McTaggart and meta-time

There is what I take to be an error common among my fellow Thomists regarding change and certain theories of time. Put tersely, this error says that the B-theory of time is committed to the Parmenidian denial of change. I had decided to write something about it, but after doing a bit of research, it occurred to me that a form of this error goes all the way back to the start of the contemporary debate about the nature of time. That is to say, a variant of this error is present in John McTaggart’s paper The Unreality of Time. Before I can write about the error of my fellow Thomists, then, it seems appropriate to first say something about McTaggart’s mistake. And that is the aim of this post.

As the name suggests, McTaggart’s paper puts forward an argument that time is unreal. In the course of doing this he introduces a distinction that is now commonplace in philosophy of time, namely the distinction between the A-series and B-series:

For the sake of brevity I shall speak of the series of positions running from the far past through the near past to the present, and then from the present to the near future and the far future, as the A series. The series of positions which runs from earlier to later I shall call the B series. The contents of a position in time are called events. The contents of a single position are admitted to be properly called a plurality of events. (I believe, however, that they can as truly, though not moretruly, be called a single event. This view is not universally accepted, and it is not necessary for my argument.) A position in time is called a moment.

The A-series is indexed by tensed terms like past, present, and future. The B-series, by contrast, is indexed by tenseless terms like earlier than, and later than. Both of these series index the same time into a time series (or timeline), but they do so differently: in an A-series there is a privileged moment we call “present,” and other moments are called past or present through reference to this moment. In a B-series, by contrast, there is no such privilege and all moments are, in a sense, on an equal footing.

McTaggart divides the time series into “moments” and calls the things at these moments “events.” Now, we need to clarify something here, because it’s key to understanding McTaggart’s mistake. We typically think of events as existing in reality and persisting across moments, as when I wave my hand for a few seconds. And at least initially McTaggart seems to talk in terms of change of reality, which aligns well with this typical way of thinking. However, he quickly switches to talking about change of the time series itself, which is what leads him into all manner of strange places. What do I mean by this? Let’s say I waved my hand from 12 to 12:05 and then stopped. Did the event of my waving go out of existence? Surely it did once 12:05 came along. But what if I asked whether the event happening at 12:02 went out of existence? On a typical reading — in which reality is the thing that changes — the answer would be the same, since I’ve just reworded the same question. But if we read it in terms of the time series rather than reality, then the question becomes whether me-waving-at-12:02 went out of existence. Do you see the difference? We are no longer asking about the event that spanned 12:02 (together with some other moments), but rather the particular slice of the event at 12:02 (rather than the slices at other moments).

As I said, in the course of his discussion McTaggart switches from talking about the event to talking about slices of the event. The confusing thing is that he refers to the latter as the event too. This is something important to keep in mind as we proceed.

With this in hand, we can formulate McTaggart’s argument as follows:

  1. Change is essential to time.
  2. Change is impossible if the moments of time are ordered only by the B-series.
  3. Therefore, the moments of time are at least ordered by the A-series.
  4. Being ordered by the A-series involves a contradiction.
  5. Therefore, the reality of time involves a contradiction.
  6. Therefore, there is no reality of time.

The relevant points here are (2) and (4). In defending both of these points, McTaggart makes unmotivated (and strange) assumptions about the nature of change and the interpretation of his distinction between the A- and B-series.

In discussing (2), he starts off well enough but quickly makes the switch that was discussed above:

If, then, a B series without an A series can constitute time, change must be possible without an A series. Let us suppose that the distinction of past, present and future does not apply to reality. Can change apply to reality? What is it that changes?

Could we say that, in a time which formed a B series but not an A series, the change consisted in the fact that an event ceased to be an event, while another event began to be an event? If this were the case, we should certainly have got a change.

But this is impossible. An event can never cease to be an event. It can never get out of any time series in which it once is. If N is ever earlier than O and later than M, it will always be, and has always been, earlier than O and later than M, since the relations of earlier and later are permanent.

