Mirror Realism
Matt Leonard
Let’s suppose that spacetime substantivalism is true. In other words, space- time regions exist independently of the objects that occupy them. They shouldn’t be thought of as mere ‘things’ or ‘events’ which happen to stand in spatiotemporal relations.1 Even with such an assumption, there is a lot to disagree about. And so, for the moment, let’s make a few more assumptions: let’s suppose that (1) material objects exist independently of spacetime re- gions (i.e., material objects are not identical to spacetime regions) and (2) material objects are exactly located at certain spacetime regions. Given this setup, it seems right to say that material objects and spacetime regions pos- sess mereological and topological properties, and enter into mereological and topological relations. Now consider the following question about the relation- ship between spacetime and material objects: are material objects a perfect mirror of their spacetime regions? We might ask: if two objects share some mereotopological relation (for instance, if one is a part of the other, or if one is connected to the other), does it follow that the regions of space at which the objects are located, share the same relation? Similarly, if two regions of space are related in some mereotopological way, are the objects located at those regions, related in the same way? If an object x is a proper part of some other object y, does it follow that the location, lx, of the former object is a proper part of the location of the latter object, ly (and conversely)? Likewise, if some spatial region, lx, is a proper extension of some smaller spatial region, ly, does it follow that the object located at the former region, x, is a proper extension of the object located at the latter region, y (and conversely)? Let mirror realism be the view that answers affirmatively.2 In other words, it
1We can further suppose that space is Euclidean. Spacetime points are instantaneous, spatially unextended concrete particulars. A spacetime region is taken to be any non-null collection of such points.
2As far as I know, the view as I am conceiving of it lacks a name. Uzquiano (2011) refers to the view, restricted to mereology, as mereological harmony, which was actually
1
is the view that, for all objects and all spatial regions, the mereotopologi- cal structure of objects and the mereotopological structure of their spatial regions are equally rich. Given that we’ve assumed a sort of dualist substan- tivalism, we can think of reality as comprised of two levels: the object-level and the region-level. The object-level is located at the region-level. And so the mirror realist will contend that, given some objects and their locations, an arbitrary mereotopological relation holds at the object-level iff it holds at the region-level.
If you think that mirror realism is correct, then one natural thing you prob- ably want to do is add some sort of mirroring axioms to your theory of location. In other words, you might think that it’d be nice if you had an ax- iom, instances of which capture cases of mirroring in the world. It turns out that this is really hard to do without eliminating certain metaphysical pos- sibilities. Now, it’s bizarre that adding mirroring axioms, which seem prima facie intuitively satisfying, rule out these cases - since the axioms seem to be irrelevant to the cases they end up ruling out. I can think of no real satisfying way of patching up the axioms, and thus, I think that we are left with the odd result that we can’t generally capture these cases of mirroring in our theory of location, with eliminating some substantial metaphysical possibilities.
The plan for the paper is as follows. First, I’ll give some preliminaries. I’ll provide some intuitive reasons for why it makes sense to be a mirror real- ist. I’ll briefly mention why philosophers have thought that (something like) mirror realism is true. Second, I’ll provide some natural axioms which cap- ture cases of mirroring. I’ll also consider some mirroring axioms provided by Varzi (2007). Third, I’ll show the logical relationships between my mirrorings axioms and Varzi’s mirroring axioms. Fourth, I’ll show what metaphysical possibilities are ruled out by the axioms. I take these excluded cases as, collectively, a reductio against the idea of introducing mirroring axioms. I think we are simply left in the awkward position of being unable to formally dubbed by Schaffer (2009). Saucedo (2011) refers to something like this view as alignment, though he distinguishes between internal and external alignment (which I’ll discuss later). Neither of them consider topological properties or relations. I think that in addition to mereological mirroring, there are other sorts of mirroring, i.e., topological. I take mirror realism to be the stronger view that the mereotopological relation holds at the object-level iff it holds at the region-level.
