - insofar as these facts have a spatial component. Now, geometrical theories of shadow abound, and it is not our concern here to rehearse them - I am not going to dwell on how to draw a shadow, given some facts about a source of light, a surface over which the shadow is cast, and an object situated somewhat between the two. What interests us is a close study of the main characters here.

The basic idea is that the visible shadow is not the whole of the shadow, indeed it is just a boundary of it, and that we have to take into account a shadow-body (the technical term 'shadow-body' translates here approximately the English term 'shade', retaining of the latter only the geometrical and morphological connotations). A shadow-body has at least two bound, visible faces (the casting object face and the object face), and at least one unbound, invisible face. In a first approximation - but we shall see some important exceptions - the free, invisible face of a shadow-body is defined by the last array of the photon shower from the light source.

 

 

7.The true metaphysics of shadows

Up to now we dealt with common sense - albeit in a refined, mildly regimented fashion. The time has come to present the true metaphysics of shadows, their real nature, and to assess the adequacy of the common sense picture.

Common sense represents shadows as quasi-objectual entities. If one were to assign types of objects a position on a continuum from the thingy (chairs and tables) to the unthingy (wind, sounds), shadows would probably appear relatively close to thingy entities. One very thingy feature is that shadows seem to exist continuously over time, in such a way that one can at times judge that the shadow that is now over there is the same that was before in another place. Judgements of identity over time are usually grounded in some conception of an underlying metaphysical structure of the objects at stake. Consider the case of standard spatio-temporal individuals (tables and chairs). Here it seems that three basic metaphysical factors co-operate in ensuring the identity of the object that is now over there with the object that was before in another place. The first two factors are spatial and temporal contiguity. The object a that is now over there may not be spatiotemporally contiguous with the object b that was before in another place, but if one is able to track the history of these objects one will see that a=b only if the spatiotemporal histories of a and b are just the same. The third factor is causal. In the literature one finds a distinction between internal causality and external causality, which can be displayed in the following way. External causality summarises the influences that the environment exerts on the object; internal causality summarises the dependencies of the present state of an object from past states (the distinction should not be conflated with the one between self-propelled and etero-propelled objects). As an example, consider what is the case when you have finished to hammer a nail into a piece of wood: the state of affairs holding now (the nail is almost completely inside the wood) depends partly upon the chain of external causality (collisions between the hammer and the nail) and partly upon the internal causality: is the case at the region of space now occupied by the nail depends heavily on what was the case at adjacent spatial and adjacent and previous temporal regions (at least, it depends more heavily on that than on what happened at non-adjacent such regions).

8.Shadows and internal causality

Now, the nature of shadows seems to forbid that portions of a shadow-body (or of shadow-patches, for that matter) are linked to one another across time by immanent causality. That there is shadow now at region r does not depend on there having been shadow anywhere at a previous time.

Even if a shadow remains constant through a period or undergoes only regular variation, its condition at times through that interval does not causally depend on its condition at earlier times. (...) Rather, the condition of the shadow at any time depends on the way things are with the light source, occluders and surfaces. It does not depend of how things were with the shadow earlier. So unlike physical objects, shadows are not internally causally connected (Campbell 1994: 28 ff.).

Thus, given the fact that we do consider shadows as provided with an identity that they (often) retain over time and change, the question is whether our judgements are grounded in anything more than a mere cognitive illusion and have some hope to be true. (The cognitive illusion I have in mind is the so called f -phenomenon, which is a consequence of gestalt grouping principles in dynamic situations. As a typical example, consider the movement of the cursor on the screen of a computer: there is no moving object there, just a series of pixels that are switched on and off according to a regular pattern. If the pattern is gestaltically sound, the impression of unity across movement arises. The point extends to shadows in a qualified way; for the perceptual unity of a shadow over time is a f -phenomenon even if the shadow does not move. That is, metaphysically speaking there are instantaneous shadows only.)

Therefore, what we would describe as the continuous movement of a shadow over a wall hides in fact two metaphysical weaknesses: there is no thing as a single (one and the same) shadow moving from one region to another, and there is no thing as a single (one and the same) movement involving the shadow. The temporally extended shadow is a pseudo-object, and the movement of the shadow is a pseudo-process.

The truth of our identity judgements about (pseudo)shadows and the (pseudo)processes they might undergo is grounded on something external to shadows themselves. It is the identity of the source and of the casting object that appear to be the most promising candidates - if anything at all is responsible for the unity of the shadow (space forbids here to consider some controversial cases in which the shadow seems to be the same even though the source of light and the obtruder do not retain their identities: consider a shadow produced by a mirrored source of light and imagine smoothly replacing the light in the mirror with a real light).