The question in the first paragraph is about the application of change to reality. But by the time we get to the second paragraph he’s switched to talking about events as slices of real events. He correctly notes that if the slice of the event ceased to exist or came into existence, then “we should certainly have got a change.” But he incorrectly assumes that this change would be of the reality he was talking about in the previous paragraph. That he’s thinking in terms of slices becomes very clear in the third paragraph, when he evaluates the suggestion in the second. He says, for instance, that “An event can never cease to be an event. It can never get out of any time series in which it once is.” Now for a normal event to go out of existence, all that is required is for it not to exist at a later time. As soon as we start talking about getting out of the time series, then we’ve moved to talking about slices of events rather than real events. And once we start talking about that, then we’ve moved away from talking about change in reality toward talking about change of the time series.[1]

So we have that McTaggart intends to account for change in reality by reference to change in the time series. But now we have a problem, for this account seems to really only push the question of change up a level: from reality to the time series. What, then, accounts for the change in the time series? If we were to apply McTaggart’s account again, then we would have to posit a meta-time series which tracks the change of the time series. Changes in the time series would be accounted for by reference to changes in this meta-time series. But now we have the same problem again, for which we’d need to posit a meta-meta-time. And so on. Eventually we’d need be led to an infinitely nested collection of times and meta-times, and we wouldn’t have gotten any closer to accounting for the change we started with. In other words, McTaggart’s account of change leads to a vicious infinite regress.

This conclusion comes up in the paper a number of times in different forms, but it’s not clear to me whether McTaggart realizes that they’re all instances of the same problem. In the present section he rejects the possibility of the time series changing in this way (“the relations of earlier and later are permanent”), and thereby rejects the adequacy of the B-series given his account of change. Indeed, the same conviction continues through his entire rejection of the B-series being able to account for change. Consider, for instance, the paragraphs that follow:

Or shall we say that one event M merges itself into another event N, while preserving a certain identity by means of an unchanged element, so that we can say, not merely that M has ceased and N begun, but that it is M which has become N? Still, the same difficulty recurs. M and N may have a common element, but they are not the same event, or there would be no change. If therefore M changes into N at a certain moment, then, at that moment, M has ceased to be M, and N has begun to be N. But we have seen that no event can cease to be, or begin to be, itself, since it never ceases to have a place as itself in the B series. Thus, one event cannot change into another.

Neither can the change be looked for in the numerically different moments of absolute time, supposing such moments to exist. For the same arguments will apply here. Each such moment would have its own place in the B series, since each would be earlier or later than each of the others. And as the B series indicate permanent relations, no moment could ever cease to be, nor could it become another moment.

This entire discussion is predicated on the confusion between the change of reality and the change of the time series. Needless to say I think his criticism of the adequacy of the B-series to track change misses the mark, and this becomes fairly clear once we fix up this confusion. All one needs for change is that things can have different properties at different times, and nothing about this requires that we talk about these times in terms of past, present, and future (that is, the A-series). If time t1 is earlier than time t2, and I have different properties at these times, then I have changed. The B-series, then, is more than capable to track change.

McTaggart continues with his confusion when moving to discuss the A-series:

Since, therefore, what occurs in time [event-slices] never begins or ceases to be, or to be itself, and since, again, if there is to be change it must be change of what occurs in time (for the timeless never changes), I submit that only one alternative remains. Changes must happen to the events of such a nature that the occurrence of these changes does not hinder the events from being events, and the same events, both before and after the change.

Now what characteristics of an event are there which can change and yet leave the event the same event? (I use the word characteristic as a general term to include both the qualities which the event possesses, and the relations of which it is a term — or rather the fact that the event is a term of these relations.) It seems to me that there is only one class of such characteristics — namely, the determination of the event in question by the terms of the A series.

Here he envisages what we might call a “spotlight” theory of time. The ordered events permanently exist in the time series, and the present moment has a temporal spotlight shining on it. This enables us to track the progression of time by following the progression of this spotlight. Dropping the metaphor, this means that each of the event slices has one of three properties: it is past, it is present, or it is future. Only one such event slice is present, everything earlier than it is past, and everything later than it is future. McTaggart is forced into this because (1) he assumes change of reality involves change of the time series, and (2) he is convinced from preceding argumentation that this change cannot consist in the creation or destruction of event slices in the time series. Thus, his proposal: moments remain permanent without being created or destroyed, and the march of time consists in them merely changing their properties.

But this A-series proposal, just like the B-series proposal above, requires that there be some kind of meta-time. What, after all, would track the changes of the time series represented by the spotlight moving along it? Or, without metaphor, what would track the changes of the time series that occur when the event slices change their properties?

After much discussion about this McTaggart realizes that the way he’s construed things entails that there will be an infinite regress of times and meta-times. We might summarize the process as follows: we sought to give an account of change, and so we proposed the spotlight account. But this requires that we postulate a meta-time to track the change of the spotlight. If our original proposal was correct, then in order to account for this meta-time we need to propose a meta-spotlight. But this will in turn require a meta-meta-time. And so on. So we have a vicious infinite regress on our hands.

