Time

Time stretches back to infinity, therefore it cannot have reached NOW {let 2009 = NOW}. Manifestly, however, it has reached NOW. How can this be?

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As a warm-up exercise, consider the following two infinite ordered sets of numbers. Firstly, take the negative and positive integers in their 'natural' ordering

... -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

trailing off unendingly to the left and to the right. Second, take all the negative numbers, in increasing size, followed by zero and all the positive numbers:

-1, -2, -3, -4, ... o, 1, 2, 3, ...

Now, in both orderings, any positive number is preceded by an infinity of numbers (including all the negative numbers). But there is a very important difference between the two cases -- they have, as mathematicians say, different 'order types'. One big difference is this: there is no first member of the first ordering (i.e. for any given element of the ordered series, there's an earlier one); but there is a first member of the second ordering (namely, -1). To bring out another difference, suppose in each ordering we take one of the negative numbers, and we ask: can we start with that predecessor of 0 and by taking finite number of steps to the right, to successors in the ordering, eventually get to 0 and on to the positive numbrs? In the first case the answer is 'yes'. Pick any predecessor, e.g. -43. Then after a finite number of steps (43 of them!), we'll reach o, and we can keep on marching throuugh the positive numbers. In the second case the answer is 'no'. Pick any predecessor, e.g. -43 again. Then after a finite number of steps, we'll just reach another, bigger, negative number.

OK, now let's turn to the case of time. Suppose (to keep the argument simple, but without losing anything essential to the present issue) we take time to be discrete, with moments ordered by the 'before/after' relation. We could tag the moments with numbers, and suppose we use some positive number like 2009 to mark the present moment, NOW. And let's ask: is the 'order type' of the temporal sequence like our first ordered sequence of numbers or like the second?

If time were whackily ordered like the second number series, there would be a first moment in the ordering (the one tagged -1), though the presentmoment 2009 would still have an infinite number of predecessors. But in that case,we couldn't have got to NOW by a finite number of steps from any given moment, wherever in the past. But of course, if we do suppose that 'time stretches back to infinity', then the natural view, still assuming discreteness, is that time is ordered like the first number series. So NOW is again preceded by an infinity of earlier moments, but there is no first moment, and from any past moment, a finite number of steps from one moment to the next reaches NOW.

Of course, it is another question whether we do have to accept that time does stretch back to infinity: but the point I'm making is that there is no paradox in supposing that it does, if you give time a sensible order-type.

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