Here’s an easy-to-think-about argument against the infinite past. It bears some resemblance to arguments discussed in other recent threads. Though obviously inspired by the Grim Reaper discussion, it does not feature a series of fractions converging toward zero. I don’t think it’s a good argument, but I think there is a lesson to be learned.
Here goes… Imagine an on-off Switch and a Switcher such that:
1. Switcher checks at one minute intervals to see whether Switch is ON. If it is, Switcher does nothing; if it is not ON, Switcher turns it ON.
2. Switch is ON iff Switcher has turned it ON immediately after one of its checks.
Now let’s look at two scenarios.
We add to 1 and 2 that Switch and Switcher have existed for exactly ten minutes.
We add to 1 and 2 that Switch and Switcher have existed forever, without beginning.
Is it now the case that Switch is ON?
In Scenario A, it is. On its first check (at zero minutes), Switcher turned Switch ON, and (given 1 and 2) it must still be ON.
What about Scenario B? Well, it is ON, and… it is not ON.
It is ON, because for any n, if Switch was not ON n minutes ago, Switcher turned it ON and it stayed ON.
It is not ON, because given 2, Switch is not ON unless Switcher turned it ON immediately after one of its checks. But none of Switcher’s checks can have satisfied the condition for turning Switch ON specified in 1. In other words, for any n, Switcher did nothing following the check it did n minutes ago. Why so? Because for any m > n, m minutes ago Switcher turned the Switch ON iff if was not ON m minutes ago. If it was already ON m minutes ago, it was still ON n minutes ago, and if it was not ON m minutes ago, Switcher must have turned it ON m minutes ago, so that it was still on n minutes ago. So either way, 1 tells us that Switcher did nothing n minutes ago.
But of course n was any number of minutes. So there is no time at which Switcher turned the switch ON. From 2, it follows that Switch is not ON.
Presumably everyone will agree that Scenario A is possible, whereas Scenario B is not.
Here, then, is the argument against the infinite past.
- It’s obviously possible for Switch and Switcher to satisfy the conditions specified in 1 and 2. So if the series of minutes had no beginning, then Scenario B would be possible. But Scenario B is not possible. Therefore, the series of minutes must have a beginning.
Is this a good argument? I think not. What the impossibility of Scenario B shows is only that the conditions specified for Switch and Switcher in 1 and 2 cannot consistently be combined with the assumption that Switch and Switcher coexist throughout a beginningless past. As I see it, then, we should deny that if the series of minutes had no beginning, then Scenario B would be possible.
It’s true, of course, that every finite sub-series of the beginningless series of minutes in Scenario B is possible. But nothing interesting follows from that.
I have a serious point here. We may sometimes be tempted to take a bunch stuff that looks possible, combine it with a beginningless series of temporal intervals, derive a contradiction from the resulting combination, and then conclude that a beginningless series of temporal intervals must be impossible. Without some way of showing that a beginningless series of temporal intervals is possible only if it can be combined with the other stuff, nothing interesting is going to follow. This is something to watch out for in discussions about the possibility of an infinite past.