The Impoverished Representations of Brains in Vats Jan Almäng Karlstads universitet Faculty of Arts and Social Sciences 651 88 Karlstad jan.almang@gmail.com Penultimate version. Final version is published in Grazer Philosophische Studien, Volume 97 no 3 2020, pages 475-494 https://brill.com/view/journals/gps/97/3/article-p475_475.xml Abstract In the present paper, the notion that brains in vats with perceptual experiences of the same type as ours could perceptually represent other entities than shapes is challenged. Whereas it is often held that perceptual experiences with the same phenomenal character as ours could represent computational properties, I argue that this is not the case for shapes. My argument is in brief that the phenomenal character of a normal visual experience exemplifies shapes – phenomenal shapes – which functions as the vehicle for our perceptual representation of shapes. Due to the unique mereological structure of shapes, phenomenal shapes are unable to reliably track any property but shapes. In so far as reliable tracking is a necessary condition for perceptual representation, phenomenal shapes can consequently and contrary to received wisdom only represent shapes. Keywords: Perception, Shapes, Twin-Earth, Externalism 1 Introduction A very popular position in philosophy of mind during the last decades has been the view that intentional content is external to the perceiving or thinking subject. On such accounts, it is only in virtue of reliably tracking features of the surrounding environment that internal states of the subject represent these features. Externalism regarding perceptual content is consequently the view that the intentional content of perceptual states depends on the tracking relations that the perceptual experience bears to the surrounding environment. One of the most important externalists, Tyler Burge, has expressed this doctrine as the claim that perceptual intentional relations depend on non-intentional relations. Externalism, or "antiindividualism" as Burge calls it, is on his view the notion "that a range of non-representational 2 relations, including causal relations, between environment and individual must constitutively be in place, if there are to be perceptual states." (Burge 2010, 71) Burge argues that the phenomenal features of perceptual experiences are not in themselves representational. (Burge 2010, 71; see also Burge 1986) It is because they bear various relations to the surrounding environment that they can represent features in this environment. These relations are normally conceived of as causal relations. Thus, for example, in the typical case, an experience of a certain type is taken to represent a kind or a property because the kind or property normally causes experiences of the said type. I shall express this in terms of perceptual experiences reliably tracking features in the surrounding environment. Intuitions to the effect that perceptual content is external often rest on various kinds of Twin Earth thought experiments (see Putnam 1973). These are characterized by the fact that two different perceivers have the same type of experience but are located in different environments where they reliably track different properties. "Phenomenal twins" consequently have perceptions with the same phenomenal character. Their perceptions, however, have different intentional contents. Recent decades have seen attempts to twin-earth colours (see Block 1990, Chalmers 2006), shapes (see Hurley 1998, Thompson 2010, Chalmers 2012) and sizes (see Thompson 2010, Bennett 2011, Chalmers 2013, Peacocke 2013). In the case of colours, the phenomenal property normally associated with a certain colour is taken to reliably track a different colour. So in the case of Block's "Inverted Earth", the colour spectrum is inverted such that the phenomenal property which reliably tracks redness on earth reliably tracks greenness on Inverted Earth. We can express this in terms of phenomenal redness reliably tracking redness on earth but greenness on Inverted Earth. Phenomenal greenness, on the other hand, reliably tracks greenness on earth but redness on Inverted Earth. Attempts to twin-earth sizes normally focus on a world where everything has a different size than on earth, yet which is populated by our phenomenal twins. In such cases, phenomenal sizes, or the phenomenal property reliably tracking sizes, would reliably track different properties than it does on earth. (See Thompson 2010.) Convincing examples where shapes have been twin-earthed are rarer in the literature. Early attempts to twin-earth shapes were made by McGinn (1989) (who went on to criticize the example) and Martin Davies (1992, 1993). The worlds in these scenarios are also populated by our phenomenal twins, but shapes are distributed in another way than on earth. Perceivers on this world exemplify the same phenomenal shapes – the phenomenal property associated with shapes – in the same way that we do. But phenomenal shapes reliably track different shapes on 3 the twin-earthed worlds than on earth. Thus, for example, the phenomenal shape reliably tracking spheres on earth might be associated with ellipsoids on Twin Earth. A convincing critique (at least in my opinion) of these ideas was given by Segal (1991, 448), who pointed out that in all such cases, the geometrical properties of the objects perceived would be misrepresented. Recently, attempts have been made to circumvent Segal's objections (see Thompson 2010, Chalmers 2012). In my opinion, these attempts are no more successful than previous attempts to show that type-identical phenomenal shapes could represent different shapes, but I shall not attempt to argue thus in this paper. In the present paper, I shall focus on a quite different attempt to show that shapes could be twin-earthed. Chalmers (2012) explicitly and Putnam (1981) implicitly suggest that phenomenal shapes could reliably track properties