The A and B theories of time

McTaggart’s argument for the thesis that the A-series involves a contradiction is more complicated, and we will be only able to give its rough outline here. The argument starts with the unquestionable assumption that the three spheres constituting the A-series – the past, present and the future – are mutually exclusive. And yet McTaggart claims that the existence of the A-series requires that each event be past, present and future; thus a contradiction ensues. A typical response to this claim is that events are past, present and future not simultaneously, but in succession. The battle of Waterloo is past, but was present, and had been future. My current lecture is present, but was future and will be past. But McTaggart demands that we explain precisely what we mean by the words “was”, “had been” or “will”. One possible explication is as follows: an event x was present means that x is present at some past moment. Similarly, an event x will be past if and only if x is past at some future moment. But now we can observe that we have applied the distinction among the past, present and the future to moments, and again it can be claimed that the A-series requires that every moment is past, present and future. To avoid this conclusion we can only repeat the same procedure: we can distinguish between the situation in which a moment m was present (past, future), is present (past, future) and will be present (past, future). In order to explain the use of grammatical tenses, we have to appeal yet again to the second-order past, present and future moments at which the first-order moments can be classified as present, past and future without a contradiction, so it should be clear now that an infinite regress looms large.

It is not clear whether the above regress is vicious. Some commentators claim that the regress can be avoided without falling victim to a logical contradiction. For instance J. Lowe insists that the A-series can be expressed in a language that employs temporal adverbial modifiers “presently”, “pastly”, and “futurely”. Lowe notes that each event x has to satisfy three disjunctions: (1) x is either pastly past, or presently past, or futurely past; (2) x is either pastly present, or presently present or futurely present; and (3) x is either pastly future, or presently future or futurely future (note that the disjunctions (1) – (3) are not necessarily exclusive). Lowe’s point is that we don’t need to explain the adverbial modifiers in a way that leads to a regress, and he also thinks that the A-theorist should be content with such a characteristic of the A-series. This last claim can be questioned, though. The partition of events into Lowe’s nine temporal spheres falls short of making time move. The truth of (1)-(3) is logically compatible with a completely stationary time, in which a given event x always belongs to the same spheres. McTaggart can repeat his main point: in order for the A-series to exist, every event has to be pastly present, presently present and futurely present, pastly past, presently past and futurely past, and pastly future, presently future and futurely future.

McTaggart’s distinction gives rise to two theories of time: the A-theory and the B-theory. The main difference between them lies in their approach to the idea of the passage of time. The A-theory accepts the existence of the objective passage of time, while the B-theory rejects it. The B-theorists invoke two arguments against the passage of time. Firstly, if the passage of time existed, it would make sense to ask how fast time flows. The rate of time’s flow would have to be measured in seconds per second, which is a dimensionless quantity. Secondly, the movement of time requires some stationary background against which it can happen (similarly to ordinary motion, for which the background is precisely time itself). But this implies the existence of a second-order time, which presumably requires yet another, higher-order time and so on. B-theorists insist that we can translate our ordinary way of speaking about time into the language of the B-theory, based on the fundamental relation “earlier than”. The main challenge for the B-theory is how to express grammatical tenses in the tenseless B-language. A typical suggestion goes along the following lines. The temporal expressions, such as “past”, “present”, “now”, “yesterday”, “tomorrow”, ten days ago”, belong to the category of the so-called indexicals, i.e. expressions whose meaning depends on the context of utterance. Other words in this category are “here”, “there” “I”, etc. When I utter the word “here” while standing in Trafalgar square, this word refers to a different place than when I utter it under the Eiffel tower. Similarly, when I say “It is cold now”, I mean something like “It is cold at the moment of my utterance”. The expression “Napoleon was defeated at the battle of Waterloo” can be interpreted as “Napoleon’s defeat at the battle of Waterloo is earlier than the moment of utterance”.

The proponents of the A-theory of time do not give up easily. They point out that the experience of the passage of time is too fundamental to dismiss it as some sort of illusion. They accuse the B-theorists of interpreting time as an extra spatial dimension (so-called spatialisation of time). The defenders of the passage of time claim that it is possible to meet the B-theorists objections. The passage of time does not have to be literally interpreted as a kind of motion, to which the ordinary notion of velocity would apply. Rather, it consists in the fundamental fact that events come into being successively. Responding to the argument from the rate of flow Tim Maudlin claims that it misses the point. He points out that there is nothing fundamentally wrong with dimensionless quantities, quoting the example of an exchange rate of one currency for itself (dollars for dollars). But we may note that even if Maudlin is right and the notion of the velocity at which time passes is not meaningless, still it is quite unsettling that in his approach this velocity can assume only one value (one second per second) as a matter of conceptual necessity. In other words, time cannot speed up, nor can it slow down. A different solution to this problem has been proposed by Peter Forrest. According to his approach, time passes by adding new layers of spacetime of positive thickness to the already existing universe. The thickness of the successive new layers is the measure of the rate of the flow of time. This picture of the passage of time dispenses with the stationary background against which the passage is supposed to occur.

The A and B theories of time are naturally associated with particular positions regarding the reality of temporal spheres of events. The B-theory is typically connected with the view known as eternalism (or the block universe view). According to eternalism, all events, past, present and future, enjoy the same fundamental status of reality. The battle of Waterloo did not vanish – it exists but in a different part of spacetime than the region occupied by us. Past events are analogous to events that occur in spatially remote regions of the universe: they happen elsewhere, but are not less real because of that. The unintuitiveness of this position is best exposed in Arthur Prior’s “Thanks goodness it’s over” argument. He points out that the eternalist cannot satisfactory explain why we feel relieved when something bad comes to an end. For instance, when my teeth stop aching after taking a pain killer, I feel relieved, but why should I, given that my past pain did not cease to be real? You may reply that the pain belongs to the past now. But according to the B-theory, this means that my (real) pain is earlier than the moment of utterance. Why should I be happy about this?

The A-theory of time is compatible with more than one ontological position regarding the reality of past and future. The most radical is the view known as presentism, which claims that only present events exist. Both past and future events are not real (the former are no longer real, the latter not yet real). The universe consists just of one three-dimensional layer of events which moves as time passes. Presentism is threatened by two main arguments: one from science, and the other from semantics. It is commonly accepted that presentism is incompatible with the special theory of relativity. According to special relativity, the relation of simultaneity is relative with respect to the frame of reference (we will talk about this later). Consequently, the set of events simultaneous with my current “present” depends on the selected frame of reference. But presentism requires that only one set of mutually simultaneous events be real, hence it privileges one particular frame of reference, and this fact violates the principle of relativity. The argument from semantics turns on the fact that some statements about past events (and future events too) are true. But what is the true sentence “Napoleon lost the battle of Waterloo” about, if neither Napoleon, not the battle exists? What is its truthmaker?

Two alternative views compatible with the A-theory are: the growing block theory and the shrinking block theory. The first assumes that past and present events exist, but not future events. The second accepts the opposite: present and future events exist, but past events do not. Both theories are susceptible to similar objections as presentism, although the argument from semantics is now limited to the case of future statements for the growing block theory, and the case of past statements for the shrinking block theory.

Readings:

E.J. Lowe, Chapter 17 "Tense and the reality of time", pp. 307-324, A Survey of Metaphysics.
M.J. Loux, Chapter 7 "The nature of time", pp. 205-228, Metaphysics: A Contemporary Introduction.