Extended Simple Objects

Extended Simple Objects∗ Jonathan Erhardt October 9, 2012 Abstract In this paper I explore the possibility of spatially extended mereological simples, i.e. of objects which have no proper parts and occupy an area of space which consists of more than one point of space. Various metaphysicians have defended the possibility of such simples recently. Such accounts face an objection which goes back at least to Descartes. Two different strategies have been pursued to defend the possibility of extended mereological simples against this objection. One strategy is to give up the Doctrine of Arbitrary Undetached Parts (DAUP)1 which roughly states that any spatially extended object has parts that correspond to the parts of the region that it occupies.2 The other strategy is to give up the idea that space is continuous and instead adopt the idea of smallest discreet packages of space. This second strategy only saves a specific type of extended simples, namely simples of the size of those smallest packages of space. In this paper I propose a third way to respond to the objection. This way is based on a positive account of how a simple object can be extended and allows us to retain DAUP and leave the question open whether space is continuous or discreet. And it allows simples of any size. The cost is that it introduces modifications of the geometry of space. Roadmap: In the next section I suggest why it might be interesting to reflect on the possibility of extended simple objects. This should motivate the defense of extended simples enough to have a look at the most famous objection in section two. In section three I sketch the two traditional ways of rejecting the objection. In section four I develop a positive account of extended simples and demonstrate how it can withstand the objection. ∗ I’m very grateful to Dominik Aeschbacher and Adriano Mannino for valuable comments. The last section is an answer to an objection by Adriano Mannino. Thanks also to Simon Dürr for discussions on the mathematical possibilities of modeling non-extended simples. 1 Van Inwagen 1981, p. 75. 2 Or some other principle to the same effect, e.g. Simons geometric correspondence principle, Simons 2004. 1 1 Extended Simple Objects Spatially extended simples are fascinating and interesting for several reasons. Some people (myself included) are mereological nihilists and believe that only a big number of simples exist. But they would rather not give up the idea of extended objects and are thus forced to accept the possibility of extended simples. Then there are the monists who think that the one big thing, the universe, is extended. Yet others think it’s an interesting feature of the physical world that extended simple objects are the basic stuff of the universe and thus want to investigate them in a metaphysical fashion. All those and certainly many others have a strong interest in demonstrating the possibility of extended simples and defend them against objections. 2 The Standard Objection Those of us who think there are extended simples in our world and those who simply cherish the possibility of such simples are troubled by arguments which purport to show that it is impossible that there are such things. In this section I will discuss the perhaps most famous objection against the possibility of spatially extended simple objects and try to come up with the strongest version of this objection. The argument can be put like this: (1) If there were an extended simple, then it would have two halves. (2) If it had two halves, then the simple would have proper parts. (3) Hence, there can’t be any extended simples. Versions of this argument have been discussed by Descartes3 , McDaniel4 and Markosian5 . Let us have a short look at the premises of this argument. McDaniel suggest that (2) is to be understood as an analytic truth. This seems plausible and in accordance with the usual meaning of the terms ‘halve’ and ‘proper part’: Being a halve of y means being a proper part of y and having some additional properties which need not concern us here (perhaps occupying half of the volume of y or something alike). Thus every halve of y is a proper part of y but not every proper part of y is a halve of y. If (2) is an analytical truth and (3) is supposed to be a substantial metaphysical statement, then (1) has to be a substantial metaphysical statement. But given our and McDaniel’s understanding of (2) we have a problem with the analyticity of (1). If our above considerations about the meaning of halves were correct, then the following proposition is equivalent to (1): (1a) If there were an extended simple, then it would have two proper parts, which both have some additional properties. 3 Descartes 1985, p.231. From which I borrow this exact formulation of the objection, McDaniel 2007, p. 137. 5 Markosian 1998. 