1. Boundary Settings

A common geometric model holds that the boundary of a region,Footnote ¹ R, is the set of points, {p₁, p₂,…, p_n}, such that, all points within the limits of that set of points are part of R, whilst all points that are without those limits are not part of R.Footnote ² Regions of space (or spacetime) are then classified topologically as closed, open, or some partial mixture of the two. Closed regions include all their boundary points, open regions include none of their boundary points, whilst partially open/closed regions include some, but not all, of their boundary points. It is then assumed, by at least many commentators (e.g. Hudson, Reference Hudson2006), that for any of these kinds of regions, there can be a material object that might occupy that region. So, objects exactly occupying a closed region are closed objects, those exactly occupying an open region are open objects, and those exactly occupying a partially open/closed region are partially open/closed objects. This topology of material objects has been an irresistible lure for those attracted to metaphysical obscurities, drawing increased attention these last few decades, especially regarding puzzles concerning contact (cf. Zimmerman Reference Zimmerman1996; Casati and Varzi Reference Casati and Varzi1999, Ch. 5; Sider Reference Sider2000; Hudson Reference Hudson2006, Ch. 3; Weber and Cotnoir, Reference Weber and Cotnoir2015).Footnote ³ Yet, discussions have presupposed sense can be made of extending these topological distinctions from regions to material objects, such that, these peculiarly open or partially open shapes are not just space (or spacetime) oddities, but oddities of material objects too. This is unsurprising, since it would be strange to posit regions that could not be exactly occupied by material objects. Nevertheless, I am sceptical about the coherence of open and partially open/closed kinds of bounded material objects.

To understand why, consider density, a structural property of ordered sets such that, between any two distinct members, there is a third member distinct from them. Given the density of space, it may be said that a boundary forms the limit of the spatial region, R, occupied by a material object, such that, for any point, p, in the boundary set, {p₁, p₂,…, p_n}, there is no closest point, or set of points, to those limits (either inside or outside of R).Footnote ⁴ That is because, by definition, spatial density ensures there is always a further point between any two points (within, without, or between, the boundary points, or indeed, any other points). This means that, for open objects, there is no last point before their boundary that falls within the region they exactly occupy; likewise, for partially open objects at their open bounds. It is this feature of fully and partially open material objects that baffles me. (From this point on, I will treat fully and partially open material objects as akin, referring to them inclusively as ‘open objects’. For, what I have to say about the one, will apply equally to the other, lest otherwise stated.) Despite my best efforts, I can only glean the syntax, and not the semantics, of these modelled boundaries. I do not understand what it would mean for there to be no set of last point-locations up to which, and not beyond, a material object occupies, where this is not a matter of indeterminacy.Footnote ⁵ Of course, many philosophers claim they do understand this.

Is the failing my own? I think not, but such matters are notoriously hard to settle. Ideally, I need an argument to elaborate those features of open material objects impeding my comprehension. Moreover, since my complaint is restricted to material objects, and not regions, there must be some aspect (or aspects) of the former making them particularly difficult to reconcile with open topologies. One thing distinguishing regions from material objects (which will be my focus here) is that the latter can move, whilst the former cannot. Yet, as I now show, open material objects cannot move.Footnote ⁶

2. The Argument from Miraculous Transformation

Suppose there are two ways objects can occupy regions: i) directly, by being simply located at that region (i.e. not occupying that occupied region in virtue of having proper parts located there), or ii) indirectly, by occupying that occupied region in virtue of having proper parts located there.Footnote ⁷ When open material objects occupy a region in either way, we are presented with insurmountable difficulties with respect to their motion. On the first horn of this dilemma, suppose an arbitrary open material object were to occupy directly whatever region it occupies. If that open material object were to move towards its open boundaries, it must first occupy those initial boundary points before occupying any region beyond them. This follows from the continuity of motion: that an object's motion necessarily traces a continuous (without gaps) path through space from its origin to its destination (lest it not have moved, but rather teleported).Footnote ⁸

Indeed, despite space's density, a material object's open boundary points are immediately adjacent to it. Accordingly, those initial open boundary points towards which the object moves would be the first new positions occupied by the material object via an uncharacteristically discrete movement.Footnote ⁹ But then, in its first movement, the open material object must undergo a topological transformation from being open to closed at those boundary points.Footnote ¹⁰ (Otherwise, the object, in its first movement, will have moved beyond (further than) those initial boundary points, and thus failed to move continuously through space; it would, in some sense, have jumped over those initial boundary points so that those boundary points were infinitely many points deep within the region that the object now occupies.) Moreover, having closed those boundaries, it is difficult to envisage how they might ever be reopened in the material object. It is simply incredible that such slight of movement (a relational change) could affect the structure of the material object (an intrinsic change, for it is the object itself, not the region it occupies, that undergoes these changes) so dramatically. To help illuminate the kind of change that the object undergoes in this simple movement, it will be fruitful to distinguish two kinds of parthood: what John Heil calls substantial and non-substantial parthood. The former parts are independent from the composites they help compose, or should at least be treated as entities in their own right.Footnote ¹¹ And the latter parts merely describe the spatial or temporal region of a thing fully occupying it regardless of whether that region of the thing corresponds to any substantial part of it. In Heil's words:

We can then say that, before moving, open material objects have no last non-substantial punctiform parts at their open boundaries. But after moving towards one of its initial open boundary points, it miraculously gains a last non-substantial punctiform part at that initial boundary point. Likewise, for every other initial open boundary point it approaches. For clarity, the argument on this horn of the dilemma can be neatly summarised, thus:

Call this the Argument from Miraculous Transformation. My suspicion is that the main criticism of this argument will be made against P1. Some might think that obviously an open object cannot come to occupy its initial boundary points and no region beyond it, since qua open object, it cannot be that there are points it occupies such that it does not occupy any points beyond them; by definition, open objects have no last points they occupy. I agree that this is obvious, and in a sense trivial. But I am not being uncharitable; this is simply what is required for continuous motion, since, otherwise, those spatial points corresponding to the object's boundaries will be surreptitiously skipped over by the object as it traverses them.

To see that this is so, consider a one-dimensional partially-open material object, LINE. LINE occupies the region [1, 10).Footnote ¹³ For all regions that LINE can occupy in the direction of its open boundary, whilst maintaining its topology, namely any region [1 + m, 10 + n),Footnote ¹⁴ LINE has already passed 10 at that region – for any value of ‘n’, LINE occupies a region in excess of point 10. And it is not just the one point that LINE will have had to skip passing over on its journeys; for any journey it will have to have jumped over infinitely many other points between 10 and 10 + n. Since, all numbers between 10 and 10 + n will at some time in that journey have been LINE's open boundary point. So, though the argument is in some sense rather trivial, that should not make its conclusion any less compelling. Indeed, given the nature of my task – which as I earlier explained was simply to draw out the incoherencies (impeding my comprehension) already present in the concept of open-bounded objects – the triviality of the arguments that I need to employ is almost an inevitability; this should not dissuade some of the successfulness of the arguments, but rather of the coherence and grasp they may have thought they had of the concept of open-bounded objects.Footnote ¹⁵

3. The Argument from Miraculous Transportation

This leads to the second horn of the dilemma, where it is alternatively supposed that an arbitrary open material object were to occupy indirectly whatever region it occupies. Now, if the substantial proper parts in virtue of which the arbitrary open material object occupies its region are themselves open objects, then we can go back to the first horn of the dilemma, where the same problem arises for those substantial proper parts (and infectiously for their composite). However, it could be that, whilst the composite is an open object, its substantial proper parts (upon which its location supervenes) are each closed. In that case, the composite would have an infinite number of closed proper parts approximating its open boundaries, with no nearest closed substantial proper part to those open boundaries (lest they themselves be open). This possibility may perhaps (though I am somewhat sceptical here) reasonably be presented as a challenge to premise P3 of the argument from miraculous transformation above by deflating the extent to which the changes in an object's topology are substantial intrinsic changes of a distinctive kind from the non-intrinsic mere relational changes of the object's movement itself. After all, the kind of open composite object currently under consideration would indeed only undergo mere relational changes (in the arrangement of its substantial proper parts) when it transforms its topology, from open to closed, as a result of its moving towards its initial open boundary points (as envisaged in the first horn of the dilemma). Some may even claim that what really matters is what goes on at the level of simples (entities lacking proper parts), since only they, and not their composites, are fundamentally real and feature in the ultimate story of reality.

Accordingly, P3, and likewise the conclusion C2, ought to be restricted to a certain subclass of open objects: namely, those directly occupying the region they occupy. This can be done fairly straightforwardly, thus:

It is because of this possibility of composite open objects with closed substantial proper parts, in virtue of which they indirectly occupy the region they occupy, that the argument from miraculous transformation needs to be restricted accordingly. And with this restriction the need for a second horn to our dilemma to rule out those composite open objects not challenged by the now restricted argument from the first horn. Though, it should be noted that the restricted argument does indeed in itself pose severe limitations on the possibility of open objects, not to be unduly dismissed.