As far as it goes this is a valid conclusion, and one McTaggart could have raised when discussing the B-series earlier had he seen the implicit dependence upon meta-time there. As we’ve already said, however, instead of taking this as a reason to reject his account of change (in terms of the change of the time series), he takes it to mean that neither the A-series nor the B-series can be real. And since these are exhaustive options for time, he concludes that time cannot be real.

In fact, the problem isn’t with either the A-series or the B-series. As we did earlier, we can frame the infinite regress argument directly against McTaggart’s account of change without ever mentioning either series.

To summarize, then, McTaggart’s mistake was in confusing change to reality with change to time. Once this error is corrected, neither the A-series nor the B-series precludes the possibility of change, since both can track things having different properties at different times. And neither series necessarily leads one into an infinite regress.

Notes

  1. Initially McTaggart raises it as a hypothetical (“Could we say that…?”), and so long as it remains option that might admit of alternatives there is no problem in his argument. The only caveat would be that he cannot conclude to the unreality of time if he limits himself to only one option among others. The problem is that this is exactly what he does conclude.

Craig’s timeless moment sans creation

William Lange Craig’s model of how God relates to time can be stated succinctly: God is timeless sans creation, and temporal since creation.[1] The reason we word it like this is obvious: he can’t be timeless before creation, since before-ness is a temporal relation and creation includes time itself. Craig holds this view largely because he is a presentist,[2] believes that time is relational,[3] and that the past is finite.

Ok, now let’s talk about “states.” Let’s say that a state is constituted by a collection of things exemplifying properties, and that an event is a change from one state to another.[4] We’ll say that a state is maximal if it is not properly contained within any other state. We’ll use the word moment as synonymous with maximal state. Finally, we’ll call the moment of God existing sans creation the timeless moment.

The central problem of this post comes when we try and answer the question, “What makes a state temporal?” Or, in different words, what is a moment of time? One is tempted to say something like the following:

1. The moment S is temporal if and only if there is another moment T such that S is causally prior to T or T is causally prior to S.

There’s an interesting consequence of (1): combined with finitism (of the past), relationalism, and presentism, it entails that God began to exist. To see this, picture the scene: God exists and nothing else exists. We’re in the timeless moment, call it t1. God creates something, bringing about the first change, and therefore the first event, and therefore time itself. Let t2 be some moment later than the beginning of this first event. How are t1 and t2 related? Well, there have been a series of changes that lead from t1 to t2, so either they’re the same moment, or t2 is later than t1. They’re not the same, so t2 must be later than t1. But, given (1) it follows that t1 is a moment of time. And because God didn’t exist before t1 (since there is no “before”), it follows that God’s existence is completely contained within time. And since the past is finite, God’s existence (extended temporally backward) is finite, and thus he began to exist.[5]

Such a conclusion is certainly worrying for theists. But regardless of whether one is a theist or not, surely it’s absurd to think that the timeless moment is temporal, or that it somehow went from being timeless (sans creation) to temporal (since creation)!

So, what’s wrong? It seems to me that the entire approach to time seems to start in the wrong place. On relationalism, time is understood as a relation between events, not states.[6] Furthermore, it seems that a necessary condition for a moment being a moment of time is that there be an event occurring at that moment. After all, surely it always makes sense to ask what is happening at a given moment of time? Moments are temporal, then, only by virtue of being “part of” or “contained within” an event.[7]

Now, go back to our timeless moment. Certainly, no events are happening at this moment: things only start happening at the first moment of creation, and surely the moment sans creation is not the same as the first moment of creation. So the problem doesn’t arise once we start in the right place. However, I’d still like an account of what makes a moment temporal, in terms of just moments (like we had in (1)). This time, of course, taking into account the fact that in reality it is their relation to events that makes moments temporal. Assuming that “instants” of time are merely potential,[6] and that in reality all temporal intervals are open, the following might work:

1′. The moment S is temporal if and only if there is another moment T such that S is causally posterior to T.

That is, there is a series of changes that lead from T to S.