4 2 But is (1a) a substantial metaphysical statement? One might argue that given any plausible understanding of ‘simple’ this statement is an analytical falsehood. And such a falsehood is clearly not a substantial metaphysical statement and also ruins the argument before it gets off the ground.6 Or one might say that (1) is an unintelligible statement. In both cases (3) is itself not a substantial metaphysical statement. Is there a better way of formulating and understanding the argument? The following reformulation avoids this problem:7 (4) For every object x, if x is extended, then it has two halves. (5) For every object x, if x has two halves, then x has proper parts. (6) A is a simple and is extended. (7) A is a simple and has proper parts. In this reformulation of the argument (4) is clearly an understandable and synthetic statement. (7) is on any plausible understanding of ‘simple’ a logical contradiction and hence the argument is a reductio of at least one of its premises. (5) may now be understood as an analytic statement without rendering the denial of either (4) or (6) metaphysically uninteresting. Since (6) is the assumption to be reduced, (4) is the substantial premise. So the defender of the possibility of extended simple objects has to reject (4). 3 Two Ways Out Unfortunately (4) is motivated by some fairly plausible metaphysical principles. One of these principles is the Doctrine of Arbitrary Undetached Parts:8 [DAUP] For every material object M, if R is the region of space occupied by M at time t, and if sub-R is any occupiable sub-region of R whatever, there exists a material object that occupies the region sub-R at t. 6 McDaniel says that (1) isn’t an analytic claim and isn’t necessarily false, McDaniel 2007, p. 138. Perhaps he is evaluating (1) solely according to the truth table of propositional logic, then his assessment concerning the possible truth of (1) is correct: (1) is true if any only if there are no extended simples. But ‘analyticity’ isn’t a notion of logic and some statement may not be necessary false for logical reasons but still be an analytical falsehood, e.g. the statement ‘if Doug is a bachelor then he is married’. It is not clear to me how we can understand (1) as a synthetic statement. Be that as it may, the following reconstruction of the argument will avoid this problem and leave the rest of his considerations intact. 7 Alternatively we could read (1) in such a way that it is an analytic truth and then let (2) be the metaphysically substantial premise. Which road we pick does not matter for the rest of the paper, both avoid the problem of an either analytically false or else unintelligible premise (1). For the rest of the argument I will work with the reformulation of the argument consisting of propositions (4)-(7) because its terminology is closer to the metaphysical principles which motivate the metaphysically substantial premise. 8 I will use van Inwagen’s formulation of the doctrine, van Inwagen 1981, p. 75. 3 DAUP does no logically imply the truth of (4), for it only implies that there are material objects which occupy subregions of the region occupied by x. But it seems highly plausible that those material objects are proper parts of x and thus DAUP strongly motivates (4). To me DAUP seems initially plausible due to the following modal considerations (MC): [MC] If a material object M is extended and occupies the region of space R, then we can reflect about that portion of M which occupies some subregion of R and isolate it in our mind. It seems intuitively plausible that God could have created only that portion of M instead of creating all of M. Why should the existence of that portion of M somehow depend on whether some other adjunct region of space is occupied by something? We might be tempted to conclude that it is possible that only this sub-region of R could be occupied by M and that the other portions of M could be destroyed. And if the existence of those individual portions of M is independent in this way, then it seems to follow that those portions are themselves numerically distinct material objects. The defender of (the possibility of) extended simples now faces the following situation: He has to reject proposition (4) in order to resist the standard objection. But (4) is motivated by DAUP which is in turn based on the modal consideration in MC. What are his options? The first way: A first way to reject (4) and thereby resist the standard objection is to reject DAUP in cold blood. This is the path that has been chosen by van Inwagen9 and McDaniel10 . Rejecting DAUP entails that something is wrong with MC. Metaphysicians taking this route might be adherents of some form of modal skepticism (as is Peter van Inwagen) and think that in this case our modal intuitions simply go wrong. After all this is one of the trickier philosophical applications of our modal intuition and if we are prepared to admit that sometimes our modal intuitions go wrong, then we should admit that this is potentially one of these cases. The case for this option can be strengthened by adding further arguments against DAUP, such as McDaniel’s case against DAUP based on the extrinsic theory of shapes11 or van Inwagens argument based on the transitivity of identity12 . The second way: Another option has been proposed by Braddon-Mitchell and Miller.13 They point out that DAUP has the logical form of a conditional and that one part of the antecedent is that there is a region of space sub-R which is a occupiable sub-region of R, the region occupied by M. But what if there is no occupiable sub-region of R? Braddon-Mitchell and Miller draw our attention to a two-dimensional extended simple object M1 . This object is a little square of the dimension Planck length by Planck 9 Van Inwagen 1981. McDaniel 2007, p. 141. 