which are not shapes at all, for example computational properties. This might, for example, be the case if the perceiving subject were a brain in a vat. In the present paper, I shall argue that this is not possible. The phenomenal shapes of brains in vats or perceivers with visual systems hooked up to computers cannot reliably track any property but shapes. Phenomenal shapes can only reliably track shapes. Or so I shall argue. It is to be noted that if we assume – as seems reasonable – that shapes can only be exemplified in spaces, it follows that the argument also shows that our visual perceptions can only represent spaces of some kind. The next section introduces the views of Chalmers and Putnam. The subsequent eight sections detail an argument in eight steps to the effect that phenomenal shapes cannot reliably track other properties than shapes. While this does not show that perceptions do not have their contents in virtue of bearing reliable-tracking relations to objects and properties, it undercuts one of the main motivations for such a view. For it shows that the phenomenal shapes of our perceptual experiences cannot reliably track other entities than shapes. The first three steps suggest that perceptual experiences have a spatial structure and that "phenomenal objects" bear parthood relations to each other. Steps four and five introduce the notion of phenomenal shape and suggest that phenomenal shapes can also bear parthood relations to each other. Steps six and seven argue that if phenomenal shapes reliably track any entity, they reliably track shapes. The reason is that the parthood relations obtaining between shapes (including phenomenal shapes) can only obtain between shapes. It is a unique kind of parthood relation. To the extent that phenomenal shapes reliably track shapes, they reliably track their parthood relations as well. The eighth step concludes that this shows that if phenomenal shapes reliably track any entity at all, they reliably track shapes. The eleventh section of the paper 4 responds to an objection. The paper is concluded in the twelfth section with a discussion of the scope of the argument in the paper. 2 Envatted perceivers and phenomenal shapes Putnam (1981) and Chalmers (2012) argue that we could have phenomenal twins who were brains in vats. However, their experiences would reliably track different entities than our experiences. Hence, even brains in vats would have perceptions with representational content and would be able to have veridical perceptions or think true thoughts. Only in their case, their perceptual experiences and thoughts would reliably track other properties and objects than in our world. Here is, for example, Putnam (1981, 14) outlining his view: By what was just said, when the brain in a vat (in the world where every sentient being is and always was a brain in a vat) thinks "There is a tree in front of me", his thought does not refer to actual trees. On some theories that we shall discuss it might refer to trees in the image, or to the electronic impulses that cause tree experiences, or to the features of the program that are responsible for those electronic impulses. [...] On these theories the brain is right, not wrong, in thinking "There is a tree in front of me." Given what "tree" refers to in vat-English and what "in front of" refers to, assuming one of these theories is correct, then the truth-conditions for "There is a tree in front of me" when it occurs in vat-English are simply that a tree in the image be "in front of" the "me" in question – in the image – or, perhaps, that the kind of electronic impulse that normally produces this experience be coming from the automatic machinery, or, perhaps, that the feature of the machinery that is supposed to produce "the tree in front of one" experience be operating. And these truthconditions are certainly fulfilled. Putnam does not explicitly discuss shapes in the quote above. Indeed, he does not more than mention perceptual experiences. His primary interest is reference and truth in language and thought. It is, however, quite clear that his argumentation can easily be extended to cover the perception of shapes as well. Chalmers (2012), on the other hand, explicitly endorses the view that perceptual experiences could reliably track computational properties instead of spatiotemporal ones. He argues that there is a possible scenario in which we are hooked up to a computer simulation. In this case our 5 perceptual experiences are caused by computational properties and relations. (Chalmers 2012, 335f) Consequently, "our spatiotemporal expressions will pick out the computational properties and relations that serve as their causal basis, and those properties and relations really are present in the computer." (Chalmers 2012, 335)1 By implication, a phenomenal shape will reliably track whatever computational property that is its causal ground. In this paper, I will challenge the notion that phenomenal shapes could reliably track other properties than shapes. Consequently, unless computational properties are shapes, we cannot reliably track them. Let us now move on to consider the argument against Putnam and Chalmers. 