There’s another interesting perplexity that is solved by starting the right place is this: the timeless moment is causally prior but not temporally prior to creation. This does seem strange at first glance. I suspect it seems strange because we try to make sense of this cause as an instance of event-event causation. But, obviously, since the timeless moment is timeless, it is not contained in any events, and so we simply can’t make sense of this as an instance of event-event causation. And of course, since the effect is an event, we can’t make sense of this as an instance of state-state causation. What we need is some sort of state-event causation, and this is what leads Craig to introduce agent-causation as the solution.Actually, thinking of agent-event causation as an instance of state-event causation can be quite helpful: the state in question is the agent being impressed by various reasons for an action combined with the causal powers they possess in that state, and the event in question is the agent freely choosing to act in accordance with some of these reasons.[8]

Notes

  1. Here we are including time and all reality apart from God in the notion of “creation” and ignoring concerns about Platonic abstract objects.
  2. Presentism is the A-theoretic view that only the present exists. That is, the past no longer exists, and the future hasn’t but will exist.
  3. The relational view of time holds that events or change is explanatorily prior to the passage of time. Thus, if there were no events, there would be no progression of time.
  4. This definition allows for states to be constituted by events: the state of me waving my hand consists of the event of me waving my hand.
  5. “Having one’s existence completely contained within a time range finitely bounded in the earlier-than direction” is, for me, a defining characteristic of “beginning to exist”. A cool paper to read about defining “beginning to exist” is Adolf Grunbaum and the Beginning of the Universe by David Oderberg.
  6. Indeed, the whole idea of thinking of events as collections of instantaneous moments seems wrong, for Zeno paradox-like reasons. I’d refer the interested reader to another of David Oderberg’s papers, Instantaneous Change Without Instants, particularly section 3.
  7. I might need to be a bit more precise than this: considering that in note 4 we said that states can be constituted by events, it’s also possible for a moment to be temporal insofar as an event is part of it. This nuance is not relevant to the upcoming discussion, so I’ve left it here as a sidenote.
  8. This suggestion is highly influenced by Timothy O’Connor’s paper Agent Causation, and Alexander Pruss’ paper Divine Creative Freedom (particularly section 4).

A Case for Chronons

Chronons are the discrete quantum of time. In other words they are the smallest (or indivisible) length of time. Naturally, if we think chronons exist, then we must hold that time is discrete.

Admittedly, if we say chronons exist we have the following weird result: Two balls, each with a 10cm diameter, are moving in opposite directions each at the speed of 10cm per chronon.

The two balls at some time, t1

It’s possible, then, that the two balls pass each other without ever being next to each other.

The two balls one chronon later, at t2

Because the balls moved between these two positions at the speed of a chronon, there wasn’t point between t1 and t2 at which A and B were next to each other. We’ll call this the “strangeness argument” for later reference.

Consider the following argument in favour of the existence of chronons based off the following 2 premises:

  1. Time is continuous, not discrete. In other words, time is infinitely divisible (there isn’t any minimum span of time that cannot be further divided).
  2. Let t0, t1 and t2 be points in time with t0 before t1 before t2. If we were a time t0 and are now at time t2, then at some point we were at time t1

Now assume we’re at some point in time, tn. Let tm be some point in time after tn. Since time is continuous (by 1. above) there exists some time tk with tm after tk after tn. By 2. we need to pass tk before we get to tm. But since tm was some arbitrary point after tn, it follows that we can never move forward in time past tn, since we’d always have to pass another point in time first. In other words, there isn’t any next point in time, so we can’t move to it in order to move forward in time.

Now of the 2 premises I think the first is the least likely to be true. Either way, if we deny the second premise, we keep time continuous, but it may as well not be. Let me explain.

Assume 1. is true and 2. is false. So we skip points in time when moving through it. In fact, we must always skip some range of points every time we move through time, because if we didn’t, 2. would be true for that range and then we wouldn’t be able to move through that range. Let’s even assume, for generality sake, that we don’t always skip by the same amount every time we move. But now, assuming 1 and not-2, time is almost indistinguishable from discrete time isn’t it? To see this we can note that the strangeness argument still applies. Say the balls are diagonally next to each other (like in the first picture above) at tn and the next point (the point we skip to) in time is tm. Now assume that the balls travel at 10cm per tm-tn moments of time. Then we have the same strange outcome we did when time was discrete. So rejecting 2. hasn’t done anything to remove the strangeness of discrete time.

It might be said that if we exist at ranges of time instead of points in time we might remove the problem. However, when the range is moving we still need to know how to move it’s boundaries which raises exactly the same problem (with a single point) we originally had. So that won’t help.

One of the premises must be false if we are to move through time, but both are susceptible to the strangeness argument. Although, because not-2 is slightly more strange than not-1 I think we should deny 1. Which means chronons exist!