Robert Koons has published a version of the ‘Grim Reaper paradox‘, also popularised by Alexander Pruss. One of the assumptions that sets up the paradox must be abandoned in order to avoid the contradiction. Koons and Pruss’ conclusion is that the assumption that an infinite past is possible should go.
It turns out that there is a potential solution to the paradox which was proposed by its originator Jose Benerdete, and more recently brought out in a paper by John Hawthorne (2000). I’m going to explain that solution here, and then (hopefully) develop it to the Koons/Pruss version in a part 2.
- The Grim Reaper paradox
The Grim Reaper paradox was first put forwards by Jose Benardete in his 1964 book ‘Infinity‘ (p. 259). Although the developer of the paradox, Benardete was a passionate defender of the actual infinite, announcing in the blurb on the cover of his book that:
“This book is an attack on finitism in all its forms … A metaphysics of the actual infinite is offered as the solution to the contemporary crisis in the foundations of mathematics”
It is perhaps a consequence of the clarity with which he engaged with the topic was that his examples have since been able to be used to argue for precisely the opposite thesis; that is, for finitism. This is how the grim reaper paradox has been used in the contemporary philosophy of religion debate, primarily by Alexander Pruss and Robert Koons, who argue for a version of finitism (‘causal finitism’).
Here is how Benardete states the paradox:
“A man is shot through the heart during the last half minute by A. B shoots him through the preceding 1/4 minute, C during the 1/8 minute before that, &c. ad infinitum. Assuming that each shot kills instantly (if the man were alive), the man must already be dead before each shot. Thus he cannot be said to have died of a bullet wound” (p. 259)
Koons describes this story as follows:
“The story leads quickly to a contradiction, on the assumption that Fred [i.e. A] does not die unless one of the Reapers kills him. At least one Grim Reaper must act, since if all of the Reapers whose numbers are greater than 1 do nothing, then Reaper #1 will act. However, it is impossible for any Grim Reaper to act, since, for any n, Grim Reaper #n cannot do so unless Fred survives until its assigned deadline at 1/2n seconds after midnight. It is impossible for Fred to survive that long, since Fred’s surviving until Reaper #n’s deadline entails that no Grim Reaper with a number larger than (n+2) has acted, but, in that case, Reaper #(n+1) must have acted.” (Does the Universe Have a Cause?, p. 4)
The contradiction lies in the fact that Fred will surely die before the end of the minute, but that also there is no Grim Reaper who will kill him. What this paradox seems to show, at least according to the finitists, is that there could not be finite duration of time (such as a minute) that is actually divided into infinitely many sub-regions.
2. The original solution, part 1: Benerdete.
Benardete did not draw the same conclusions as Pruss and Koons from the grim reaper paradox, and remained committed to the possibility of actually infinitely divided durations and lengths. Shortly after stating the paradox, he makes the following comments about it:
As to the dead man, although he did not die of any single bullet wound, his death was certainly caused by the infinite fusillade of shots. Here, again, although he is already dead prior to each shot, he remains alive at any assigned instant which is prior to them all. Thus he cannot be said to have died at any moment of time whatever! (Infinity, p. 260)
These brief comments offer only a hint of how to understand the response. The outline can be made out however.
Firstly, Fred is alive at every moment prior to all the shots. Yet, for each shot he is already dead before that one is fired. It follows from this that “he cannot be said to have died at any moment of time whatever!” We can spell this out by saying that although there is no first moment in which it is true that Fred is dead, there is a last moment at which Fred is alive. And this is to be expected. The series of bullets (reapers) is an open infinite sequence; while there is a last bullet at 12:01, there is an endless sequence of bullets ever closer to 12:00, and in particular there is no first bullet. Given the continuity of time, we can think of the transition from 12:00 to later times as a Dedekind cut on the real numbers:
In the above diagram, the left line is time as it approaches 12:00, during which Fred is alive. This is a closed set (which is what the square join indicates), in that 12:00 is a member of that set, but it is the final member. So Fred is alive at every time up to and including 12:00. However, the line on the right is an open set (which is what the curved join indicates). So Fred is dead at every time strictly after 12:00. This is what Benardete means by “he cannot be said to have died at any moment of time whatever”. There is no first moment at which he is dead (rather, there is just a last moment at which he is alive).
This much of the solution seems fairly straightforward. It is addressing the time of death, and we have the following information. When is Fred alive? All the way up to and including 12. When is he dead? At all times later than 12. If time is continuous and actually infinitely divisible, this is what it is like to transition from one state to another (it is like a Dedekind cut on the real number line).
The second bit is where the difficulty lies. It involves the cause of death. Benardete has it that no bullet (reaper) is the individual that causes the death of Fred (“he did not die of any single bullet wound”). Yet, the totality of all of them does cause Fred to die (“his death was certainly caused by the infinite fusillade of shots”). But how can it be that the totality of the bullets causes him to die, when none of them individually causes him to die?
3. The original solution, part 2: John Hawthorne
This is where Hawthorne comes in. His paper Before-Effect and Zeno Causality develops Benerdete’s solution, spelling out the principles in greater detail.
