Consistent mathematical definitions of McTaggart's A- and B- series

Consistent mathematical definitions of McTaggart's A- and B- series

Here are consistent definitions of McTaggart's A-series and B-series. [refs.?] This does not address the question of whether a mathematical representation of time is sufficient to capture everything we know about time.

Let T be a timeline: a set of lineraly-ordered elements (which represent moments of time, or the state of the universe at moments of time, it doesn't matter), and T is isomorphic to the real numberline. Let t1, t2, ... be variables that range over all times t in T.

Start with the B-series. Define a (transitive) two-place relation B = _B_, such that for each pair of distinct times t1, t2, we have

(1) t1Bt2 if and only if t1 happens before t2

The A-series is more subtle. As McTaggart famously noted, an element t1 of T cannot be past, present, and future, simultaneously.

One may start by picking some one time t1 in T, and declaring it present. t1 is now(?...). Then all the times t2 before it (in the B-series) are past and all times t3 after it (in the B-series) are future. These assignments are a particular (homomorphic) map Mt1:{past, present, future} to T.

The twist is that, given that the moment or time that is present is (the value of the variable) t1, we need more, since other moments need time to be present as well, when it's their turn, in the linear ordering.

Perhaps the most natural thing to do is to construct a map like Mt1 for each time in T. For each time t in T we have the map Mt:{past, present, future} to T such that t is present, times before it are past and times after it are future. Then the set S of all maps Mt, S = {Mt for all t in T}, is a consistent assignment of (many of) each of the A-states (past, present, and future) to each time t in T. In this sense McTaggart was wrong: each time t is associated with past by some maps Mt', present by some Mt'', and future by the remaining Mt'''. You have to pick which time t is present. Since there is no reason to pick one time over another (so far as the mathematical representation is concerned) we needed the set of all homomorphic maps (past, present, and future) to T.

Conversely, one can start with the A-series and derive the B-series (or B-relation. I take a relation to be defined ultimately in terms of sets). Basically, if variable t4's future is a subset of variable t5's future, then t4 is after t5, i.e. t5Bt4. The point is the B-series (relation) can be defined in terms of the A-series. Therefore

(2) The mathematical structure of the A-series and the mathematical structure of the B-series can be defined in terms of each other.

The question is whether a mathematical representation of time is sufficient to capture everything we know about time.