Two more arguments against an infinite past
October 6, 2009 — 8:23

Author: Alexander Pruss  Category: Existence of God Free Will  Tags: ,   Comments: 26

In my earlier post, I gave a Grim Reaper based argument against an infinite past. Here I want to give two more arguments. Unlike the earlier argument, these two arguments are not going to be useful for arguing for the existence of God, since they make use of premises that the atheist is likely to deny (in one case, a version of the Principle of Sufficient Reason, and in the other, the existence of God). But they are useful in a broader sense, namely they help show what might be wrong with an infinite past.

Argument 1. 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.

Argument 2. If there is an infinite past, the following scenario should be possible. The universe contains nothing but bobs, and at no time is there more than one. A bob is an asexually reproducing person who lives for a century. At the end of the century he dies, but at the end of his existence he has a choice whether to reproduce or not, and can choose either way. If he freely chooses to reproduce, a new bob comes into existence out of the old bob’s body after death. So, this is a universe where every bob has always chosen to reproduce, though they could have chosen otherwise. But now consider the following very plausible Thesis:
(*) Necessarily, if a world contains at least one contingent being, then there exists something in that world determined into existence by God’s will.
But the story in Argument 2 seems to violate (*), since each bob’s existence is partly dependent on the free choice of the preceding bob. Maybe God has determined, then, not the fact that there is a bob, but that there is some initial infinite sequence of bobs, without determining which initial infinite sequence there is. But even that there is an initial infinite sequence of bobs already depends on bob-made choices.

Argument 2 won’t impress theological compatibilists.

  • Mike Almeida

    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.
    This is interesting, but puzzling. How does it not follow from PSR in this case that God created the rabbit-creators, and there was no time at which those rabbit-creators were not in creation? This looks oddly like an argument for God’s existence; otherwise you have uncreated artifacts, which are not possible. I understand that this is meant as a sort of reductio on infinite pasts, but it works, I think, only if you assume that God (or some other god-like being) does not do any creating.

    October 6, 2009 — 11:39
  • Joshua Rasmussen

    Nice ideas! Here’s another argument:
    Suppose there are and always have been two races of people, the giants and the midgets (every giant was produced by parent giants, and every midget was produced by parent midgets, and the giants and midgets are mortal). Every Jan 1., the existing midgets move 1 mile in a direction away from the city of the giants. Before they make their move, they leave behind statue of a midget (perhaps floating in outer space). This has been so from all eternity. A consequence of this scenario is that there is now, and always has been, an infinite distance separating the midgets from the giants. (If the distance were finite, then there would have been a start to their moving away a finite time ago.) But it seems extremely weird that anything could ever be infinitely far away from anything else. Think of it this way. Along the direction from the midgets toward the giants there are infinitely many midget statues each a mile apart. How can there be giants “on the other side of” the statues? Strangely, if the giants send a probe toward the midgets, this probe will never encounter a single midget statue!
    (Interestingly, if there are two particles moving toward each other from all of eternity, then there is no incoherence in supposing that those particles have never been infinitely far apart: the particles are simply a larger finite distance apart the further into the past we consider. Isn’t it odd that if two things have always been moving apart, then they must always have been infinitely far apart, whereas if they’ve always been moving closer together, then they can have always been merely finitely far apart?)

    October 6, 2009 — 12:27
  • Josh:
    I am not convinced by this, because it seems that the right thing to say is that either infinite spatial separation is possible or not. If it is, there is no absurdity. If it is not, then they couldn’t have always been moving apart, since d years ago, they were in the same place, where d is the distance.
    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 woork 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.
    I don’t think a principle of recombination will yield your case. Nor will it yield the p-widgets, perhaps. But it will yield GRs. rabbit-makers and bobs.
    “How does it not follow from PSR in this case that God created the rabbit-creators, and there was no time at which those rabbit-creators were not in creation.”
    God created the rabbit-creators, sure. But what about the rabbits? Where did they come from?

    October 6, 2009 — 13:53
  • Mike Almeida

    God created the rabbit-creators, sure. But what about the rabbits? Where did they come from?
    Rabbits comes from the rabbit-makers (i.e. a breeding pair of rabbits, as I read the example). But 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 (again, as I read the example), so, where is the tension with PSR?
    But you could run a similar argument with humans and basketballs. We reach the conclusion that there is a world in which humans never create basketballs, but there are basketballs all over the place. Where do the basketball makers come from? God creates them (i.e., us) and there was never a time when there weren’t such basketball makers. But where do the basketballs come from? One available answer is that there was never a time when a basketball maker had not already made a basketball or several. It’s true for all times t that
    (Vt)(Et’)((t’ is earlier than t) & (there are created basketballs at t’)). Again, no problem with PSR, or none that I can see offhand.