3 Step 1: Perceptual experiences have a spatial structure Putnam and Chalmers assume that envatted perceivers could be our phenomenal twins. They could have perceptual experiences of the same kind as we have. If this is correct, the phenomenal character of a perceptual experience must be an internal state of the perceiver. Unlike the intentional content of the perceptual experience, the phenomenal character does not depend on the relations the subject bears to the surrounding environment. It is important to note that this is a consequence of their view. It is not an assumption made in this paper that is independent of their account. The phenomenal character of a perceptual experience is consequently not identical to the object represented. The object of perception is external to the perceptual experience, whereas the phenomenal character is an internal state of the perceiver. On this conception, perceptual experiences represent what their phenomenal characters reliably track. One possible way of construing the relationship between phenomenal character and the intentional content in an externalist framework has been suggested by Ned Block when articulating the Inverted Earth scenario. According to Block, the phenomenal character of perceptual experiences functions as a kind of vehicle for the intentional content. The possibility of an inverted spectrum shows, according to Block, that we must distinguish between the phenomenal character and what it represents: "your experience and my experience could have exactly the same representational content, say as of red, but your experience could have the same phenomenal character as my experience as of green". (Block 1996, 27f) On earth, phenomenal redness reliably tracks redness. By implication, phenomenal redness is the vehicle for the content representing redness. On Inverted Earth, however, phenomenal redness is the vehicle for the 1 For a view very similar to that of Chalmers, see Prosser 2016, 60. 6 content representing greenness. So on Inverted Earth, phenomenal redness is the vehicle for the content representing greenness. Now, the first step in my argument claims that the phenomenal character of visual perceptual experiences has a spatial structure. This is a claim that pertains to the structure of the phenomenal character. Clearly, the phenomenal character of a visual experience must, just like any other mental state, have a structure of some kind. This claim is a claim to the effect that the structure is a spatial one. This does not mean that the phenomenal character is external to the mind. It is quite consistent with the notion that the phenomenal character is an internal state of the mind. It simply means that visual experiences have a three-dimensional geometrical structure. We shall call this structure a phenomenal space, a term that to my knowledge was first used by Barry Dainton (2000). He notes that we are in perception "immediately presented with [...] a closely integrated three-dimensional world, albeit a wholly phenomenal world." (Dainton 2000, 61)2 What does it mean to say that our phenomenal character has a spatial structure? In this context, it is an assertion to the effect that the phenomenal character has a geometrical structure. The phenomenal character has a structure that contains positions or locations which may or may not be occupied by various entities. The structure of phenomenal character has a metric such that the positions in the structure are related to each other by certain distances; and the structure itself has a certain number of dimensions. So we end up with four characteristics of a spatial structure. The first characteristic is that a spatial structure contains different locations or positions. I think it is easy to see why at least visual phenomenal characters have this characteristic. We can after all distinguish between the objects on the left side of the visual field and the objects on the right side of the visual field. If you perceive multiple entities, the vehicles for perceiving these entities are located at different positions in the phenomenal structure. The second characteristic is that the positions in a spatial structure may or may not be occupied. In the case of visual experiences on earth, what occupies these locations are the vehicles for perceiving ordinary physical objects. Once again it is, I think, easy to see that the structure of our phenomenal character has this characteristic. If you see an object to your left but not to your right, it makes sense to say that in some sense of the word there is a phenomenal 2 A similar claim is possibly to be found in Smith (2002). He claims that perceptual consciousness is characterised by "phenomenal three-dimensional spatiality" (Smith 2002, 133), which he describes as a "matter of the intrinsic character of certain sense-fields" (ibid.). It is however not clear whether or not Smith is claiming that perceptual consciousness itself is three-dimensional or whether he is claiming that perceptual consciousness is such that it presents us with a three-dimensional spatial world. 7 object that occupies a position at the left side of your visual field but not at the right side of your visual field. There is no vehicle that represents anything at your right side.3 The third characteristic of a spatial structure is that it has a metrics. Once again it is easy to see that the structure of visual experience has this characteristic. Let us imagine that you are perceiving an object following a straight path from the left side of your visual field to your right side. As it is moving, it is passing through three positions in physical space, A, B and C, such that B lies between A and C. If it is doing this, we may assume that the distances AB and BC are shorter than the distance AC. If we assume that A, B and C are represented by the positions in phenomenal space A', B' and C', it is natural to assume that the distances A'B' and B'C' in phenomenal space are shorter than the distance A'C'. The fourth characteristic is that geometrical structures have a number of dimensions. There is no consensus on how many dimensions phenomenal space could possibly have. But in my opinion it must have either three (as Dainton 2000 seems to suggest) or two and a half, as Marr (1982) suggests that at least a stage in the computing of visual information has. For present purposes, I shall simply assume that the structure is three-dimensional. Structures with these four characteristics will also exemplify certain shapes. So we might add that as a fifth characteristic, but it is really one that is entailed by the other four. Structures which have these geometrical characteristics will in this