11 McDaniel 2007, p. 140. 12 Van Inwagen 1981, p. 77. 13 David Braddon-Mitchell 2006. 10 4 length. They point out that physicists tell us that we cannot divide up space into any finer-grained regions than those constituted by Planck squares.14 This allows them to deny that the antecedent of DAUP is ever satisfied for entities like M1 : “[I]n principle, there cannot be anything that occupies the sub-regions of such a square. For space-time is a macroscopic property at the scale of Planck squares and up.”15 Therefore they may endorse DAUP or any principle to the same effect and still reject (4) and resist the standard objection. It should be noted though that this second way also entails some form of modal skepticism: The modal considerations in MC do not stop at Planck length, nothing concerning conceivability changes at regions below the size of objects like M1 . This means that defenders of this defense also have to reject MC. 4 The Third Way: Simples and The Geometry of Space What both of these ways have in common is that they are not positive accounts of how extended simples are possible.16 They are mere defenses against the standard objection. In what follows I develop a positive account of how extended simples are possible. We can then deduce ways of avoiding the standard objection from this positive account. And this account will also explain our mistaken modal intuitions in MC. 4.1 A Geometrical Formulation Figure 1 shows a two-dimensional physical space. The vertical and horizontal lines represent paths which moving physical object take if no physical force is applied to them. The space depicted in figure 1 is euclidean and the axiom of parallels holds: through a given point there is one and only one parallel to a given straight line which does not go through the given point, i.e., one straight line which lies in the same plane with the first one and does not intersect it.17 Now let us add a non-extended simple object O in the center. Any physical object traveling on the black lines will not collide with O. Only paths leading directly through the point of space occupied by O (e.g. the line m) will lead to a collision between a moving physical object and O. Any path parallel to such a path, no matter how close it is, will not lead to a collision. Let us assume that the following will happen if a physical object collides with O: It will take the same way back it took to get to O and O itself will not be affected by this. O will not change its position, it is a fixed object. Let us also assume that collisions are the only physical interactions in this world. In such a world we have two simple ways to measure the extension of O. We can either shoot infinitely many unextended objects on horizontal lines from L towards O and measure which ones of these will be reflected back to us. If more than one object will be reflected 14 See David Braddon-Mitchell 2006, for further references. David Braddon-Mitchell 2006, p. 224. 16 Although it has to be mentioned that McDaniel’s extrinsic theory of shapes, which has not been discussed here, allows something like a positive account of how extended simples are possible. McDaniel 2007. 17 Reichenbach 1958, p. 2 f. 15 5 Figure 1: A two-dimensional physical space with a non-extended simple object O at the center then O is extended. If only one will be reflected then O is a non-extended object. The other way is to shoot infinitely many unextended objects on horizontal lines towards O and measure on the other side which ones make it past O and collide with R. If more than one object does not collide with R, then O is extended, else O is not extended. Measuring the extension of O in figure 1 will tell us that O is a non-extended object, because the only object which will collide with O is the one moving along the straight horizontal line m through the point of space occupied by O. Now suppose the following happens: O acquires the ability to change the space around it from a euclidean to a non-euclidean space by collapsing the space in its vicinity onto a single point. In other words: O acquires the ability to reduce the distance between distinct points of space in its vicinity to zero. This would result in a rejection of the axiom of parallels: For some points and a line through the point of space occupied by O, there is no parallel going through said point without intersecting the line going through the point of space occupied by O. This situation is illustrated in figure 2. Figure 2: A two-dimensional non-euclidean physical space with a simple object O at the center The lines in figure 2 still represent the paths of objects which are not subject to any physical force. But now several lines which used to be parallels intersect at O. As a result of this our two methods for measuring the extension of O would tell us that O is 6 an extended object. But O is still a simple object and so we have an account of how O can be both extended and simple. Let us call this way of being extended ‘non-classical extension’. We can sum up the difference between those two views in the following slogan: An object has classical extension in virtue of having different proper parts at different points of space, while an object has non-classical extension in virtue of being completely present at more than one point of space. The difference is again illustrated