    October 6, 2009 — 17:16
  • Joshua Rasmussen

    I see what you mean. It does seem weird to me, though, that things could have always been moving toward each other but not away from each other… I’ll keep thinking about it.

    October 6, 2009 — 17:39
  • Joshua Rasmussen

    Mike or Alex,
    I’m not sure the first scenario needs something as general as PSR (I assume this is why Alex called it a causal version of PSR). Even many atheists will accept that no rabbit could exist wholly uncaused. If “Nobody and nothing but one of these potential rabbit-makers makes a rabbit,” then it would seem that all the rabbits are wholly uncaused. I think even many of those skeptical of PSR would balk at that.

    October 6, 2009 — 18:02
  • Joshua Rasmussen

    In the above comment, I’m building into the scenario that the rabbits don’t themselves reproduce.

    October 6, 2009 — 18:26
  • Joshua Rasmussen

    and that no rabbit ever dies…

    October 6, 2009 — 18:38
  • Joshua Rasmussen

    How about this?
    An attracter is a particle that attracts all existing particles toward itself at a finite rate.
    A repeller is a particle that repels all existing particles away from itself at a finite rate.
    Now if it is possible for any contingent thing to be infinitely old (which I’m assuming is possible if it is possible for there to be an infinite past), then it should be possible for there to be an infinitely old attracter. And if it is possible for there to be an infinitely old attracter, then it should be possible for there to be an infinitely old repeller. What do you think?
    If this last inference is doubted, then consider instead a toggler, which is a particle that randomly alternates from attracting all existing particles to repelling them and from repelling to attracting… If an attracter could be infinitely old, then surely so could a toggler. But if a toggler can be infinitely old, then by a principle of recombination, it should be possible for there to be an infinitely old toggler than happens to repel more than 50% of the time. Call such a toggler a repelling togler.
    If either a repelling toggler or a repeller could be infinitely old, then assuming there could be more than one infinitely old repeller or repelling toggler, then
    it would be possible to for there to be particles that are infinitely far away. But that’s a very counter-intuitive result.

    October 6, 2009 — 19:51
  • Joshua Rasmussen

    Oops: the “What do you think?” should go at the end. 🙂

    October 6, 2009 — 19:52
  • Mike Almeida

    Forget the rabbit-maker. I’m beginning to think that Josh produce infinitely many posts, and in a finite period! 🙂

    October 6, 2009 — 20:06
  • Mike Almeida

    If “Nobody and nothing but one of these potential rabbit-makers makes a rabbit,” then it would seem that all the rabbits are wholly uncaused. I think even many of those skeptical of PSR would balk at that.
    I’m less sure. The main objections (say, Rowe’s objections) to PSR are largely to the causal version. He doesn’t balk at all at the idea that, say, it is not apriori (or otherwise necessary) that causal PSR is true. So there are worlds (perhaps not ours) in which uncreated rabbits exist. My point is that the c-e presents no worries for PSR.

    October 6, 2009 — 20:12
  • Joshua Rasmussen

    Yes, though there are different causal versions. My sense with Rowe is that he’s more worried about the principle that every positive contingent fact about the existence of concrete things has an explanation (which is different than the principle that every contingent thing has a cause.) I recall van Inwagen somewhere saying that it may be plausible that every contingent thing has a cause, but it is not plausible that every contingent fact has a causal explanation…
    ok, I should maybe stop posting so rapidly. 🙂

    October 6, 2009 — 20:42
  • Gordon Knight

    Re: your second argument, could not the contingent thing be the infinite series of Bobs?

    October 7, 2009 — 7:44
  • Mike Almeida

    My sense with Rowe is that he’s more worried about the principle that every positive contingent fact about the existence of concrete things has an explanation (which is different than the principle that every contingent thing has a cause.)
    The version of PSR that Rowe discusses is perfectly general, covering states of affairs generally (positive facts, in his terms) and objects. It is certainly true that not every contingent fact does not have a cause unless omissions are causes. I don’t think we want that (though of course there’s lots of dispute here) since it entails that I brought about infinitely many states of affairs. A state of affairs for every one of those infinitely many bird houses that fail to exist in my backyard, for instance.

    October 7, 2009 — 8:51
  • John Alexander

    Does not 1 rest on the assumption that there are an infinite number of January 1sts? If there are not then on the first January 1st there were no rabbits, etc. There does not seem to be a contradiction in maintaining that there is an infinite past without there being an infinite number of January 1sts.

