Response to Pruss’ Argument against an Infinite Past

In a blog post entitled, Two More Arguments Against an Infinite Past, Alexander Pruss offers an argument for the impossibility of an infinite past. The argument is as follows:

If there is an infinite past, we could imagine that each January 1 in the infinite past somebody looks around and checks if there are any rabbits. If there are, she does nothing. If there aren’t, she makes a breeding pair. Of course, once a breeding pair of rabbits exists, there will be rabbits forever. Nobody and nothing but one of these potential rabbit-makers makes a rabbit. The setup entails that there have always been rabbits, and the rabbits have not been made by anybody or anything, contrary to a causal version of the Principle of Sufficient Reason.

From the outset, I will confess to not being at all convinced by this argument, but explaining precisely why will take some setting up. So, for the sake of clarity, before offering my response to Pruss’ argument, I will begin by laying out the formal structure of the argument, which can be characterized as a ‘Benardete paradox-inspired’ argument for finitism. In our case, Pruss seeks to demonstrate temporal finitism.

Formal Structure

Following Shackel (2005), we can say that all Benardete paradoxes share a formal structure involving a pair of conditions which are jointly unsatisfiable:

1. Condition ANB: For all x in S, E at x iff E nowhere before x
2. No Beginning Condition: The set S in condition ANB is an ‘unbegun set’

Allow me to explain each condition: first, let an ‘unbegun set’ be any set that has no first member, and let the ‘unbegun condition’ be the condition of having no first member; second, let the ‘ANB condition’ be the condition which states that something happens at a place or time iff it does not happen anywhere before that place or time.

Now, for those of you who are interested in philosophy of religion, and particularly in debates revolving around the Kalam cosmological argument, you may already be familiar with a popular ‘Benardete paradox-inspired’ argument for finitism called ‘the Grim Reaper Paradox’. This is an argument which seeks to demonstrate causal finitism, and it runs as follows:

Imagine that there are infinitely many Grim Reapers, and that you are alive at 12:00 a.m. Grim Reaper #1 will strike you dead at 1:00 a.m., but only if you are still alive at that time. Grim Reaper #2 will strike you dead at 12:30 a.m., but only if you are still alive then. Grim Reaper #3 will strike you dead at 12:15 a.m., and so on. We can now ask, “which Grim Reaper strikes you dead?” Clearly some Grim Reaper must have struck you dead since the clock does not stop ticking. The point is that before any Grim Reaper can strike you down, you will have already been dead, since prior to any Grim Reaper’s action there is an actually infinite number of earlier Grim Reapers, each ready to kill you. But, as a result, no Grim Reaper strikes you dead; and yet some Grim Reaper must have struck you dead. Contradiction!

As we can see, the ‘ANB Condition’ is that for any Grim Reaper to activate, no prior Grim Reaper can have already struck you dead. And the ‘No Beginning Condition’ is met because the set of all Grim Reapers is an ‘unbegun set’ because it satisfies the ‘unbegun condition’; there are infinitely many Grim Reapers lined up to strike you dead.

In view of this, we can lay out the formal structure of these ‘Benardete paradox-inspired’ arguments for finitism:

1. If there could be an ‘unbegun set’, then there could be a set which satisfies both ‘Condition ANB’ and ‘No Beginning Condition’
2. But there could not be a set which satisfies both ‘Condition ANB’ and ‘No Beginning Condition’
3. Therefore, there could not be any ‘unbegun set’

Against this background, it goes without saying that the formal structure of Pruss’ Rabbit Maker (RMA) argument follows the argumentative scheme laid above:

1. If there could be an infinite past, then there could be a Rabbit Maker story
2. But there could not be a Rabbit Maker story
3. Therefore, there could not be an infinite past

The idea behind (1) is that if there could be an infinite past, then it could be the case that any amount of rabbits — which can only be caused by Rabbit Makers, each of which create a breeding pair of rabbits iff no earlier Rabbit Maker creates a breeding pair of rabbits — can exist uncaused. But such a set satisfies both ‘Condition ANB’ and ‘No Beginning Condition’, which is impossible. Hence, infinite pasts are impossible.

A significant challenge for these ‘Benardete paradox-inspired’ arguments, however, is answering why we should accept (1)? Why would the possible joint satisfaction of ‘Condition ANB’ and ‘No Beginning Condition’ follow from the possible individual satisfaction of ‘No Beginning Condition’ for some set? This is acknowledged by Pruss when he writes, “All of these examples are kind of tricky because they attempt to derive a contradiction from a situation that should be possible if there were an infinite past. But the opponent can just go back and say: ‘Look! The example is, in fact, contradictory.’ I think to get around something like this, one needs to have a principled story about why the construction should work if there were an infinite past. Some sort of appeal to a principle of recombination seems to be the best move, combined with some principle that one can recombine beings with causal powers, and then predict what will happen from these causal powers.”