paper be known as spaces. It is important to note that I am not claiming that these are the only characteristics spaces can have – they can also be bounded or unbounded, have a certain curvature, and so on and so forth. But for present purposes, they are the most important characteristics. It is also important to note that there can be many different kinds of spaces. We have already encountered phenomenal spaces – the structure characteristic of at least the phenomenal character of vision – and physical spaces, which is what phenomenal spaces represent on earth. But there is in principle nothing that precludes that there might be other spaces. Tyler Burge, for example, suggests that the "geometrical structures in sensory-motor memory correlate with the structure of physical space" (Burge 2010, 513). If that is the case, sensory-motor memory might also instantiate a space of some kind. Phenomenal space and physical space are obviously different kinds of spaces. Physical space is a fundamental part of reality in a sense that phenomenal space is not. The phenomenal character of a visual experience presumably supervenes on physical facts, and so its structure – 3 At this point someone may protest about the notion of a "phenomenal object", but this notion is no more controversial than the idea that the phenomenal character functions as the vehicle for the intentional content. When we perceive mind-independent objects, the vehicles of this content are what I opt to call "phenomenal objects". 8 phenomenal space – will also supervene on physical facts. Physical space, however, is best conceived of as an essential feature of the physical world and not as anything that supervenes on yet further physical facts. Another crucial difference between phenomenal space and physical space is that phenomenal space cannot be occupied by physical objects,4 whereas the locations in physical space can be occupied by physical objects. The spaces are, however, similar with respect to at least some other characteristics –in particular, they are similar in the sense that they both exemplify the same kinds of shapes. There is nothing ontologically different about a sphere in phenomenal space and a sphere in physical space. The difference is that the sphericality is the sphericality of a phenomenal object in phenomenal space but of a physical object in physical space. But that is not an intrinsic difference between the shape-properties but of what they are the properties of. They can still be instances of the same universals. Shapes are properties, and they can be the properties of many different kinds of objects – not merely physical objects but also collections of physical objects and phenomenal entities. But there is nothing that precludes these different kinds of entities from instantiating the same shapes. A shape is a geometrical property, and as such it can be instantiated in many different kinds of spaces. Phenomenal shapes are on earth the vehicles for representing physical shapes. My claim, as it is developed in this paper, is that they can only represent shapes but not that they can only represent physical shapes. 4 Step 2: Phenomenally spatial objects can bear parthood relations to each other The second step claims that phenomenal objects can bear parthood relations to each other. By a "part" I mean a proper part. Consequently, if an object is a part of another object, then it is not identical to it. Thus, for example, if you visually perceive a wall composed of bricks, you will perceive the wall and the bricks. In this case, the phenomenal counterpart of the wall – i.e. that part of the phenomenal character which functions as the vehicle for the representation of the wall – will be composed of the phenomenal counterparts to the perceived bricks. Let us unpack this claim in some detail. First of all, we assume as before that features of phenomenal character function as vehicles for the intentional content, regardless of whether we 4 We are assuming now that the phenomenal character of a visual experience cannot be reductively identified with ordinary physical objects. 9 accept that intentional content requires a tracking relation or not. If this is granted, it seems reasonable to assume that if we perceive three different objects, then there are at least three different parts of the phenomenal character which function as vehicles for these perceptual representations. Let us say that these are "phenomenal objects". Now, if we in our world perceive bricks as being parts of a wall, then the phenomenal objects which function as phenomenal counterparts to the bricks in the phenomenal character are parts of the phenomenal object which functions as the phenomenal counterpart of the wall. In other words: the parts of the phenomenal character which function as the vehicles for the representation of the bricks jointly form a whole which functions as the vehicle for the representation of the wall. In the example above, I have argued from an ordinary kind of visual perception to the nature of the phenomenal character. A normal case of visual perception of bricks shows that we are not merely able to perceive objects as being parts of other objects. Phenomenal objects can also be parts of other phenomenal objects. 