2
account for object-region mirroring without ruling out plausible (to some at least) metaphysical possibilities. Fifth, I’ll mention an interesting note about supersubstantivalism and mirror realism. Above I made a few assumptions. In addition to substantivalism, I assumed (1) and (2). A number of philoso- phers have recently denied such assumptions. In particular, philosophers have maintained that material objects are in fact identical to their spacetime regions. This is often dubbed supersubstantivalism (or, monistic substanti- valism). Now, one common objection to supersubstantivalism is that it rules too many metaphysical possibilities. Interestingly, it turns out that intro- ducing mirroring axioms rules out the same set of metaphysical possibilities that supersubstantivalism rules out. Thus, if you’re a dualist about space- time and material objects, it looks like if you want some mirroring axioms then you’re stuck with the same problem as the supersubstantivalist.
1 Preliminaries
I begin with some preliminaries. I’ll first say a little bit about why the view might seem attractive. Then, I’ll mention a few technical notes.
1.1 Why be a mirror realist?
Why be a mirror realist? You might wonder whether or not mirror realism is a desirable view. It’s not my intent in this paper to defend the view. But let me try to provide the view with a first pass defense. The first reason is based on our intuitions about the relationship between material objets and spacetime regions. If this doesn’t sell the view, I’ll at least mention why some philosophers have been interested in the view.
1.1.1 The initial (though controversial) intuition
When we ask, What is the relationship between the mereotopological structure of objects and the mereotopological structure of their locations?, one intuitive answer is the following: you have the object-structure iff you have the region- structure; two objects are mereotopologically related iff their locations are mereotopologically related. One direction seems plausible enough. Consider the following case.
3
Case 1. Consider my arm and my body; intuitively, if my arm is a proper part of my body, then it seems to follow that the spatial region which hosts my arm is a proper part of the spatial region which hosts my body.
If my arm is a proper part of my body, then how could it not be the case that my arm’s location is a proper part of my body’s location? This direction at least seems initially plausible. Now consider the converse.
Case 2. Consider a hockey stick and a puck; intuitively, if the spatial region which hosts the puck and the spatial region which hosts the stick are connected, then the puck and the stick should be connected, as well.
At least in this case, this seems intuitively true. Though this case seems harmless, the careful reader will notice that this direction is not so easy to swallow. Perhaps the most substantial problem for the mirror realist concerns her commitment to so-called expansions of objects. An expansion of an object o is an object o′ which fills a superregion of o. This direction of mirror realism seems to commit us to all sorts of such expansions. Consider the spatial region which corresponds to the volume of my body and the spatial region which corresponds to the volume of my body together with an additional outward two feet in every direction. Since the former region is a proper part of the latter region, does it follow that I am a proper part of the object which the latter regions hosts? “No!” you might say. “There just isn’t an object occupying that spatial region; my ontology isn’t that bloated.”3 There is another immediate problem for the mirror realist. In addition to mirror realism’s commitment to expansions, it also seems to commit us to all sorts of contractions of objects. A contraction of an object o is an object o′ which fills a subregion of o. Consider an arbitrary spatial region, which we’ll call Tim, located within the spatial region which corresponds to my body, say a 3-dimensional spherical region in the center of my body’s location. Since Tim is a proper part of my body’s location, it follows that there is an object located at Tim, which is a proper part of my body. But must I believe that there are all sorts of contractions? This is strikingly similar to van
3[Note: What to do with this note?] There are quite a few other results which might seem implausible. Varzi (2007) notes that the view seems to exclude the possibility of extended simples and Uzqiuano (2011) indicates that the view secures that atomistic space is an inhospitable environment for material gunk and that Whiteheadian space is an inhospitable environment for unextended material simples.
4
Inwagen’s Doctrine of Arbitrary Undetached Parts. Some might find both such expansions and contractions totally plausible. But most, I presume, will not.
1.1.2 Why have philosophers cared about mirror realism?
In light of these commitments, many philosophers have recently considered something like mirror realism: see Schaffer (2009), Skow (2005), Varzi (2007), Uzquiano (2011), and Saucedo (2011).
The first motivation for thinking that mirror realism is true comes from su- persubstantivalists. If, with Schaffer (2009) and Skow (2005), you think that material objects are simply identical to their spacetime regions, then mirror realism seems to trivially follow. On a supersubstantival picture of the world, it seems that two objects will be mereotopologically related iff their locations are.4 A lot of people think supersubstantivalism is an attractive view, and so a lot of people will think that mirror realism is an attractive view, as well.