in figure 3. Figure 3: Classical extension on the left and non-classical extension on the right 4.2 A Spatio-Mereological Formulation Some might object that non-classical extension is not real extension at all, it’s merely a form of ersatz extension: By modifying the geometry of space we made a point-like simple object look like it is extended, but as a matter of fact it is not. To answer this objection we have to get clear on the conditions of being truly extended. I take it that an object O is truly extended if and only if it stands in the occupy relation to more than one point of space. In what follows I want to argue, partly on mereological grounds, that non-classically extended objects fulfill this condition and are thus truly extended. I will first explore whether Classical Extensional Mereology (CEM) will let us demonstrate that non-classical extension is true extension. I argue that CEM does not have the resources to demonstrate the (classical or non-classical) extension of simples and perhaps not even the extension of objects with proper parts. I then expand CEM into a more powerful system and argue that this new system lets us demonstrate that non-classical extension is true extension. CEM contains a number of concepts which suggest an intimate relation to spatial concepts, e.g. overlapping and disjointness. Perhaps we can use those concepts to demonstrate that an object is truly extended. Let us have a closer look at those concepts. In the notation of Leonard and Goodman’s Calculus of Individuals “x overlaps y” is written as x ◯ y and “x is disjoint from y” as xíy. In CEM those concepts can be defined in terms of parthood:18 18 Simons 1987, p. 28. I will follow Simons notation and use corners as scope-marking brackets. I will 7 D1 x ◯ y ≡ ∃z⌜z < x ∧ z < y⌝ D2 xíy ≡∼ x ◯ y Parthood (“x is a part of y” is written as x < y) itself can be defined in terms of identity and proper parthood (“x is a proper part of y” is written as x ≪ y), which are then taken to be primitive concepts: D3 x < y ≡ x ≪ y ∨ x = y Could identity and proper parthood help us determine whether an object is extended? In the case of identity the answer seems to be “no”. Even entities which supposedly exist outside of space and time, such as universals or numbers, are taken to be self-identical. What about the concept of proper parthood? Does the proposition x ≪ y entail that y is extended? One way to answer this question is by finding a situation in which an object has a proper part but is clearly or possibly not extended. One candidate is the following situation: An object O has only one proper part O’ which overlaps O. In such a scenario there are two co-located objects, O and O’, which stand in the proper parthood relation. Nothing then guarantees that O occupies more than one point of space: They both occupy the same space, but whether that space is a single point of space or more than one point of space is not determined by the proper parthood relation in this scenario. In this scenario it is thus possible that an object has a proper part but is not extended. But such a situation is usually ruled out as unintelligible or at least not compatible with the usual notion of proper parthood: an object cannot have only one proper part. There are various ways to rule out such situations and for the present purpose we will use an axiom which is known as the Weak Supplementation Principle:19 A1 x ≪ y ⊃ ∃z⌜z ≪ y ∧ zíx⌝ A1 guarantees that every object with proper parts has at least two proper parts. Does A1 also guarantee that objects with proper parts are extended? The general intuition seems to be that it does20 and it is easy to see why: If an object has at least two disjoint proper parts, then, so the intuition, those parts have to be at different places. And an object is where its proper parts are.21 It’s not clear to me whether this is true. As mentioned above disjointness is itself defined in terms of parthood and it thus not to be confused with the ordinary language concept of disjointness (although Simons mentions only an example from ordinary language: Most human beings are disjoint from each other22 ). Perhaps CEM could also describe a universe without physical space, it’s not entirely clear to me whether non-spatial entities could stand in the proper parthood occasionaly omit the brackets if it helps the readability and if there is no danger of misunderstanding. It turns out that A1 is too weak to rule out other objectionable constellations, but those constellations will not matter for the present purpose. I mention A1 because it is clearer and more intuitive than its stronger rivals. 20 At least that’s the result of an informal survey conducted among friends of mine. 21 E.g. Yagisawa 2010, p. 32. 22 Simons 1987, p. 13. 