An obvious definition of having a beginning is:
- x has a beginning provided that x exists at some time but there is an earlier time at which x does not exist.
- x has a beginning provided that x exists at some time at which it has finite age.
There is a somewhat recondite potential counterexample to (2). Suppose that the universe has an infinite past, and object x has a temporally gappy existence, such that last year x existed only for half a year (the other half is the gap), the year before x existed only for a quarter of a year, and you see where that's going. So x's current age is something like 1/2+1/4+1/8+... = 1 year. So x is one year old. By definition (2), x has a beginning. But it doesn't seem like x has a beginning.
But perhaps this case is not fatal to (2). Maybe we should agree that x has a beginning. For the relevant time sequence for saying whether something does or does not have a beginning is internal time. And x has a finite internal time past. If we say this, then we will also say that a person y that has a slowed-down past of the following sort also has a beginning: over the last year, y functioned (in all respects, mental and physical) at half of the speed of a normal person; the year before, at a quarter of the speed of a normal person; the year before, at an eighth. Thus, y experienced 1/2+1/4+1/8+... = 1 years of internal time. Yet y has always existed. While I might tolerate saying that x has a beginning, to say that y has a beginning is very awkward. (Maybe it's true, though? It's worth exploring. But for now I shall dismiss this.)
The above cases show that age in (2) must be reckoned in an external manner. To be more precise we should revise (2) to read:
- x has a beginning provided that x exists at some time T and there is a number N of years (or other units) of time such that x did not exist more than N years (or other units) before T, where the times are reckoned externally.
Note that this is very much an extrinsic characterization of x's having a beginning. We could imagine two objects whose intrinsic careers are exactly alike, one of which has a beginning and the other does not. Take, for instance, our slowed down y from the last counterexample, and then a person who lives through y's past in one ordinary external year. The slowed down y has no beginning, but the other person does, even though their internal lives could be exactly alike. This seems unsatisfactory.
Also, intuitively, having a beginning is more about the order properties of time rather than metric properties of time. But (3) (as well as (2)) makes it be a feature of the metric properties of time.
I am not quite sure what to do with these thoughts. Maybe this: The notion of a beginning isn't actually all that natural a notion. Perhaps the natural notion in the vicinity is the notion of having a cause?