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The river of time implicitly conflates two ways of representing time and its passage that were made salient a century ago by J. M. E. McTaggart (1866–1925) in the course of a discussion of temporal passage. Something that occurred yesterday is in the past, an occurrence today is in the present, and an occurrence tomorrow lies in the future. Tomorrow will be, before long, today, and then, some hours later, yesterday. Representing time in this way is to represent time’s passing. Yesterday, today, and tomorrow belong to what McTaggart calls the “A series.”

Suppose you are here, and here is San Francisco. St Louis is there, to the east. When you travel to St Louis, St Louis is here; San Francisco is no longer here, but there to the west. This is to represent spatial locations indexically, that is, by reference to here, here being wherever you happen to be.

The A series orders temporal locations indexically by reference to now, to the present. If now is Tuesday, Monday lies back there in the past and Wednesday is ahead in the future. When Wednesday comes, Wednesday is in the present, Wednesday is now, and Tuesday lies in the past. Here and now travel with you as you move through space and time.

The A series affords a now-centered representation of locations in time analogous to here-centered representations of locations in space. Spatial locations can be specified without a here, however. St Louis is 2,816 km (1,750 miles) east of San Francisco. This is so whether you are in San Francisco or in St Louis, or anywhere else for that matter. Might there be an analogous way of ordering occurrences in time?

Imagine that, on Tuesday, May 19, 2020, you had lunch five hours after you had breakfast and six hours before you sat down for dinner. In putting it this way, you would be locating your actions at intervals along a temporal dimension comparable to your locating St Louis 1,750 miles east of San Francisco. This, McTaggart’s “B series,” is fixed and unchanging. The thought that, on Tuesday, you have lunch after breakfast and before dinner is “eternally” or “timelessly” true: it is true on Monday, and remains true on Tuesday and Wednesday.

McTaggart observed that the B series leaves us with a frozen “block” universe. The B series admits of no change. For change to occur, for a tomato to ripen and change from green to red, for instance, you need something that is green today and red tomorrow. In a block universe, however, objects do not move through time and undergo changes. Everything is fixed, once and for all.

The same holds for motion: change in spatial location. For that you need something to be here – that is, wherever it happens to be – at one time, there, and not here, at a later time. Motion through both space and through time, then, appear to require the A series.

McTaggart argued that anything deserving to be called time would have to include temporal passage, and temporal passage involves the A series. Certainly, the A series is what comes to mind when we think of our experience of time and its passage. The A series is internally inconsistent, however. So, McTaggart concluded, time must be unreal. (Metaphysics in action!)

To appreciate McTaggart’s reasoning, focus on the time of a particular occurrence – the moment Neil Armstrong set foot on the moon (July 21, 1969, 2:56 UTC). You, the reader, can say truly that this occurrence is past. For his part, Armstrong, stepping onto the Moon, can say truly of this moment – July 21, 1969, 2:56 UTC – that it is present, and anyone, prior to July 21, 1969, could have said truly that it is in the future. As far as the A series is concerned, it is true of this moment that it is past, present, and future! Were time real, were time “out there,” every moment of time would have to be past, present, and future, and that, McTaggart insisted, makes no sense.

You can see how the argument works by considering a spatial analogue. The A series places you in the boat, a spectator on the passing scene as you float down the river. Upstream, the willow is afore (ahead of you), later it is abeam (directly opposite you), and, still later, abaft (behind you). Its being true of the willow that it is ahead, abeam, and abaft reflects the fact that, in describing the willow as ahead, abeam, or abaft, you are not describing features of the willow. Subtract you from the picture and it would make no sense to say that the willow, or anything else, is ahead, abeam, or abaft.

This leaves us with the B series, which, in effect, places you on the riverbank, adding you to the scene, undermining any sense of temporal passage. The appearance of time’s passing must be in us, a mere appearance with no foundation in reality. So says McTaggart.