Third, you might think that mereotopological relations on material objects should be explained by (or, are just reducible to) spatial relations on the regions at which they are located.5 You might just think that what it is for some material objects to enter into a mereotopological relation is for their exact locations to enter into that relation. At any rate, the mirror realist has a lot to explain and a lot to defend.6
4A supersubstantivalistsmight entertain the idea of there being empty regions, i.e., regions of spacetime which are not identical to material objects. On this softer super- substantivalist view, all material objects are identical to spacetime regions, but not all spacetime regions count as material objects. What differentiates between material objects and empty regions is that the former instantiate the intrinsic property of ‘being suitably matter-filled’. See Uzquiano (2006) for a discussion of this possible supersubstantivalist option. Nonetheless, the more natural form of supersubstantivalism would not make this distinction.
5See Uzquiano (2011) for a brief discussion of this motivation, though as mentioned above, restricted to mereological relations.
6However, some sort of moderate mirror realism might be more tenable. A more moder- ate position might claim that for all objects and all spatial regions, the mereotopological∆ structure of objects and the mereotopological∆ structure of their spatial regions are equally rich, where the mereotopological predicates in question are restricted somehow to some set of predicates ∆. In my opinion, the best restriction (for the moderate mirror realist) is yet to be formulated; there is a lot of work to be done in determining the scope of ontological
5
1.2 Technical preliminaries
Let me quickly make a few technical introductory remarks regarding (i) the material world, (ii) mereotopology, (iii) location theories, and (iv) mirroring principles.
1.2.1 The material world
For simplicity, and without loss of generality, in what follows let’s assume we’re modeling a three-dimensional world and not worry about spacetime, hyperspace, or any other (for present purposes) unnecessary complication.7 In particular, let’s not worry about temporal-indexing.
1.2.2 Mereotopology
I don’t want to assume too much about the mereotopology. In other words, I don’t want to assume too much about the parthood and connection (primi- tive) relations. I’m more interested in providing general mirroring principles, which can in principle, be applied to most mereotopologies.8 However, let me provide a brief introduction to mereotopology.
Very roughly, mereotopology is the theory of parts, wholes, and their contact. It is built up by adding a topological component to a mereological theory. Having such a theory allows us to say much more about parts and wholes; we can talk about the boundaries and interiors of wholes, the relations of contact and connection, the concept of surface, point, etc. In this section, I will rehearse one mereotopological theory. Not only will briefly spelling out a formal mereotopological theory provide some necessary background for the paper, but it will allow us to see the sort of formal theory to which we can add our mirroring axioms. There are quite a few mereotopological theories in the literature: Whitehead (1929), Grzegorczyk (1960), Smith (1993), Smith (1996). I will lay out Casati and Varzi’s (1999) theory, which is arguably mirroring. 7In principle, ‘L’ can take a four-dimensional spacetime slice or a five-dimensional hypertime slice in its first argument slot, and a spacetime or hypertime region in its second argument slot. But for the purposes of this paper, let’s keep things simple.
8See Whitehead (1929), Grzegorczyk (1960), Smith (1993), Smith (1996), and Casati and Varzi’s (1999) for some theories of mereotopology.
6
one of the most intuitively satisfying.9 They dub their theory General Ex- tensional Mereology with Closure Conditions, or GEMTC.
The mereological component of the theory is just general extensional metrol- ogy, or GEM. Before I give the axioms of GEM, let me provide a few mereological definitions. In GEM, we take as primitive the predicate ‘P’, and let it denote the parthood relation. Once we take ‘P’ as primitive, we can define all sorts of interesting mereological notions:
Oxy=df ∃z(Pzx∧Pzy) Dxy =df ¬Oxy PPxy =df Pxy ∧ ¬Pyx PExy=df ¬Pxy∧Pyx
Overlap Disjointness Proper Parthood ProperExtension
GEM is the (A.1) (A.2) (A.3) (A.4) (A.5)
(A.1) − (A.3) assure us that P is a partial order. (A.4) assures us that if you are not a part of something x, then you have a part which does not overlap x. (A.5) assures us that every specifiable non-empty set of entities has a sum.