Nevertheless, composite open objects, with closed substantial proper parts, face an equally recalcitrant problem. In particular, if composite open material objects were to move towards their initial open boundaries, in virtue of the movements of their closed substantial proper parts, then they must still first occupy those initial boundary points before occupying any points beyond them. But then we are compelled to answer which of its closed substantial simple parts reaches those initial boundary points first. The problem is, given that there is no closest substantial proper part to those initial boundary points – indeed, for each substantial proper part, there will be infinitely many other substantial proper parts that are closer to the initial boundary points – there would seem to be no plausible candidate substantial proper part of the composite open object to first occupy those initial open boundary points at the moment when the composite open object first reaches those initial open boundary points and no further. And to this my fictional accused must surely feel some compunction, for it is clear that no serious answer is forthcoming. Or, if there is a good answer, we need to hear it. In short, if there is no nearest part, there ought to be no first arriver!

That presentation of the argument may have been a bit quick for some. So, let us outline the argument a little more formally as follows:

Call this the Argument from Miraculous Transportation. P4 follows from the same reasoning for P1 of the dilemma's first horn. P5 follows from the definitions of closed objects and boundaries together with the indirect occupation contention that the open composite object's location supervenes on the location of its proper parts. P6 is just an ex hypothesi description of the starting condition of the considered scenario. And P7 rests on a presumption of what I take to be a plausible mechanics, wherein movement of composites does not require substantial proper parts jumping past an infinite number of closer substantial proper parts.

Together, I think the arguments from miraculous transformation and miraculous transportation make a compelling case against the possibility that fully open objects can move and that partially open objects can move towards their open boundaries. And I take the denial of this possibility to leave our understanding of open objects in a plainly absurd position. At the very least, I think answers need to be provided here, and it would help all of us understand open-bounded objects much better if some response to these challenges were made.

4. A Boundary to Movement

This dilemma also highlights how the movements of open material objects aggravate a well known puzzle regarding the moment of change. Richard Sorabji nicely summarises the puzzle thus:

The moment of change is when something stops being true of a thing, whilst something else starts being true of it. Pertinent to our present case is the change from rest to motion. I shall take for granted that things cannot be simultaneously at rest and in motion – contrary to a proposal by Graham Priest (Reference Priest2006, Chs. 11–12). So, if there is a last-moment-of-rest and a first-moment-of-motion, they must be distinct instants. However, assuming time's density, between any two instants there is a third. What then should we say of instants between the last-moment-of-rest and first-moment-of-motion? If they are genuinely what they are claimed to be, the thing cannot there be either in rest or in motion, and this too is deemed unacceptable. So, seemingly there cannot be both a last-moment-of-rest and a first-moment-of-motion. Yet, there must at least be either a last-moment-of-rest or first-moment-of-motion, since otherwise, there would be no moment of change, and consequently, no change at all. But which moment must we exclude? As Sorabji notes, the choice seems arbitrary.

Thankfully, as Sorabji explains, the choice is not as arbitrary as it first appears. For, given the continuity of motion, there would be a decisive asymmetry settling the matter:

According to Sorabji's solution – a solution he argues is shared by Aristotle – there is no first-moment-of-motion, only a last-moment-of-rest. We pick out the moment of change via the instant which is either the initial or terminal stage of the change. If the former, it is a change from how things are at that instant. If the latter, it is a change to how things are at that instant. But given time's density, there is no first change from or last change to how things are at any instant. In short: ‘…in a continuous transition, there is no first or last instant of being away from the initial or terminal stage. But there is a first instant of being at the terminal stage.’ (Ibid., p. 413).

This implies that, when objects change from rest to motion, there is a last-moment-of-rest, but no first-moment-of-motion. However, at this point, material objects with open boundaries do not conform. Since, despite time's assumed density, their initial motion is discrete, not continuous. For, upon moving, they must first occupy their initial boundary points before exceeding them (lest it not be motion, but teleportation) – this following from the same reasoning given for premises P1 and P4 in the arguments from miraculous transformation and miraculous transportation above. And, ex hypothesi, there are no intervening points between the open object and its initial boundary points. So, upon its moving, there must be a first position – which includes those initial boundary points, but not points beyond them – occupied away from its starting position. Moreover, given this initial discrete change in position, unless the open object rests at its initial boundary points for some period, there must be a first moment when it reaches its initial boundary points but no further. More specifically, to avoid there being a first moment when it reaches its initial boundary points but no further, it must have moved a point's length, only to have immediately, and inexplicably, ceased moving. And if it were then to start moving again, unprovoked, when that movement begins would be completely arbitrary and inexplicable. Consequently, this suggestion simply defies plausibility. So, there must indeed be a first-moment-of-motion.

Accordingly, we are again saddled with both a first-moment-of-motion and a last-moment-of-rest. Yet, given time's density, this is impossible, since time's density entails there must be a third instant between the first-moment-of-motion and last-moment-of-rest. And either the object is in motion or at rest then. If the former, what was said to be the first-moment-of-motion would not be that, since there is a preceding moment-of-motion after the rest. If the latter, what was said to be the last-moment-of-rest would not be that, since there is a succeeding moment-of-rest before the motion. Therefore, once again, open material objects are rendered strangely immovable.