Hawthorne first considers the case of a ball rolling towards an open-infinite Zeno-sequence of walls. 2 miles away there is a wall; 1 and 1/2 miles away is another wall; 1 and 1/4 miles away is another wall; 1 and 1/8 miles away there is another wall, etc. Thus, there is an infinite sequence of walls, ever closer to the point that is exactly one mile away. There is no wall which is the ‘closest’ to the one mile point (which makes it an open sequence). Suppose the walls are impenetrable and cannot be knocked over (etc). The ball is rolled towards the walls. What happens as it arrives at the one mile mark? Hawthorne’s answer is as follows:
“The ball does not proceed beyond a mile and it does not hit a wall.” (p. 625)
We are puzzled by this combination partly because we have fuzzy intuitions about what happens when there is ‘contact’ between objects and open series in this sort of setting (where space is continuous and actually infinitely divided). But we can spell it out by specifying what we mean by ‘contact’ in such a way that it makes sense. Hawthorne calls ‘open-closed contact’ what happens when “A closed surface contacts an open surface insofar as there is no unoccupied space in between the two surfaces.” When the ball arrives at the one mile point, it has achieved open-closed contact, in the sense that the closed surface of the ball has no unoccupied space between it and the infinite series of walls. Here is what Hawthorne says:
Consider the fusion of walls. Call it Gordon. On reflection it is clear that the sphere contacts Gordon. Gordon has an open surface. When the ball stops proceeding at the one mile mark, there is no unoccupied space between the sphere and Gordon. Contact occurs … So the ball is stopped by contact: The ball hits something, though the thing that it hits is not one of the walls. (p. 626)
If we find this puzzling still, Hawthorne has the diagnosis at hand. It is because we are assuming what he calls the ‘contact principle’, namely:
If y is the fusion of x’s and z contacts y, then z contacts one of the x’s.
That principle holds for the finite case. But it is false if the x’s are infinite in number. Once we are clear about this, there is no residual puzzle, nor anything further to learn about the wall case. It is clear what happens in worlds that satisfy the original description: At a mile, the ball makes contact with the fusion of walls, which is rigid and impenetrable. As a result, it does not proceed further. The ball does not, however, make contact with any wall. (Ibid)
It is weird that the ball can be stopped by making contact with the totality (or ‘fusion’) of all the walls even though it does not contact any particular wall, but that is partly because we are taking an intuition which is applicable to the finite cases only, and trying to apply it to the infinite case. In finite cases, there will be a first wall, and contact would be defined in relation to that wall. However, this is precisely what we do not have in this case, and relying on an intuition that presupposes that there is one will lead us into trouble.
4. From walls to reapers
“If x is the fusion of y’s and y’s are individually capable only of producing effect e by undergoing change, then x cannot … produce effect c without undergoing change”
This principle is also true in the finite case but false for the infinite case. Each of the bullets (reapers) are only capable of producing the effect of killing Fred by undergoing change (by being shot, or by swinging their scythes, etc). The assumption is that this applies to the totality of bullets (reapers). And it does, in finite cases. If there were only 10 bullets, then the totality would have to change, in the sense that one of the bullets would have to be fired at Fred, for it to bring about its effect. Yet, in the case of the infinite sequence of bullets (reapers), this is not the case. The totality can bring about the effect even though none of its elements (or the totality itself) changes in any way.
Going right back to Koons, we can apply this to his comments. In the quote I gave of his from above, he begins by saying:
“The story leads quickly to a contradiction, on the assumption that Fred does not die unless one of the Reapers kills him.” (Emphasis added)
What Hawthorne’s approach questions is this assumption. It is false that one of the reapers kills Fred, but that doesn’t mean that the effect is not brought by the totality of reapers. In the infinite case at hand, that is what happens. Once again, the weird behaviour of the infinite confounds our intuitions. Yet, giving up this intuition saves the situation from contradiction, and thus avoids the paradox.
Of course, some will argue that a ball cannot be stopped by a bunch of walls unless one of those walls makes contact with the ball, and the move from finite to the infinite context should make no difference to this.
Hawthorn is sensitive to the seemingly radical nature of the conclusion, admitting that it is a “big metaphysical surprise”. He goes on to finish by saying that:
“The Contact Principle, in full generality, could be given up fairly readily on
reflection. The Change Principle has a rather deeper hold on us. It seems to us scarcely thinkable that mundane causal powers—say that of killing with a machete—could combine so as to logically entail the causal power of producing some effect without the agent of the effect undergoing change. Nevertheless, surprising as this may be, the Change Principle should be rejected. The diagnosis is complete. The logic of each case is very much in order. And our puzzlement has been traced in each case to some faulty principle relating fusions to parts. Once we discard those principles, we will have no problem in accepting the required conclusions about what happens in each case” (p. 630-631)
I intend to write a part 2 to this, where I apply Hawthorne’s analysis to the version of the Grim Reaper paradox in which the target is not the actually infinitely divided duration, but the actually infinite past.