5 Step 3: If a phenomenal object is a part of another phenomenal object, then it is a spatial part of it The third step in the argument claims that if a phenomenal object is a part of another phenomenal object, then it is a spatial part of it. If an object is a part of another object, then the first object is located within the spatial region occupied by the second object. But the converse need not be true. The same is true of phenomenal objects. If a phenomenal object is a part of another phenomenal object, then the first object is located at the same place in phenomenal space as the second object. Thus, when we perceive the bricks in the wall, the phenomenal objects that function as the vehicles for the representation of the bricks are located at the same position in phenomenal space as the phenomenal object that functions as the vehicle for the representation of the wall. Since the part of the whole is located within the spatial region of the whole object, there is obviously no phenomenal distance between them. If object a is a spatial part of object b, then the size of b is bigger than the size of a. This is obviously true of phenomenal objects that jointly compose objects. The phenomenal objects that function as the vehicles for the representation of the bricks have exactly the same size as the phenomenal object that is the vehicle for the representation of the wall. So each phenomenal 10 object that is a proper part of the whole phenomenal object representing the wall is smaller than the whole phenomenal object. We have seen that phenomenal objects that are part of other phenomenal objects are located within the regions of phenomenal space occupied by the object of which they are a part. It is thus fair to conclude that phenomenal objects that are parts of other phenomenal objects are spatial parts of them. Let us now move on to consider phenomenal shapes. 6 Step 4: Perceptual experiences exemplify phenomenal shapes If phenomenal objects have a location and a size in a three-dimensional geometrical structure, then they will also have a shape. Phenomenal objects can no more than ordinary material objects have locations and sizes without having shapes. The shape of the object is what delimits the object from the positions in space that it does not occupy. The perception of shapes is typically taken to be characterised by perceptual constancy. This means that we perceive objects as having the same shape when we look at the objects from different perspectives. In some sense, the object is in these cases taken to "look" differently from different perspectives. Now, it is quite clear that a perceptual experience of a shape may vary in the sense that one's experience is changing when an object is rotating in front of one. Let us, for example, assume that we are perceiving a rotating coin. In this case, there is something that is constant in perception, which accounts for the perceptual constancy, but there is also something that is changing. Is the phenomenal vehicle for our representation of the shape of the coin successively changing when the coin is rotating? Or is the fact that the "looks" of the shape are changing to be explained in some other way? This is a difficult question to answer, and I cannot hope to give a satisfactory answer here – that would require a separate paper. I will simply assume that the phenomenal shape is what accounts for the constancy in this case, and that it is some other phenomenal property that is varying. A very simple argument to this effect is that the shape of the object does not in this case appear to be changing; and the easiest – if not the only – way to account for this is by assuming that the phenomenal shape is invariant, whereas some perspectival property or other is changing. 11 7 Step 5: Shapes actually or possibly have other shapes as parts The fifth step claims that shapes can have other shapes as parts, and, by implication, that phenomenal shapes can have other phenomenal shapes as parts. Many properties have other determinate properties of the same determinable as parts. Johansson (2004, 41ff) points out that this is true not only of shapes but also of mass and volume. Thus, for example, the volume 2 m3 may have two volumes of 1 m3 each as parts; and the mass 2 kg might similarly have two parts, each consisting of 1 kg. It is also often assumed – not least in geometry – that shapes can also have other shapes as parts. Mandelbrot, for example, famously defined a fractal as "a shape made of parts similar to the whole in some way". (Feder 1988, 11) But we need not only consider such relatively sophisticated shapes as fractals in order to see that shapes can have parts. Consider a two-dimensional figure such as a quadrangle. If two straight lines are drawn on the quadrangle, so that a cross is formed inside the quadrangle, the quadrangle has four other shapes as parts. Depending on how the cross is drawn, we can say that the quadrangle has four quadrangles as parts, four rectangles as parts or four triangles as parts. It is important to note that shape tokens need not actually have other shapes as parts. Even when they don't, however, they are type-identical to other tokens which have other shapes as parts. Thus, for example, a token of the quadrangle mentioned above may not have any actual parts. But it may be type-identical to a quadrangle that has other parts. One quadrangle may, as we have seen, have four other quadrangles as parts, and a third quadrangle may have four triangles as parts. At this point, it might be objected that a quadrangle that is composed of four triangles is necessarily different from a shape that is composed of four quadrangles. It could not be typeidentical to a quadrangle that was not composed of our triangles. In reply, I would like to make two observations. First of all, if this were the case, then we would also have to say that a line of 9 metres, consisting of two lines that were 6 metres and 3 metres, would not be type-identical to a line of 9 metres that consisted of two lines of 5 metres and 4 metres. We would in other words have to reject the notion that any property with parts could be type-identical to a property with parts of different types. In my opinion, this is absurd. Secondly, even though our two quadrangles are numerically different tokens, they might still be tokens of the same type. There is nothing that prevents a property-type from having tokens with mutually inconsistent sets of possible parts. Thus, even though it might well be the case that a 12 token of a shape with four triangles as parts could never in any sense be