The Cheshire Cat problem and other spatial obstacles to backwards time travel.

'Did you say pig, or fig?' said the Cat.

'I said pig,' replied Alice, 'and I wish you wouldn't keep appearing and vanishing so suddenly: you make one quite giddy.'

'All right,' said the Cat, and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.

'Well! I've often seen a cat without a grin,' thought Alice; 'but a grin without a cat! It's the most curious thing I ever saw in all my life!'

Lewis Carroll, Alice's Adventures in Wonderland

1. The double-occupancy problem

Are there difficulties raised by the idea of backwards time travel--travel to earlier times--that are peculiar to objects (as opposed to, say, information)? By 'object' in this context I mean something that takes up space, that typically prevents other items in the same category from occupying the same space, and for which it is generally thought appropriate to talk in terms of persistence conditions. One such problem is raised in H. G. Wells' The Time Machine, but highlighted in the philosophical literature only very recently, by William Grey. Consider Tim, who boards his time machine and, at 12 noon precisely, presses the button to send him back into the past. At any moments after noon, both Tim and the machine have ceased to exist (or at least to have any location in space at those moments).

    But what of the moments just before  noon? At those times there seem    to be not one but two machines--one  going backwards and the other    forwards--each apparently occupying (or  attempting to occupy) the    same location! That is, it seems that as  the machine sets out into    the past it will collide with itself, or on  a four-dimensionalist    view the Tim-at-noon stage has earlier and  later stages at the same    space-time location. (Grey 1999: 60f)  

This conundrum Grey dubs 'the double-occupancy problem'. Objects are rarely, if ever, completely impenetrable, but it is part of the ordinary conception of an object that two objects of the same kind (persons, chairs, iron spheres) cannot occupy the same space at the same time. Moreover, an object prevents objects of certain other kinds occupying the same space at the same time (a person cannot occupy exactly the same space as an iron sphere). This, in part, is what is supposed to distinguish objects from events, for there seems to be no difficulty in imagining one event occurring in precisely the same space-time region as another event, for example a solution's changing colour and temperature at the same time. Apparent counterexamples to the 'no coincident objects' principle involve sortal terms whose applications differ over times or worlds: the lump of clay that is moulded into a statue, for example. Lump and statue have different histories, and perhaps also different modal properties, yet they coincide for a certain period of time. But we are not forced to view these cases as genuinely involving numerically distinct and coinciding objects. Various alternative descriptions offer themselves. In the case of the clay and the statue, the four-dimensionalist will describe the statue as a temporal part of the four-dimensional object, or space-time 'worm' that is the lump of clay. (See Sider 2001 for a full account and defence of four-dimensionalism.) In the case of Tim's time machine, however, the four-dimensionalist is already committed to the numerical distinctness of the earlier and later temporal parts of the machine.

The three-dimensionalist, who talks instead of objects enduring through time, will insist that the time machine that has not yet been activated is numerically identical to the one that is travelling backwards in time. So there is only one machine, not two, at (e.g.) five minutes to noon. On the face of it, it seems that it is the four-dimensionalist alone that faces the double-occupancy problem. Since it has been argued that a four-dimensionalist approach to persistence through time is needed to make sense of backwards time travel (Lewis 1976, Sider 2001: 101-109), any plausible argument to the effect that there is a difficulty peculiar to four-dimensionalism would be of considerable interest. However, the three-dimensionalist does not escape paradox. If there is only one machine at five minutes to noon, then it has incompatible properties: it is both travelling backwards in time and not travelling backwards through time. It is reasonable to suppose that there will also be intrinsic differences. Qua backwards-travelling machine, it has a certain temperature, as a result, let us say, of being put in motion. Qua non-backwards-travelling machine, it has a different temperature. So we face a dilemma: we can opt for a four-dimensionalist description that violates the no coincident objects principle, or we can opt for a three-dimensionalist description that violates the most anodyne version of the principle of the indiscernibility of identicals, viz.

[for all]x [for all]y [for all]t (At t(x = y) [right arrow] (At t(Fx) [left and right arrow] At t (Fy))).

In addition, the three-dimensionalist who believes in tropes faces a version of the double-occupancy problem. The not-yet activated time-machine possesses a mass trope [m.sub.1] at some given moment before the button is pressed. At the same place and moment, the same, but backwards-travelling, machine possesses a numerically distinct mass trope [m.sub.2]. (I am assuming that the machine changes its mass, if only minutely, throughout its career, and qualitatively distinct tropes are necessarily numerically distinct.) But how can both [m.sub.1] and [m.sub.2] exist at the same time and place? It seems that pressing the button has resulted in the machine doubling in mass.

It looks, then, as if both four-dimensionalists and three-dimensionalists face spatial difficulties in accommodating backwards time travel, though quite how we express those difficulties depends on whether we represent persistence in four-dimensionalist or three-dimensionalist terms, something we shall draw attention to at various …