 

9.A further extended theorita

If the true metaphysics of shadows construes them as selective gaps in processes, as just 'holes in light', we should revise our common-sense intuitions and stretch the theorita a bit further. The resulting theorita is not an extension of the TDV theorita. Shadow-bodies are infinitely extended away from the light source, and have only one face, which does not correspond to the casting face (the self-shadow) but to the hidden face of the enlightened surface of the obtruder. The shadow-body, that is, starts immediately under the enlightened surface of the obtruder (exactly where it physically starts could be a conventional matter at this stage; one can e.g. assume that this face of the shadow-body is topologically open) and extends indefinitely. It 'permeates' all the bodies that are on its way, generating (in correspondence to these encountered bodies) surfaces that are internal to the shadowbody and are oriented. In the extended theorita the target body loses its importance and the obtruder reduces to its enlightened surface. This theory is metaphysically purified (but it is not clear that it serves well the purpose of representing common-sense shadows) and it differs from the common-sense theorita discussed at the beginning on its rejection of principle II. A shadow can be cast through an opaque object. This is indirectly proven by considering opaque objects as limit cases of less-than-opaque objects. Obtruders can be non-opaque, or semi-transparent. Putting a non-opaque obtruder in the weak shade of another non-opaque obtruder has the effect of 'strengthening' the shade (i.e. of weakening the light).

Common sense, on the other hand, endorses principle II (possibly) because it mistakenly attributes intrinsic causality to shadows.

The original TDV theorita had a place for the notion of shadow-patches that could allow for a distinction between object- and cast-shadow-patch. In the new theorita the distinction is modal. Shadow-patches are superficial parts of a body that are internal to a shadow-body. Cast shadow patches are shadow patches that would not have been in the dark if the obtruder were removed.

 

10.The structure of shadows

To sum up the best theory up to now: shadow patches are internal boundaries of shadow-bodies; shadow-bodies are identified by a source of light and an opaque obtruder. The obtruder is not transparent to light but is transparent to shadow and permeated by it. This is because shading is not a process - it is the absence of one. As a consequence, real shadows are instantaneous, and temporally extended shadows are only pseudo-objects (and processes involving shadows are only pseudo-processes). However, our judgements about pseudo-shadows and their pseudo-processes are not ungrounded, and criteria are available that make their truth depend on the truth of judgements about light sources and obtruders (to a certain extent).

 

11.Formal principles

This section articulates the theorita. If we ban mirrors, we can identify a shadow-body through the source of light and the obtruder it depends upon, and state that:

1. A shadowbody is externally connected with its obtruder.

2. A shadowbody is sure never to be connected with its light source.

3. An obtruder is wholly located within its shadowbody (unless it is located in some other obtruder's shadowbody relative to the same light source).

4. Successive obtruder's contributions to a shadowbody do not diminish previous obtruder's contributions, unless the following Minimalist Theory is true:

The Minimalist Theory: given an obtruder and a source, there exists only one shadowbody relative to them (which is the possibly scattered sum of all holes generated by the possibly scattered obtruders in the light of the source)

5. In particular, a shadow-body does not share parts with any material body - not even with those that cast it or onto which it is cast.

6. Only the lightened parts of an object can cast a shadow. It follows that

-6a- Only superficial parts of an object can cast a shadow.

-6b- Not all parts of an object that cast shadows cast shadows

-6c- An object can cast a shadow only in a derivative sense, i.e. only if some superficial parts of it cast a shadow.

7. No two shadows (cast from the same source) ever overlap

12.Generalised shadows

I take the question of dependency as representative of a class of more general question about the existence and the identity of 'incomplete' or 'abstract' objects that are nevertheless clearly localised in time and space and have a curriculum vitae of sorts (objects such as nations and geopolitical entities at large). As we have seen, obtruders can be non-opaque, or semi-transparent. This gives us a first generalisation.

It is tempting to consider ordinary shadows as just one among other cases of shadowing phenomena. Ordinary shadows are light-shadows (the first variable in the quadruple that identifies a shadow-body specifies a light source). But light is one among other examples of causal flows. A generalised shadow-body is thus whatever area is shielded from a given causal flow. Particularised again, the notion gives us objects such as shadow-bodies in rain (the dry area under your umbrella), but also in crowds, avalanches or glaciers. What really matters is the presence of a flow of sorts, ad the possibility of finding shelter.