token-identical to a shape with four quadrangles as parts, it is perfectly possible that it is type-identical to a shape with four quadrangles as parts. This is no stranger than the fact that the number 9 is identical to both the sum of four and five and to the sum of three and six. 8 Step 6: The mereological principles that apply to shapes do not apply to any other entities The sixth step claims that the mereological principles governing how shapes qua parts form shapes qua wholes are unique to shapes. These principles do not pertain to any other kinds of entities. The kind of parthood relation that obtains between shapes can only obtain between shapes. Or, more specifically, the kind of parthood relation that obtains between n-dimensional shapes can only obtain between n-dimensional shapes. There are two crucial principles which separate the parthood relations obtaining between shapes from other parthood relations. While it is possible to find other entities where one principle governs the formation of wholes out of parts, it is not possible to find other entities where both principles govern the formation of wholes out of parts. The first principle is that the kind of whole formed by different shapes depends on the relative location of the shapes in a spatial structure. The second principle is that shapes do not conform to the formal-mereological axiom or theorem that a whole cannot have itself as a proper part. Or, to be more precise, a shape-type can have itself as a proper part. But a shape-token cannot have itself as a proper part. Thus, for example, a particular sphere can have another particular sphere of the same type as a proper part. But a particular sphere cannot have itself as a proper part. It is quite obvious that the shape-whole formed by two shape-parts depends upon the relative location of the parts in space. Let us say that shapes located so that they form a shape are placed in a certain configuration. Which shape they form then depends upon the configuration of the shapes. Eight cubes may for example form a cube when they are placed in one configuration but a rectangular cuboid of some shape when they are placed in a different configuration. The dimensionality of the shape is in this respect crucial. N-dimensional shapes must be placed in spaces with at least n-dimensions. Three-dimensional shapes are located in spaces with at least three-dimensions. They can in principle be located in spaces with more than three-dimensions, but for the sake of simplicity, we will focus on the three-dimensional example. Thus, the relative location of the parts of a shape 13 can vary in at least three dimensions. If one of the parts that compose a shape is "moved" in one dimension, then a new whole is formed. It is true that this first principle also applies to ordinary material objects. This is because ordinary material objects exemplify shapes. This principle, however, does not apply to objects which cannot be described in geometrical terms. For these objects cannot be said to form different wholes depending on their location in space. The second important characteristic is that shapes can have themselves as proper parts. We have seen that this is what characterises fractals. It can also characterise less sophisticated shapes such as quadrangles or cubes. A cube may, for example, have eight other cubes as parts. Stephen Kearns (2011, 91) has put this point nicely in a recent paper: Some (abstract) shapes are self-similar and some are not. That is, some shapes (such as fractals) are (qualitatively) identical to parts of themselves while other shapes are not qualitatively identical to any part of themselves. Furthermore, if two abstract shapes are qualitatively identical to each other, then they are in fact numerically identical (and vice versa). Therefore, some shapes have themselves as parts and other shapes do not have themselves as parts. We need not follow Kearns in claiming that qualitative identity entails numerical identity for shapes. Shapes are different from other spatial objects, such as material objects. Material objects cannot have themselves as proper parts. It makes no sense to assume, for example, that a wall of bricks can be composed of itself plus some other bricks. The reason for this is that material objects exemplify other properties than shapes, for example mass and size. Mass-properties and sizeproperties cannot have themselves as proper parts. The weight 7 kg cannot have itself as a proper part. Consequently, a wall that weighs, for example, 50 tons, cannot have itself as a proper part, for this would entail that the wall would be heavier than 50 tons, which automatically generates a contradiction. At this point it might be objected that shapes are just abstract entities. So what I have said so far has no applicability to phenomenal shapes or the objects of perception. Here, I would emphatically disagree. Properties such as shapes are exemplified in the world by both perceptual experiences and objects in the surrounding environment. In fact, a very good case could be made that properties only exist as exemplified in the spatiotemporal world. There is no platonic realm 14 that is separate from the spatiotemporal world and populated by abstract properties. Shapes such as cubes and quadrangles exist in the objects that exemplify them. If the present analysis is correct, there are two crucial characteristics of the mereology of shapes. The kind of whole formed by the parts depends on the parts' relative location in space. This means that shapes are dissimilar from all other entities than geometrical entities such as material objects. The second characteristic is that shapes can have themselves as proper parts. This differentiates shapes from all other geometrical entities such as ordinary material objects and volumes. The mereology of shapes is, in other words, governed by a unique set of principles that applies to no other entities than shapes. 