19 8 relation to each other. If proper parthood is also possible for such objects, then A1 does not guarantee the spatial extension of objects with proper parts and the common intuitions would be mistaken. But even if we assume that the concept of proper parts guarantees extension for objects with proper parts this is of no use for our case. Simples are objects without proper parts and thus we can’t use this way of demonstrating the extension of objects within the framework of CEM. We need a stronger language to deal with simples. Let us work out a sketch23 of such a language and let us call it SM for spatial mereology. We first need a sufficiently strong logical foundation. Which axiom set for first-order predicate calculus we pick does not matter: SMA1 Any axiom set sufficient for first-order predicate calculus with identity. Then we add definitions of parthood, overlapping and disjointness: SMD1 x < y ≡ x ≪ y ∨ x = y SMD2 x ◯ y ≡ ∃z⌜z < x ∧ z < y⌝ SMD3 xíy ≡∼ x ◯ y We take proper parthood as a primitive concept and add a new primitive concept: ‘x ⊚ y’. The variables x and y take either individuals or points of space as values. We signify this by using the letters x, y, z for variables taking individuals as values, and x’, y’, z’ for variables taking points of space as values. Depending of the combination of these two types of variables we read ‘⊚’ differently: x ⊚ y - “x and y are co-located” x ⊚ y ′ - “x occupies y’” x′ ⊚ y - “y occupies x’” x′ ⊚ y ′ - “x’ and y’ are collapsed/co-located” Given this understanding of the four different combinations we can lay out some axioms: SMA2 ⌜x = y⌝ ⊃ ⌜x ⊚ y⌝ SMA3 ⌜x′ = y ′ ⌝ ⊃ ⌜x′ ⊚ y ′ ⌝ SMA4 ⌜x < y ∧ x ⊚ z ′ ⌝ ⊃ y ⊚ z ′ According to SMA2-SMA3 if two objects are identical or two points of space are identical, then they are co-located. SA4 says that if an object O occupies a point of space and is part of another object O’, then O’ also occupies that point of space. It 23 We will not go beyond a sketch of this language because it is not the aim of the present paper to develop a logical system. We have to develop this language only up to the point where we can prove the theorem(s) necessary to demonstrate that non-classical extension is true extension. 9 is a formal formulation of the idea that composed objects are where their (proper or improper) parts are. The next axiom might seem more controversial, but I think it is not the axiom itself that is problematic, but the question whether the antecedent is possible. I will address this question in the last section. SMA5 ⌜x ⊚ y ′ ∧ y ′ ⊚ z ′ ⌝ ⊃ x ⊚ z ′ SMA6 ⌜x ⊚ y ∧ x ⊚ z ′ ⌝ ⊃ y ⊚ z ′ According to SMA5 if an object occupies a point of space y’ and y’ and z’ are collapsed or co-located, then the object also occupies z’. The most uncontroversial instance of this is the case in which y’ and z’ are identical (see also SA3). The same seems to hold if two distinct points y’ and z’ of space are co-located, it seems clear to me that the object occupying y’ also occupies z’. It might be less clear though whether it is possible that two points of space can be collapsed in such a way. We will discuss this question later. SMA6 is the mirror-image of SMA5. In this case two objects O and O’ are co-located, and if O occupies one point, then O’ occupies that point also. With this more powerful language we can prove some interesting theorems24 : SMT1 ⌜x ⊚ x′ ∧ y ⊚ y ′ ∧ x′ ≠ y ′ ∧ x ≪ z ∧ y ≪ z⌝ ⊃ ⌜z ⊚ x′ ∧ z ⊚ y ′ ∧ x′ ≠ y ′ ⌝ SMT1 expresses what we couldn’t express in CEM, namely that an object with different proper parts at different points of space does itself occupy these points of space, i.e. it is extended according to our criterion of extension. But the following theorem is more important to us: SMT2 ⌜x ⊚ y ′ ∧ y ′ ⊚ z ′ ∧ y ′ ≠ z ′ ⌝ ⊃ ⌜x ⊚ y ′ ∧ x ⊚ z ′ ∧ y ′ ≠ z ′ ⌝ It is of particular interest to us because there is only one variable which takes objects as values and no proper or improper parthood is involved. And yet the consequent of the conditional says that this object is extended according to our criterion of extension: x occupies more than one point of space. This means that SMT2 can be applied to find out whether a simple object is extended. Let’s apply SMT2 to a case of non-classical extension and see what happens. In cases of non-classical extension we have a simple object which occupies a single point of space. But that object then collapses the points of space in its vicinity onto a single point of space. For the purpose of demonstrating the extension of that object we can restrict ourselves to one additional point of space besides the one originally occupied by the object, for an object is extended if it occupies more than one point of space. This means that we have three premises following from the scenario: a ⊚ a′ , a′ ⊚ b′ , a′ ≠ b′ . Together with SMT2 they yield the desired result: We connect our three premises with conjunction introduction and then apply conditional elimination to the resulting conjunction together with SMT2 to arrive at a ⊚ a′ ∧ a ⊚ b′ ∧ a′ ≠ b′ where ‘a’ is the name of the simple object. Thus a fulfills our condition for true extension. 24 I will not actually go through the proof because it is simple enough and should be rather clear. 