A-theory of Time Section 2 -Ian Dunbar's Philosophy Pages

2.02	A Defence of the A-theory of Time

How to construe statements containing temporal reference will be addressed in a parallel essay, “Time and Modality”.

The question of the appearance in physics of the real numbers is addressed in "The Problem of Quantity".

4 A-THEORY: EXPOSITION AND PARADOX

6 A-THEORY: REVISED EXPOSITION

2 TECHNICAL DEVELOPMENTS

Before developing the two rival theories, it is useful to clarify the issues by making some simple technical developments. The first is to generalise McTaggart’s two qualitative series. Both of them can be extended to quantitative series with the structure of the real line:

A-series: x seconds ago - now - in x seconds time

B-series: x seconds earlier than - simultaneous with - x seconds later than

for all real x greater than zero. Since McTaggart initially introduces the A-series as “that series of positions which runs from the far past through the near past to the present, and then from the present through the near future to the far future ...”, this generalisation to a continuous series is consistent with the development of his thoughts about time.

By using zero and negative reals we can reduce the terms in each series to a single family of terms parametrised by real numbers:

for all positive x : x seconds ago = in (-x) seconds time , and

x seconds earlier than = (-x) seconds later than ;

now = in 0 seconds time

simultaneous with = 0 seconds later than.

The second terms in each of the series is just the same as the corresponding item in the original qualitative series. The first and third terms in the qualitative series can be reconstructed using the operation of continuous disjunction - if coherent sense can be made of this operation. Informally at least, we can write, for example

future = {in x seconds time}

and similarly for “past”, “earlier than” and “later than”.

All this is based on the assumption that time, whether interpreted A-theoretically or B-theoretically, has the structure of the real line. This assumption will not be questioned in this essay, but it is worth making it a little more precise. Time has almost all the structure of the real line. It lacks the multiplicative identity, that is, the number 1. To make a mapping onto the real line we have to make an arbitrary choice of unit; in the above we have chosen the second, which means that the items “in 1 seconds time” and “1 second later than” are associated with the multiplicative identity of the real line. There are as many mappings from each series to the real line (choices of unit) as there are points in the real interval (0,¥). Whether or not time has a unique additive identity (a zero) is a matter at issue between A- and B-theory, to which we shall return.

Next we have to look in more detail at what the terms in each series actually are. They appear in the form of one-place predicates (A-series) or two-place predicates (B-series). The expressions which go in their argument places are the names of events. Let us introduce the following abbreviated notation for each of these.

Tense-xs (x) = x occurs in x seconds time

Later-xs (x,h) = x occurs x seconds later than h.

It is a simple matter within B-theory to produce a family of one-place predicates referring to events, analogous to the tenses of A-theory. Once chooses an arbitrary event to be the zero of our calendar, and defines the attribution of a date to an event relative to this choice of calendar as:

Date-xs (x) = Later-xs (x, cal0) .

If one starts B-theory from dates, then the “later” relation can be reconstructed from the difference between the dates, a feature which is calendar independent.

It is worth noting at this point that here we have used the words “tense” and “date” with senses somewhat generalised from their usage in normal English. Normally we use the word “tense” to denote the three forms of the verb, corresponding to the three terms in McTaggart’s original qualitative A-series. Here we have made two changes. The first is replacing the three tenses by a continuum of them. The second is interpreting the verb modification in terms of a predicate qualifying event names. In normal English the word “date” is reserved for a location in time with a limit of resolution of a day, and is expressed in terms of day-of-the-month, month-of-the-year and year. Any finer resolution is expressed in terms of time-of-day, denominated in yet other units. This conventional representation can readily be replaced by a single scale denominated in a single unit. The choice of the word “date” for this single measure is a little eccentric, but the more natural English choice, “time”, is a word whose usage is so badly ambiguous that it cannot have a place in serious philosophical analysis. For example, when talking about “the A-theory of time”, the word “time” refers vaguely to the generality of temporal phenomena.

This analysis exposes a fundamental difference between A- and B-theory. In the latter the choice of a zero for the calendar is arbitrary. The fundamental items in the theory are the temporal relations between events. In A-theory the zero, namely the present, is a fundamental feature of reality. But trying to say just what this feature is leads us to McTaggart’s paradox. How this happens, and how we can escape paradox is the main subject matter of this essay.

Before proceeding though, it should be noted that what has been given above is one particular Fregean analysis of temporal statements. It is the one suggested most directly by the choice of terms in the original McTaggart series, and has been adopted here entirely uncritically. Whether it is the correct analysis, and whether therefore we are right to adopt the concomitant ontology of events, is a serious question, which is beyond the scope of the present essay, apart from the comment that while the ontology of events is natural for B-theory, it is far more problematic in the A-theoretic context. This separation of issues is based on the assumption that the two issues: A-theory versus B-theory, and the Fregean analysis of temporal expressions, can be solved independently of each other, and more specifically that if one can solve the problem raised by McTaggart’s paradox using this analysis, then the solution can be transferred to another one.

Yet another matter of Fregean analysis should also be mentioned. When we have a family of predicates with the structure of the real line (except the multiplicative identity, and in some cases without the additive identity as well), can we continue the analysis and extract the name of the real numbers? Are we then committed to an ontology of numbers? These questions are not exclusive to the philosophy of time, but arise generally whenever real-valued quantities occur in physics.