    October 7, 2009 — 9:21
  • Josh:
    The attractors and repellers are interesting. However, I don’t think you just get to say it’s always existed to use the kind of principle of recombination that my examples implicitly use. In my examples, I recombine the initial locations of the objects. That doesn’t work for infinitely old objects that don’t have initial locaitons.
    I am now getting worried about the principle of recombination behind my examples. Intuitively, what I’m doing is taking an entities with a coherent specification of their nature, specifying their initial spacetime positions, and then letting them do their thing.
    But on reflection one must perhaps be very careful here, lest the apparently coherent specification of the nature turn out to be subtly inconsistent with other natures. For instance, if an entity is irresistable repeller and another is an irresistable attractor, we can’t put them both close together and then put another particle far away for them to fight over.
    But maybe all possible natures are actually consistent with one another. There cannot be any irresistable repellers and attractors.
    I am now getting rather worried about the following argument against me. Take Thomson’s lamp. The obvious thing to say about it is that specifying that infinitely many pushes of the switch have happened underspecifies the outcome at the end of the supertask, and the story that infinitely many pushes have happened is, thus, quite consistent with both the lamp being on at the end and with the lamp being off at the end.
    Now, suppose we supplement the story with two assumptions:
    1. A causal principle for states of the lamp: The lamp’s being on as well as its being off is always caused.
    2. Nothing but the pushes of the switch causally affects whether the lamp is on.
    Once we supplement the story with these two assumptions, the story becomes incoherent. But the defender of supertasks can say: “Of course the story is incoherent, because you’ve ruled out, in the description of the story, the possibility of an explanation of the final state, and then said that there is an explanation. You story is just as uninterestingly incoherent as this story: ‘Imagine that God ex nihilo creates a lamp at t0 satisfying (1) and (2) above.’ This story is also incoherent because the lamp has to be either on or off at t0, but no switch-push happened prior to t0. But it would be absurd to take this as, say, an argument against omnipotence. We’ve simply taken a situation–God ex nihilo creates a lamp without determining whether it’s on or off–and then added the claim that nothing else determines this and that, nonetheless, it is determined by something.”
    I am now quite worried that this happens in the above cases, as well as in the original GR case. We’ve described a story from which we can derive that Fred is dead without there being a cause of death in the story, and then we’ve added two assumptions: (1) the story is relevantly complete, and (2) Fred’s being dead has a cause. Obviously, this yields a contradiction.
    I think the answer to this has to make use of a really carefully crafted principle of recombination. That’s going to be hard work. And it’s going to have a dialectical problem, in that we want to use this in an argument for theism, but theism is incompatible with standard principles of recombination. (Standard principles of recombination entail it’s possible to have a world all the contingent persons in which are innocent but suffer horrible torment for eternity and get no goods out of that. But this is not possible, given theism.)

    October 7, 2009 — 10:19
  • Joshua Rasmussen

    Well Alex, maybe we can make do with a certain restricted principle of combination if we also use a certain principle of “modal continuity”. Take the GR case. Even if some possible objects or states are incompatible with each other, it seems intuitive that any collection of GRs would be compatible with each other (I’ll assume there could even be an infinite collection). Now the question is, why think the GRs could be temporally ordered in the paradoxical way if there could be an infinity of events? Well, perhaps it’s because of a principle of modal continuity in combination with the very intuitive premise that the setup of one GR (the time when it would fire if Fred isn’t dead) isn’t dependent upon the setup of any others. The principle of modal continuity adds that there is no interval v, such that it would be impossible for two events to elapse within an interval of time shorter than v. All of this seems quite reasonable to me. Notice that Fred’s being dead without a cause isn’t built into this story–rather that’s an implication of it. Perhaps work needs to be done to spell out why we may believe in a principle of recombination implicit in the above story without also believing in a more general principle.
    The cases of my repellers/attractors and your rabbits might be more susceptible to the type of worry you expressed. Take a parallel of rabbits: Let M be a type of device that produces an everlasting blue ball if and only if no blue ball exists. I don’t see why an instance of M would be impossible. And if the past could be infinite, it may seem that there could always be at least one instance of M. The result is that it’s possible for there to be a blue ball that was never caused to exist if we build into our story that nothing but an instance of M causes a blue ball. But then it seems like we are building the incoherence into the story from the start. Is this making sense? I’m not sure this same problem arises in the case of GRs…
    In the case of repellers, we build into the story that there can be more than one eternally old repeller, which maybe pressupposes that there can be infinitely distant things… But this isn’t completely clear to me. Maybe I don’t have a principle of recombination for the “starting” locations, but it does seem really weird to me that there could be multiple infinitely old attractors but not multiple infinitely old togglers, and if there can be infinitely old togglers then by a principle of recombination of the natures of these togglers, I’m tempted to think that there could be infinitely old repelling togglers.
    I think the issue here is whether the incoherence that emerges was implicitly built into the scenario from the start or whether it is purely an implication of it.

