The terms “A-series” and B-series” were dubbed by J. M. E. McTaggart in an argument that he
presented as proof that time does not exist. The latter term describes an order and directionality
of events in time. The former adds to that a dynamic quality, that of a moving “now” through
the series. In his argument, McTaggart aimed to show that attempts to clarify the understanding
of each series leads to absurdity, namely that the B-series is incoherent because it lacks a
dynamic quality, thus requiring the A-series; and the A-series contradicts itself by attributing to
each moment all three mutually exclusive properties of pastness, presentness and futurity. His
conclusion was that time could neither be an A- nor a B-series; that it is unreal. Since it is
questionable that the B-series requires a dynamic element, that such an element be supplied by
“nowness”, or even that there is contradiction in the attribution of pastness, presentness and
futurity to all moments, there are good reasons to question the soundless of McTaggart’s
argument. What we do gain from McTaggart are terms and a framework for looking at time, and
the question of passage. What is not at issue is the ordering or directionality, namely the features
of the B-series. The question is whether or not there is reason to believe in a moving “now”, as
described by the A-series. I believe that there is not.
In “The Myth of Passage”, Donald C. Williams supplies a case against passage. He writes that
“as soon as we say that time or the present or we move in the odd extra way which the doctrine
of passages requires, we have no recourse but to suppose that this movement in turn takes time of
a special sort: time1 move at a certain rate in time2.” That is, assuming an ordering of events in
time, if there is a present moment that moves through the series, it does so at some rate, namely
that “the moving present slides over so many seconds of time1 in so many seconds of time2,”3 It is
clear to see that the rate of the present through time2 would need to be defined. This line of
thinking leads to an infinite regress, never yielding an answer to the question of at what rate the
present passes. It is on this basis that Williams discards the notion of a present. A response might
be that it is absurd to think of time as something that moves with a rate through time. The
confusion could be linguistic.
To show the necessity of an A-series, one might instead try to identify an absurdity in a B-series
conception of time. One such example could be the intuition that it entails a strict determinism.
This would be unappealing to anyone who has the sense that we have choice in the future; that
the future is not fixed. However, under careful examination, we will find that a B-series account
of time does not entail a deterministic universe. Let us define a world W as deterministic if and
only if every world with the same physical laws as W, and the same initial conditions as W, all
have the same history as W. Let us further define a world W as non-deterministic if and only if
there is a world W’ such that both are subject to the same physical laws and initial conditions,
but W’ has a different history from W. Consider the world within which in the next hour I opt to
go outside and to enjoy the day. Now consider the world within which I choose to sit inside and
finish this paper. Up to this point, both worlds are the same. Looking at the two options ahead,
we can define the world where I go outside as world W, and the world where I finish the paper as
W’, both worlds are subject to the same physical laws and initial conditions, but with different
histories. That I will experience one of these worlds (likely the latter), does not preclude the
non-existence of the other. It means that we can interpret these options as different worlds that
are subject to the same physical laws and initial conditions. Since we defined W as non-
deterministic in the event that there is a W’ with these conditions, then we can say that W is non-
deterministic. In this way, it makes sense to say that a B-series timeline, one in which there is an
order and direction of events without a “present”, does not entail a strict determinism. A B-
series account of time permits freedom. If it is to be considered insufficient, then it has to be on
Finally, an A-series proponent might want to point to experience as proof that the present exists.
We experience time. It is possible that it is illusory, that the experience of the passage of time
suggests that time could be a secondary property of matter. For example, we consider mass and
charge primary proprieties, and colour, flavour and smell as secondary properties. After all,
physical laws seem to have no requirement for a ‘now’4. Just because we perceive it doesn’t
mean that it exists objectively.
The conception of an “absolute” A- or B-series is built in a Newtonian model, where absolute
simultaneity is defined and an absolute ordering of events is possible. According to Einstein’s
Special Relativity theory, there is no such thing as absolute simultaneity. That the speed of light
remains constant in all inertial reference frames has consequences as the relativity of
simultaneity. A distinct ordering of events is possible within each inertial frame of reference.
There is no privileged frame of reference. Although an ordering is preserved in each reference
frame, without absolute simultaneity there is no absolute order. Without an absolute order, what
is the ‘now’ that moves through time? Does the present that we are searching for also become
relative? If there is no such thing as absolute simultaneity, then there cannot be objective facts of
the form “t is present”. But recall that this is the concern for inertial reference frames. The Earth
rotates on an axis, revolves around the sun, and the sun through the galaxy. In what sense are we
ever measuring in an inertial frame of reference? Unfortunately, when we look at time dilation
in Einstein’s General Theory of relativity which takes into consideration time within non-inertial
reference frames, the matter only appears to become more complicated.
We started out this journey trying to answer the question of whether or not there was reason to
believe in an A-series account of time, and where I’ve ended up is with concern that the problem
of passage as a question between the A- and B-series accounts appears to be more of a false
dichotomy. Neither the A- nor the B-series can give a full account of time.
Kosso, Peter. 1998. Appearance and reality : An introduction to the philosophy of physics. New York: Oxford University Press.
Savitt, Steven. 2014. Being and Becoming in Modern Physics. http://plato.stanford.edu/entries/spacetime-bebecome/.
Williams, Donald C. 1951. The myth of passage. The Journal of Philosophy 48 (15) (-07-19): 457.