9 Step 7: The parthood relations that obtain between phenomenal shapes reliably track parthood relations between the properties phenomenal shapes reliably track In the sixth step, we learned that shapes can have parts. Since phenomenal shapes are shapes too, they can also have parts. In the present section, I will argue that if a phenomenal shape reliably tracks a type of property (for example shapes), then the parthood relations that obtain between the parts of this phenomenal shape will reliably track parthood relations between the entities phenomenal shapes reliably track. The parthood relations that obtain between phenomenal shapes reliably track the parthood relations between whatever entities they reliably track. The closing part of the argument that phenomenal shapes cannot reliably track other properties than shapes is very simple. Let us bear in mind that phenomenal shapes are shapes and that consequently the principles for the mereology of shapes apply to phenomenal shapes as well. A consequence of this is that the parthood relations that obtain between phenomenal shapes reliably track the parthood relations between whatever entities phenomenal shapes reliably track. We can express this in general terms by assuming that p is a phenomenal shape that is composed of two phenomenal shapes q and r. Let us also assume that p is a token of the type P, q is a token of the type Q and r is a token of the type R, and that the phenomenal shape type Pp reliably tracks the shape P', Qp reliably tracks the shape Q' and R reliably tracks the shape R'. If this is the case, P:s that are composed of Q:s and R:s will reliably track P':s that are composed of Q':s and R':s. Let us illustrate this by giving an example from our world. If a phenomenal cube has a phenomenal sphere as a part, and if phenomenal cubes reliably track cubes and phenomenal 15 spheres reliably track spheres, then phenomenal cubes with phenomenal spheres as parts will reliably track cubes with spheres as parts. In order to see why we should accept this principle, we could ponder a scenario where it is not true. Let us assume that in this scenario, phenomenal cubes with phenomenal spheres as parts do not reliably track cubes with spheres as parts but merely cubes. If this were the case, phenomenal cubes with phenomenal spheres as parts would not reliably track spheres at all. Consequently, phenomenal spheres would not reliably track spheres. But this contradicts the assumption above that phenomenal spheres would reliably track spheres. It should be noted, in order to prevent misunderstanding, that I am not claiming that a type of phenomenal shape, such as P in the example above, reliably tracks what its parts are tracking. My claim is that those P:s that are composed by Q:s and R:s must also reliably track P:s that are composed of Q:s and R:s. For if this were not the case, we would be forced to conclude that Qp:s or Rp:s did not reliably track anything. The point is a general one. The parthood relations obtaining between phenomenal shapes will reliably track parthood relations among the entities phenomenal shapes reliably track. For a phenomenal shape cannot, according to the argument above, be composed of other phenomenal shapes without reliably tracking a whole that is composed of the entities that their parts reliably track. But that, alas, was what this section attempted to demonstrate. 10 Step 8: If phenomenal shapes reliably track anything, they reliably track shapes We are now in a position to see that phenomenal shapes cannot reliably track any entities other than shapes. According to the sixth step, the mereological principles governing shapes are unique to shapes. So it is only n-dimensional shapes that can bear the type of relation that holds between the parts and wholes of n-dimensional shapes. According to the seventh step, the parthood relations that obtain between phenomenal shapes reliably track parthood relations between the entities phenomenal shapes reliably track. So, by implication, phenomenal shapes cannot reliably track anything but shapes. I conclude that if phenomenal shapes reliably track anything, they will reliably track shapes. Insofar as computational properties are shapes, phenomenal shapes can reliably track them. But insofar as computational properties are not shapes, they cannot be tracked by phenomenal shapes. Note that phenomenal shapes cannot reliably track entities which merely exemplify shapes. 16 They can only track entities which are shapes. An ordinary material object cannot have itself as a proper part, even though it exemplifies shapes. But shapes can have themselves as proper parts. 11 An objection and a reply An anonymous referee has suggested that we can successfully model space in a computer, even though the computer is not literally using space itself in the medium of representation. The objection goes on to suggest that in this case there will be an isomorphism between the coded representations in the computer and the space that is represented; and if a brain in a vat was hooked up to such a computer, phenomenal space would be isomorphic to these computational properties and so would represent them. The argument has three steps. The first step is a claim to the effect that a computer can model space without using space itself (i.e. there need not be spheres in a space in the computer in order for it to represent spheres in physical space). I think the first step is a reasonable assumption and will without further ado accept it. The second step claims that there will be an isomorphism between the computational properties and the space that is represented. That is an assumption that I will reject. The third step claims that the first two steps entail that a brain in a vat that was hooked up to such a computer and which was our "phenomenal twin" (or had relevantly similar experiences as we have) would represent these computational