Theodore Sider - Time (Chapter 3 of Riddles of Existence)

What does it mean to move? To move is to be in one location at one time, and then at another location at a later time. So movement as a concept depends on the concept of time. Movement itself is a temporal notion. Talk of time's movement then, such as the flow or passage of time, is strange. How can time move? Just as a train moves in terms of time, does the present moment move in terms of some higher order of time, like hyper-time? But then hyper-time would move in terms of a further kind of time. We have an infinite regress.

Maybe time is not like a river that flows, but is part of a structure called space-time. Time is like space in three respects: 

1) Just as objects can be near or far in space, objects can be near (present) or far (past or future) in time.

2) Material objects take up space by having parts that extend through space. I have a leg there and an arm here. Likewise, temporal objects have temporal parts. For every moment that passes, there I am in that moment, and that snapshot of me is a temporal part of me. So just as there are material parts and wholes, there are temporal parts and wholes.

3) Just as space is relative to an observer, so is time. My here is not your here, just as my now is not the same now of someone at some other time. Now and here are relative, not objective.

So time is not like a river that flows, but is like space that objects would have temporal extension.

But your intuitions might say the following:

1) No, past and future objects do not exist. The past is gone and the future is not here.

2) There are no temporal parts, only temporal wholes; all of you exists now.

3) While space is relative, time is not. Everything exists now. (Perhaps necessary objects exist atemporally.) The present moment is privileged.

There are three further arguments against the idea that time is like space:

1) From change: As Sider states, "Change is having different properties at different times." That's a temporal notion of change. Spatial change is having different properties at different places. A pillow may be hot on one side and cold on another. The difference between the two kinds of change is that in the case of temporal change, when something changes, its total properties have changed. But for the pillow, its total properties are the same. Spatial change really isn't change at all; no old properties have been lost and no new properties have been gained. Change is fundamentally a temporal notion and not a spatial one.

A counter to this is that change is not fundamentally temporal; it just so happens that our ordinary notion of 'change' is the temporal one. On the space-time view, time is a dimension like space, and sure enough change is simply having different properties at different places or times. The space-time view does not need time to be identical to space; they're just similar.

You could also say that just as an object in space has all of its properties "now" even with different properties at different places, so too does an object in time have all its properties "now" in the sense that no reference frame is privileged and the object is a temporal whole. In other words, just as a plank of wood has a smooth surface here and rough surface there, so too does the plank of wood have a smooth surface now (having been sanded), and a rough surface there in the past (before being sanded).

2) From motion: You can move backwards, forwards, left, right, up, down, and so on, in space. But in time you only have one direction, from past to future.

A response to this is that it's not fair to say you can move backwards, forwards, etc., in space, when movement is a temporal notion. So when we translate moving back and forth in space, with a dimension reference to time, we get (t = time, p = place):

Moving back and forth in space: An object is at p1 at t1, then at p2 at t2, then at p1 at t3.

To show whether time and space are analogous, we invert the reference dimensions so that we get:

Moving back and forth in time: An object is at t1 at p1, then at t2 at p2, then at t1 at p3.

This depicts one object at two places that comes together in a single place at a later time. Sider gives the example of clapping hands: one hand is at p1, the other hand at p3, and the two come together at p2.

This doesn't make sense to me. When the hands open up after coming together, the hands are not going back to t1. That would involve traveling back in time, which is exactly what this objection to space-time is saying. Moving back and forth in space makes sense, as the locations stay the same while time moves forward. But when moving back and forth in time, it's the temporal points that would remain while spatial locations change (per the graph). But when clapping your hands, the spatial locations stay the same while time moves forward, just like in the first graph.

Besides, when I think of moving back and forth through time, I imagine, say, the TARDIS from Doctor Who appearing at the same location it left from. Moving back and forth through time doesn't require more than two spatial locations. 

Plus, someone could say you don't have one object, a pair of hands, in two places. Rather, you have two objects in two places, when we want to know whether one object can move back and forth in time.

3) From causation: Like motion, causation stretches out in all directions in space, but there's only one direction in which temporal causation moves. Future events cannot cause past events.

Sider admits that we do not observe backwards causation. But if time is like space, then in theory backwards causation is possible, such as through time travel. This leads to discussions about whether time travel is possible. If it's not, and if space-time implies it is, then we have to give up space-time. Sider concludes that there are consistent stories of time travel, and cites the Terminator as an example, and thus concludes that the objections against the space-time model of time have been answered.

But I don't see how the Terminator story is consistent. For John Connor to send his own dad, Kyle Reese, back in time, John must have already been born. But born how? By Reese being sent back in time. Sent back in time by who? By John Connor. There must have been a first John Connor to start the chain. But there couldn't have been a first John Connor, because each John Connor depends on an earlier one for their birth. It's a bootstrapping paradox.