10 This means that given the axioms of SM the ersatz-extension objection fails. The objector might of course reject some of the axioms of SM, but it’s hard to see which ones. 4.3 Non-Classical extension and DAUP In this section I want to argue that non-classical extension makes extended simple objects and DAUP logically compatible: Non-classical extension allows us to have the metaphysical cake and eat it too. DAUP is the following metaphysical thesis formulated by Peter van Inwagen:25 [DAUP] For every material object M, if R is the region of space occupied by M at time t, and if sub-R is any occupiable sub-region of R whatever, there exists a material object that occupies the region sub-R at t. DAUP is formulated as a conditional: If an extended object fulfills the antecedent of the conditional, then there exists a material object that occupies the region sub-R at t. The antecedent of the conditional deals with a sub-region sub-R of the region R, where R is the region of space occupied by the extended object M. Sub-R has to be occupiable, only then the antecedent of the conditional is satisfied. On one reading of DAUP this follows from the fact that the extended object M occupies the region of space R. If M occupies R, then it also occupies every sub-region of R.26 And if the sub-region is in fact occupied by M, then it is occupiable, for actuality implies possibility. This is especially clear if we express regions of space as sets of points of space. let R be the set of points of space which M occupies: R = {x′ ∶ x′ ⊚ M }.27 We can then make use of the axiom of specification28 and define a subset of R, for example the one occupied by the right halve of M, sub − R = {x′ ǫR ∶ x′ ⊚the right halve of M}. If sub-R is a proper subset of R, then all the members of sub-R are also members of R, which means that they are occupied by M. And again, if they are occupied, then they are occupiable. But this sense of occupiable is trivial and can’t be what van Inwagen has in mind. This understanding of DAUP would also contradict Braddon-Mitchell and Miller’s way of reconciling DAUP with extended simples, for their account rests on the claim that sub-R is not occupiable.29 It seems to me clear that ‘occupiable’ in DAUP is best understood as follows: 25 Van Inwagen 1981, p. 75. This claim rules out a type of extended objects which McDaniel calls ‘spanners’: “According to this conception, an extended simple bears the occupation relation to exactly one extended spatiotemporal region, without bearing the location relation to any proper part of that extended region.” McDaniel 2007, p. 134. I don’t think that’s a big loss. Saying that an object can occupy a region of space without occupying proper parts of that region seems to be as intelligible as claiming that you can eat a sandwich without eating its parts. 27 I assume that M doesn’t stand in the occupy relation to the set, it stands in the occupy relation to the members of the set. 28 The axiom of specifications says that to every set A and to every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. Halmos 1974, Position 101. 29 David Braddon-Mitchell 2006, p. 224. 26 11 sub-R is occupiable if and only if it is possible that there is an object which occupies sub-R and no other region of space. This understanding is very plausible because the material object in the consequent of DAUP is supposed to be an object which occupies only sub-R. But according to this understanding the individual points of space occupied by a non-classically extended object are not occupiable. According so SMA5 every object occupying a point of space y’ also occupies all the points of space which are co-located with y’. A subregion sub-R of a region occupied by a non-classically extended simple object will therefore not be occupiable in the sense required by DAUP and DAUP will not claim that there exists a material object which occupies only sub-R. DAUP will as a result of this fail to motivate premise (4) of the argument against the existence of extended simples, if those simples are non-classically extended. Thus non-classical extension allows us to resist the argument (4) to (7) by rejecting (4) while endorsing DAUP. 4.4 Non-Classical extension and MC I think non-classical extension gives a plausible account of why our modal intuitions fail in the case of extended simple objects and of what is wrong with MC. If we (or at least I) isolate some portion of an extended object in our mind as described in MC, say the right halve, then I have a visual representation of the scene in my mind: I imagine there to be an extended object in front of me and then I isolate the right halve of that object in my mind. Such a visual representation is inevitably also a spatial representation, for I imagine the extended simple to be located in front of me as extended in space. But there are well known limitations to our ability to represent or imagine noneuclidean three-dimensional space. We can easily represent two-dimensional non-euclidean space in terms of a surface on a three-dimensional object in our minds. But we lack the ability to represent three-dimensional non-euclidean space in a four-dimensional space because we have troubles representing four dimensions. I think it is this failure which leads to our mistaken modal intuitions in the case of non-classically extended simples. I think this can best be illustrated with an analogy. Figure 4 shows a situation in which an observer is put in front of a system consisting of two mirrors and an object. The mirrors are arranged in such a way that the observer from his point of view sees the same object three times, side by side, looking like one extended object. Let us assume that the observer does not know that he has mirrors in front of him and that he is not able to move his head to discern the illusion. Let us also assume that the depth of field will not reveal the illusion, perhaps because the mirrors and the object are very small. In this situation the observer will think he sees