As will be shown below, Pruss fails to get around this problem. 

Response

So, why does Pruss’ RMA fail to establish temporal finitism? I offer four reasons why his RMA fails:

1. First, we can ask, “how does it not follow from the PSR that God created the Rabbit Makers, and that there was no time at which those Rabbit Makers were not in creation?” In response, Pruss writes, “God created the rabbit-creators, sure. But what about the rabbits? Where did they come from?” Answer: rabbits comes from the Rabbit Makers (i.e. a breeding pair of rabbits), and the breeding pairs themselves do not come from nowhere. God creates them, and there was never a time at which they did not exist. Rabbit Makers make rabbits once made.
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   In fact, we can run a similar argument with human persons and teddy bears. We reach the conclusion that there is a world in which humans never create teddy bears, but there are teddy bears all over the place. So, where do the teddy bear makers come from? God creates them (i.e., us) and there was never a time when there were no such teddy bear makers. But where do the teddy bears come from? One available answer is that there was never a time when a teddy bear maker had not already made a teddy bear or several. It is true for all times, t, that (∀t)(∃t’)((t’ is earlier than t) ∧ (there are created teddy bears at t’)). Again, no problem with the PSR.
2. Second, perhaps Pruss meant to ask, “why are there any rabbits at all, ever?” Answer: first, there might be something timeless which accounts for why there are any rabbits at all, ever — perhaps Ney, Barbour, and Carroll’s universal wavefunction understood in non-spatiotemporal terms, or maybe some timeless quantum field, or maybe even a God or an impersonal neo-Platonic One; second, under branching actualism, every possible world shares a history with the actual world. But this entails that if there have always been rabbits, then it is necessarily the case that there are rabbits. But then it is no longer a contingent uncaused or contingent unexplained thing, and so does not violate the PSR.
3. Third, the argument is missing a crucial premise: if the past could be infinite, this Rabbit Maker story could obtain, and Pruss himself recognizes this fact when he writes that, “the logical form of these arguments is this: (1) if an infinite past is possible, then by a plausible rearrangement principle, P is possible; (2) but P is not possible; (3) therefore, an infinite past is not possible. One problem is that few of the arguments make the rearrangement principle explicit.” Following Shackel (2005) and Malpass (2022) we have good reason to reject this premise.
4. Fourth, consider this parody of the argument which shows that an endless future is impossible: if there is an infinite future, we could imagine that on each January 1 in the infinite future, somebody checks whether there will be any rabbits in future years. (How do they know? God reveals to them whether there will be rabbits in future years.) If there will be no rabbits in future years, the person does nothing for the whole year. If there will be rabbits in future years, the person makes no rabbits, kills all the rabbits that there are on Jan 1, and ensures that there are no rabbits up until the beginning of the next year. There are rabbits today. Nobody and nothing but one of these potential Rabbit Killers kills or makes rabbits.
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   The setup entails that there will never be rabbits in any future year.** But there are rabbits now, and only a Rabbit Killer could kill the rabbits, and yet the Rabbit Killers do literally nothing. (Remember, if there will be no rabbits in future years, each Rabbit Killer does nothing.) Thus, the current rabbits die, and yet nothing causes them to die, in violation of the causal PSR.
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   **Suppose that there will be a rabbit in some future year y. Call this SUPPOSE. Focus on y: y is either such that there will be rabbits in future years, or there will be no rabbits in future years. If y is such that there will be rabbits in future years — call this assumption RFY — then y’s Rabbit Killer kills all the rabbits that there are and ensures that none come to be until year (y+1). But no Rabbit Killer ever makes any rabbits, and since only Rabbit Killers make rabbits, if the causal PSR is to hold true, rabbits can come to be only if a Rabbit Killer makes them. Hence, no rabbit ever comes to be. But then there will never be any rabbits, in contradiction to RFY.
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   If y is such that there will be no rabbits in future years — call this NRFY — y’s Rabbit Killer does nothing. But by SUPPOSE, there’s some rabbit in y, and y’s Rabbit Killer does nothing to make it cease to exist. Per the causal PSR, it can only cease if it’s killed in some way, and yet nothing kills it at least up until the next year. Hence, the rabbit exists in the next year. But this contradicts NRFY.
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   Hence, from SUPPOSE, we derived a dilemma on each horn of which there is a contradiction. Hence, ~SUPPOSE.