structures. Given that the first two steps are correct, this would indeed entail the third step. But I think that the argument can be stopped at the second step. Let us begin by noting that the objection presupposes, or at least the referee seems to presuppose, that I have actually shown that the spatial character of our visual experience can only represent something that is isomorphic to it. Now, if my argument in this paper is correct, a geometrical structure of some kind can only be isomorphic to another geometrical structure of some kind. For shapes are unique among properties in the sense that a shape can be type-identical to one or more of its parts. No other property can have parts with which it is type-identical. Since shapes are only to be found in space, it follows that phenomenal space can only be isomorphic to other spaces and, more broadly, that spaces in general can only be isomorphic to other spaces. The argument by the referee proceeds by claiming that if a computer models a space, then the computer will be isomorphic to a space. I would deny that this is the case. Even though a 17 computer can model a space, there need be no computational state that has a spatial or geometric character. Consider, for example, a very simple spatial model. Let us assume that a computer models a space by assigning coordinates to positions in the space. Each position in space could in that way be represented as either occupied by some entity or as unoccupied. It is in that way possible to represent various shapes in a purely digital way, where the representation is not in any way isomorphic to what is represented. The medium of representation would in such a case only consist of a set of coordinates, each of which consists of a triplet of numbers and a representation of the coordinate as being either occupied or empty. The medium of representation is in this case not isomorphic to what is represented. It makes no sense to say that the representation of a shape in such a system could have something type-identical as a part. So I conclude that the fact that computers could successfully represent space does not entail that the medium of representation must be isomorphic to what is represented. A fortiori, it does not follow from the fact that we could have phenomenal twins who are brains in vats that our twins would necessarily represent a spatial structure of some kind. 12 Concluding words I have argued that the vehicle for representing shapes – phenomenal shapes – can only represent shapes. I have, however, noted that there is nothing that precludes that phenomenal shapes represent shapes in non-physical spaces. Which kinds of spaces are there? I have suggested that the phenomenal character of visual experience forms one kind of space, and we obviously know that physical space is another kind of space. It is distinctly possible that there are yet other kinds of spaces. This invites the following kind of objection. Even if I have shown that phenomenal shapes must represent shapes, would it not be possible for brains in vats to represent shapes in other spaces than physical spaces, for example a computational space? In reply, I would like to make two observations. The first observation is that we really don't know to what extent there could be other spaces than phenomenal and physical ones. In the case of computers, it seems unlikely that they actually instantiate spaces. Computers work by digital representations and not by analogue representations. Secondly, and more importantly, even if computers (or other kinds of entities that brains in vats might be hooked up to) could instantiate a space of some kind such that the visual 18 experiences of brains in vats reliably tracked occurrences in that space, we would indeed have to say that these brains in vats represented that space and that phenomenal shapes represented shapes in that space. But – and this is the important point – this is rather uninteresting from the point of view of Twin-Earth scenarios such as those involving brains in vats. Let us recall that brain-in-vat scenarios were designed precisely in order to show that our vehicles of representation could represent something different than they do on earth. If my argument in this paper is correct, this is not true with respect to shapes. Even though we may suppose that brains in vats might be able to represent shapes in these non-physical spaces, they would still represent the same shapes we represent on earth. Spaces of different ontological kinds are capable of exemplifying shapes with the same universal. Consequently, there is nothing that prevents phenomenal, physical and computational (if such can exist) spaces from exemplifying the same geometrical properties like sphericity. Being spherical is the same property in a physical and a phenomenal space. However, it is a property of physical entities in the first space and of phenomenal entities in the second space. If there can be computational spaces, it would be the same property of sphericality that was exemplified in such spaces as in physical space. It would, however, be computational entities which were spherical in the computational space and physical entities which were spherical in the physical space. Nevertheless, the physical object and the computational object would exemplify the same shapeuniversal. I conclude that the brain-in-vat scenario shows that the brain in vat either has an impoverished content in comparison to us (and hence fails as a Twin-Earth scenario) or that it represents shapes and other geometrical entities just as we do. Acknowledgments Thanks are due to Ingvar Johansson, Kristoffer Sundberg, Christer Svennerlind and an anonymous referee for valuable comments on a previous version of the paper. The writing of this paper was funded by The Swedish Research Council (Research Grant 2017-02546). References Bennett, David J. 2011. 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