one object consisting of three proper parts (at some level of decomposition). But those three proper parts correspond simply to three copies of the same object. If he is asked to apply DAUP to this situation, he will conclude that there are three subregions of the space occupied by that object and that they are individually occupiable and that there are three material objects (all proper parts of the original object) occupying those subregions. But this is a mistake, in reality the antecedent of DAUP is not satisfied 12 Figure 4: An observer thinks he sees an object with three parts, but in reality he sees three copies of the same object because the regions of space occupied by the three instances of the same object are not individually occupiable for they are all the same regions of space. Any object occupying one region of space will therefore also occupy the other two regions of space (see SMA3 and SMA5 for the axioms necessary to prove this within SM). Something very similar is going on in the case of MC and the application of DAUP to non-classically extended simples. The most important difference between the situation depicted in figure 4 and the case of non-classically extended simples is that the role of the mirrors is played by bent space in the case of non-classically extended simples. The photons are not reflected by the mirrors, they move along a straight line, but the straight lines all end at the same location because space is bent. Our visual representation of the situation in MC is compatible with both classical and non-classical extension, just as the perception of the observer in figure 4 is compatible with both what he thinks he sees and what he actually sees. Thus the cause of our modal mistake is the following: Our mental representation is compatible with two entirely different situations and only in one of these situations our modal considerations are correct. We don’t initially notice that our mental representation is compatible with two different situations (classical and nonclassical extension) because due to psychological constraints we’re not well equipped to represent one of the situations, namely the case of non-classical extension which involves non-euclidean space. Once we see that we tacitly presupposed classical extension we can fix our mental representation and try to represent non-euclidean space and non-classical extension. If we succeed in doing this, then our modal intuitions should deliver the right result in the case of non-classical extension, just as the modal intuition of the observer in figure 4 should deliver the right result once he has discerned the illusion. 4.5 Co-located Points of Space In this last section I want to address the objection that it is either false that two or more points of space can be co-located or that it is meaningless to say so. I provide a 13 sketch of two strategies the defender of non-classically extended simples might take. The first strategy defends the possibility of co-located points of space. The second strategy denies that there is a matter of fact on whether this is possible because classical and non-classical extension are two different ways to describe the same state of affairs. I leave it for another occasion to pursue these strategies and develop a defense against the objection. Strategy 1: This strategy requires two steps. In the first step a model of a space is provided in which distinct points of space have a distance of zero. Again set-theory might be used to define the set of all points of a space A and a proper subset B of A which has the points occupied by the non-classically extended simple as members. Then a distance function on pairs of points of space of A can be defined which always delivers the distance 0 if the two points are members of B and which delivers different results if either one or both points are not members of B. Then we can study the properties of that space and determine how it is related to possible laws of nature of that space. The purpose of this step is to show that the idea is coherent and not meaningless. In the next step the defender of non-classical extension has to argue against the identity of indisceribles thesis. The idea behind this second step is to get rid of metaphysical obstacles which aim to show that it is false that two or more points can be co-located. Strategy 2: The second strategy avoids the task of demonstrating that it is possible that two or more points can be co-located. Instead followers of this strategy claim that classical and non-classical extension are merely two descriptions of the same situation and not genuine metaphysical or physical possibilities. Thus if classical extension is coherent, so is non-classical extension. They then point out that the argument (4) to (7) only works in one description (classical extension) and not in the other (nonclassical extension). They conclude that the argument must be merely an artifact of our description of reality if there are other descriptions of the same situation for which the argument does not work. References David Braddon-Mitchell, Kristie Miller (2006). “The Physics of Extended Simples.” In: Analysis 66:3, pp. 222–226. Descartes, René (1985). “The Principles of Philosophy.” In: The Philosophical Writings of Descartes. 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