13.Shadows and Socio-Economic Units

Although shadows are in themselves an interesting geographic object (consider, for instance, the kind of problem that was addressed by town planners when skyscrapers began to rise), shadows are of some interest to the scholar of the geographical location of socio-economic units because they provide a powerful and intuitive toy analogy. Both shadows and SEUs are less-than-concrete objects (they are rather different from mountains or lakes); but at the same time they both are somewhat localised in space. Moreover, both shadows and SEUs are dependent objects of sorts - they are what they are because something else (a source of light and an obtruder in the case of shadows; people, their properties, their beliefs etc. in the case of SEUs) is what it is. Moreover, the way shadows and SEUs are where they are is related to this dependence they both have to the things they depend upon. The present analysis of shadows suggests some of the guidelines for construing the location of less-than-concrete entities.

As an example, consider Italy, taken here as a geopolitical entity (not as just a region of space).

· Italy is a space occupier. It takes some space, in an exclusive way (for instance, it does not interpenetrate with Switzerland). But it does so not in virtue of some 'hard' physical facts (bar ideological nonsense about a sacred physical 'Italian boden') but because of a general consensus of a large number of individuals - both Italian citizens and citizens from other countries. Italy is not a metaphysically independent entity. It is causally sustained by the individual wills of a number of people. Italy is projected. (So is any shadow.)

· Italy is thus a metaphysically dependent entity. It is a 'lesser' entity. Which does not mean that we ought to endorse an eliminative strategy for Italy. We can accept that it exists, as its citizen and the mountains in it exist, but not as an independent object. (Shadows, although dependent, still exist).

· The size and shape of Italy depend on how much of space can be controlled by the citizens of the country, and thereby sheltered from other influences or claims. (A shadow's shape and size depend upon those of the obtruder).

· Italy's shape partly depends also upon the physical geography of the space it is projected on. (A shadow's geometry partly depends upon the geometry of the target object.)

· For the purposes of the definition of Italy as a space-occupier, the boundary alone is what matters. Whatever is inside the boundary need not receive any representation on a map. (A shadow is adequately described by its profile. The interior does not represent any feature of the obtruder.)

· The boundary of Italy might be subject to negotiation in some cases - e.g. when individual property rights conflict with international politics. The blurring of the boundary that one might experience here resembles the grey area around a shadow that is projected by a non-pointlike source, and which can be resolved into the sum of many different projections from individual sources.

· Italy does not enjoy any form if intrinsic causality. Where it is now depends much less upon where it was before than upon the will of the individuals (both of Italian and of non-Italian citizenship) that project it. Barring again mythical nonsense about the magic powers of the boden, Italy has no causal inertia. Nor does any other shadow.

14.Some warnings

It is not claimed, of course, that all geopolitical entities are shadow-like. And it is not claimed that all aspects of shadows are relevant here either (see the next section for an enrichment). Finally, it is not claimed that the shadow metaphor is good or bad. A more modest claim is put forward here: that the shadow metaphor is a rather intuitive model for our understanding of SEU's way of being in space.

15.Further aspects of the shadow metaphor

In this last section I shall list some of the more common metaphorical usage of shadows. Although in the paper I concentrated mostly on the spatial and causal clusters, it might be argued that other features of the concept of a shadow might be fruitfully employed in modelling SEU's dynamics. An open question is, Which of these cluster overlaps which other? (For instance, facts about control and causality ground facts about representation.)

· The shadow-as-shelter metaphor. This is a causal/spatial metaphor, with both good and bad connotations. One can be in somebody else's shadow, meaning that one lost one's independence.

· The shadow-as-proxy metaphor. Shadows stand for something projecting them. A causal metaphor, but shadows are bestowed some independence.

· The shadow-as-representation metaphor. Partly overlaps with the shadow-as-proxy metaphor: representations are proxies. Shadows represent only by outline. Their interior is essentially hidden.

· The shadow-as-immaterial metaphor. Probably subordinate to a more general (but not clearly individuated) metaphor, the shadow-as-lesser-entity metaphor.

· The shadow-as-dependent-entity metaphor. (John follows Mary as a shadow).

· The shadow-as-obscurity, danger metaphor

· The shadow as silhouette metaphor. (Overlaps with the shadow-as-representation metaphor). One can abstract from what is inside the shadow, the only relevant thing is the profile.

· The shadow-as-controlled-entity metaphor. Obviously overlaps with the shadow-as-dependent-entity metaphor, it is nevertheless different insofar an entity can be controlled without thereby being dependent.

· The shadow-as-attached thing metaphor. ("I have got a shadow"). It is not very clear how this interacts with the dependent feature of shadows. The fact that shadows stick to one suggest that they are relatively independent (they are a nuisance, something one would like to get rid of and cannot).

 

References

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