To attain GEMTC, we add some topological axioms. We take ‘C’ as our topological primitive and let it denote the relation of connectedness. Roughly,
9Well, I will lay out most of it, leaving aside some unnecessary aspects, for our present purposes
7
theory constituted by the following axioms:
Pxx Reflexivity
(Pxy ∧ Pyz) → Pxz (Pxy ∧ Pyx) → x = y ¬Pyx → ∃z(Pzy ∧ ¬Ozx) ∃wφw → ∃z∀w(Ozw ↔ ∃v(φv ∧ Ovw)) Unrestricted Sum
Transitivity Anti-symmetry Strong Supplementation two things x and y are connected if they either overlap or are touching. No- tice how we can use our primitive to define all sorts of interesting topological notions:
Exy =df ∀z(Czx → Czy) IPxy=df Pxy∧∀z(Czx→Ozy) TPxy =df Pxy ∧ ¬IPxy
In GEMTC, ‘C’ is reflexive, symmetric, and monotonic:
Enclosed InternalPart Tangential Part
Reflexivity
Symmetry
Monotonicity
(A.6) (A.7) (A.8)
Cxx Cxy → Cyx Pxy → Exy (or, Pxy → ∀z(Czx → Czy))
To completely attain GEMTC we need to add a few more axioms regard- ing interiors of objects, which I will omit in this section because it requires quite a bit of machinery that is unnecessary for the purposes of this paper. What we’ve seen so far is how a mereotopological theory works and what kind of axioms it possesses. If we want to capture the phenomenon of onto- logical mirroring, we’ll want axioms which will fit nicely with a theory such as GEMTC.
1.2.3 Location theories
In offering theories of location, philosophers have recognized different con- ceptions of what it is for an object to be located at a region of spacetime. It is controversial which conception to take as primitive, so let me briefly note two ways of proceeding. First, you might with Parsons (2006) take weak location to be primitive, where to be weakly located at a region is to be located at a region that is not completely free of you. For example, if you are standing in a hallway with your arm hanging out of the hallway window, you are weakly located at the hallway, at the region immediately outside the hallway’s window, the particular city and state you are currently in, and so forth. Parsons defines other conceptions of location in terms of weak loca-
8
tion (when we quantify over ‘s’, we are quantifying over regions of spacetime): x@◦r x@r=df ∀s(r◦s=⇒x@◦s) x@r =df ∀s(r ◦ s ⇐⇒ x@◦s)
Weak Location
EntireLocation
PervasiveLocation
Exact Location
Given these definitions, Parsons entertains the following axioms for a theory of location: (x@r ∧ x@s) =⇒ r = s Functionality ∃r(x@◦r) =⇒ ∃r(x@r) Exactness
In this paper, I take another approach. I, along with Varzi (2007), take ‘L’ as a primitive dyadic locative predicate with the following intended meaning: x is exactly located at y, i.e., x is located at the spatial region y which exactly corresponds to the volume of x. With exact location as our primitive, we can define other notions of location in the following way:
GLxy=df ∃(Ozy∧Lxz) ELxy=df ∃(Pzy∧Lxz) ULxy=df ∃(Pyz∧Lxz)
Generic Location Entire Location Ubiquitous Location
More informally, to be generically located at a region is to be located at a region that is not completely free of you. To be entirely located at a region is to be located at a region and to not have a part that is not located at that region. To be ubiquitously located at a region is to completely fill up the region, so to speak.
However, let’s not assume a specific theory of location; in other words, let’s not assume any particular axioms. I’m interested in providing mirroring
9
principles which, in principle, can be added to just about any theory of location.10
1.2.4 Mirroring principles
I am interested in introducing mirroring axioms of a certain sort. In partic- ular, I am interested in axioms with schematic mereotopological predicates. Uzquiano (2011) and Saucedo (2011) have entertained specific mirroring ax- ioms. Uzquiano, for instance, has conceived of such principles in the following way. Assume that each and every material object has exactly one location. We can characterize mirror realism (or “mereological harmony”, as Uzquiano calls it) in terms of a total, bijective function, p, from material objects to their exact locations. Thus, Uzquiano characterizes mirror realism by offer- ing a set of mirroring axioms (let ‘≤’ be the parthood relation, ‘◦’ the overlap relation, ‘