Grim Reaper, Part 4: The Finite Past

As Alexander Pruss has observed (Pruss 2009), the Grim Reaper paradox suggests not only that no finite time period can be divided into infinitely many sub-periods but also that it is impossible that there should exist infinitely many time periods, all of which are earlier than some event. It seems to provide grounds for thinking that time must be bounded at the beginning: that there must be a first period of time. If not, we could simply construct a new version of the Grim Placer paradox. As in the original version, we postulate the possibility of a Grim Placer, who creates a particle and places it at a designated spot, if and only if no particle is already located at a spot corresponding to any earlier Placer. In this version, Placer 1 is set to act at the first moment of 1 B.C., Placer 2 at the first moment of 2 B.C., and so on ad infinitum. Once again we can generate the contradiction: some particle must be placed within d meters of the plane, but there is no finite distance from the plane such that a particle could have been placed there.
 
Let us try to be more explicit about the premises needed to generate the paradox. First of all, we must assume that a single, isolated Grim Placer scenario is metaphysically possible:
 
P1. Possible Grim Placer (PGR). There are  numbers d and m such that for every positive integer n there is a possible world W and a region R such that R has a finite temporal duration d seconds, there is a Grim Placer wholly contained within R, and throughout R the Grim Placer has the power and disposition to create a “Fred” particle and place it at a designated position m/2^n  meters from the plane P if there is no unique particle located at m/2^i meters from P for some i >n (eliminating all other particles located within m meters of P, if there are more than one), and otherwise to maintain the unique Fred particle that is located at m/2^i meters from P in its initial position.
 
Secondly, we appeal to some version of David Lewis’s Patchwork Principles (Lewis 1983, 76-7). Much, if not most, of our knowledge of possibility is based on patchwork principles, since we have little direct access to alternative possibilities. Instead, we have to rely on our direct knowledge of the actual world, as well as the license to cut-and-paste or recombine various regions of the actual world into a new arrangement.
 
Binary Spatiotemporal Patchwork. If possible world W1 includes spatiotemporal region R1, possible world W2 includes region R2, and possible world W3 includes R3, and R1 and R2 can be mapped onto non-overlapping parts of R3 (R3.1 and R3.2) while preserving all the metrical and topological properties of the three regions, then there is a world W4 and region R4 such that R3 and R4 are isomorphic, the part of W4 within R4.1 exactly duplicates the part of W1 within R1, and the part of W4 within R4.2 exactly duplicates the part of W2 within R2.
 
Following Lewis, I will assume that ‘intrinsicality’ and ‘exact duplication’ are inter-definable:
 
Definition of Intrinsicality: a property P is intrinsic to a thing x within region R in world W if and only if x is P throughout R in W, and every counterpart of x in any region R’ of world W’ whose contents exactly duplicate the contents of R in W also has P throughout R’.
 
Binary Spatiotemporal Patchwork licenses recombining region R1 from world W1 with region R2 from world W2 in any way that respects the metrical and topological properties of the two regions, so long as there is enough “room” in spacetime as a whole to fit the two regions in non-overlapping locations (as witnessed by the two regions R3.1 and R3.2 in world W2). The Binary Patchwork principle can plausibly be generalized to the case of infinite recombinations:
 
P2. Infinite Spatiotemporal Patchwork (PInfSP). If S is a countable series of possible worlds, and T a series of regions within those worlds such that Ti is part of Wi (for each i), and f is a metric and topology structure-preserving function from T into the set of spatiotemporal regions of world W such that no two values of f overlap, then there is a possible world W* and an isomorphism f* from the spatiotemporal regions of W to the spatiotemporal regions of W* such that the part of each world Wi within the region Ri exactly resembles the part of W* within region f*(f(Ri)).
 
In order to apply the Patchwork principles to Benardete's story, we must assume that the relevant powers and dispositions are intrinsic to the things that have them when they have them. Otherwise, we cannot assume that the joint possibility of an infinite number of Grim Placer scenarios follows from the possibility of a single scenario, taken in isolation.
 
Intrinsicality of the Grim Placers’ Powers and Dispositions (PDIn). The powers and dispositions ascribed to each Grim Placer are properties intrinsic to that Placer in its corresponding region and world.
 
Our hypothesis for the reductio will be the possible existence of a world with an entity that has an infinite past:
 
HIP. Hypothesis of the Possibility of an Infinite Past. There exists a possible world W´ and a spatiotemporal region R´ in W´ such that R´ has infinitely many temporally extended parts such that these parts can be put into a sequence (ordered by the natural numbers) in which each successive part in the sequence is within the backward time cone of its predecessor, and each part is large enough to contain a Grim Placer.
 
1. Start with a possible Grim Placer in world W and region R, with finite duration d. (From PGP, the Possibility of Grim Placer)
2. Next, locate a world W' with a region R' containing a non-well-founded infinite series of non-overlapping temporal parts, each of duration d and each in the backward time cone of its predecessor. (Assumption of HPIF, for reductio)
3. Find a single possible world W* with region R* containing a non-well-founded infinite series of non-overlapping temporal parts (R1, R2, etc.), with each Ri containing a counterpart of the Grim Placer. (From 1, 2, and Infinite Spatiotemporal Patchwork)
4. Assume that, in world W*, there is after period R1 no particle located at any distance m/2^n from P, for any n > 0. (Assumption for second reductio)
5. Therefore, there is after period R2 no particle located at any distance m/2^n, for any n > 1. (From 4)
6. Grim Placer #1 in period R1 in world W* placed a Fred particle at distance m/2 from P. (From 5, and the Possibility of Grim Placer)
8. Contradiction (4 and 6). So, after R1 in W*, there is some particle located at some distance m/2^n from P, for some n > 0. 
9. Therefore, no particle is located any distance m/2^j from the plane P, for any j > n. (From 8, the Possibility of the Grim Placer)
10. Therefore, no particle is located any distance m/2^j from the plane P, for any j > n+1. (From 9)
11. Therefore, Grim Reaper n + 1 placed a particle at distance m/2^(n+1) from P. (From 10, and the Possibility of the Grim Placer).
12. Contradiction (9 and 11).
13. So, there is no possible world containing a non-well-founded infinite series of non-overlapping temporal parts, each of duration d0 and each in the backward time cone of its predecessor. (Negation of HPIF)
 
From the conclusion of this argument (step 12), we can deduce premise P3:
 
P3. Every non-eternal thing began to exist at some point in time (since the past of each non-eternal thing is finite in length).
 
If any temporal thing had an infinitely long past, then that past would include an infinite series of non-overlapping periods of length d seconds, all in the past light cone of the current state of the thing in question, in contradiction to step 12. Thus, to reach the conclusion of an eternal first cause, we need only add the assumption of causal finitism. In the next section, I will argue that the Grim Placer paradox can be generalized into an argument for causal finitism.