    October 7, 2009 — 13:33
  • Joshua Rasmussen

    On second thought, in my M story (parallel of your rabbits), perhaps we don’t need to build in that only an instance of M causes a blue ball. We just build in that the only concrete things that aren’t blue balls are instances of M (putting theism to the side here). Then perhaps a blue ball’s existing uncaused is a consequence rather than implicit presupposition (analogous remarks for the rabbits). (BTW: in the GR case, I think it’s implicitly understood that Fred’s being caused to die by something other than a GR shouldn’t logically depend upon whether a GR situation is setup.)

    October 7, 2009 — 14:39
  • Thomas Feeney

    Re: the first argument, if PSR is true, then we should expect surprising restrictions on possibility – but why think that the argument shows the impossibility of an infinite past rather than the impossibility of the scenario itself given an infinite past?
    Is this the sort of worry that has turned the discussion toward principles of recombination, i.e., independent guides to what’s possible? But again, if we accept PSR, then we should frame our principle of recombination with PSR in mind. It’s not the sort of principle that can be introduced second, once other more basic principles are in place. And if we let PSR shape our principle of recombination, then perhaps rabbit makers of the sort described will be ruled out.
    This leads me to a broader question: do principles of recombination range over concrete objects only, or over spaces, times, space-times, and the like (assuming these are not concrete)?

    October 7, 2009 — 22:57
  • Joshua Rasmussen

    How about something in the neighborhood of this?
    Axiom 1: Necessarily, no contingent object is such that it exists only if certain other [jointly possible] contingent objects do not (we may want to reword this to avoid talk of “possible” objects).
    Axiom 2: Necessarily, if the members of every finite subset of an infinite set S are jointly possible (can all exist in the same world) and if there is no spatial limitations to fitting all the members of S into a single world, then the members of S are jointly possible.
    Axiom 3: Necessarily, there are no non-overlapping spatial regions r and s, such that s is occupied only if r is not occupied.
    Axiom 4: For any possible event [e.g., of killing Fred], there is no smallest possible object that can cause an instance of that type of event [modal continuity].
    Axiom 5: For any size of an object, there is no quickest possible event [e.g., of killing Fred] that results from an object of that size [modal continuity].
    Axiom 5b: For any size and function of an object and for any times t0 and t1, it is possible for an object of that size to begin to perform that function after t0 and before t1 [modal continuity].
    Axiom 6: If there can be an infinity of past events, then there can be an infinite series of events in which each event results in the existence of a GR.
    Axiom 7: Necessarily, no collection of things is such that necessarily, if none of the members of that collection cause a certain event [e.g., kill Fred], then there is something else that causes that event.
    Theorem: If there can be an infinity of past events, then there can be a paradoxical GR scenario.
    Proof (I realize this isn’t rigorous and is perhaps in need of modifications): Once a GR is produced, it can continue to exist along with any of the others to be produced [axiom 1]. Since this is so for every GR, it follows that the members of every finite set of the GRs to be produced in any infinite series of GR producing events are jointly possible. For any infinite series of GR producing events, there are no spatial limitations preventing the infinitely many GRs produced from fitting into a single world [from axioms 3 and 4]. Therefore [from axiom 2], for any infinite series of GR producing events, it is possible for all the GRs produced to exist in a single world. There can be an infinite series of GR producing events in which the earlier GR’s produced are going to begin to kill Fred if he’s still alive at times that are later than 12:00 by increasingly shorter time intervals [axiom 5b] and with increasing quickness [axiom 5]. In light of the preceding conclusion, this possibility combined with axiom 7, allows a paradoxical GR scenario to be set up.

    October 10, 2009 — 12:03
  • That’s the sort of thing I was looking for.
    By the way, I can like the symmetry of the following variant. An RGR (randomized GR) is a critter which has the following property. At noon on January 17, 2010 it chooses a random time between then and 1 pm, uniformly distributed over that interval; when that time comes, if Fred is alive, it kills him, and if Fred is not alive, it does nothing. Fred is alive at noon.
    Now, given an actually infinite number of past spacetime regions of some fixed size, we could imagine that a new RGR comes into existence in each of these regions. The RGRs are all duplicates of one another. But as they reach noon on January 17, 2010, they randomly and independently choose activation times for themselves.
    Theorem: With probability one, the wakeup times will be dense in the [noon,1pm] interval.

    October 11, 2009 — 16:39
  • My randomized scenario presupposed instantaneous action for the GRs. We can do without that assumption. An RGR acts as follows:
    1. It activates at a random activation time t0 between noon and 1 pm, uniformly distributed over that interval.
    2. When t0 comes, it executes a process which takes a random amount T1 of time (any fixed continuous distribution with non-zero density at 0 will do for the argument–e.g., an exponential decay or a cut-off Gaussian). This process has the following properties:
    2a. If Fred is dead at all the times in [t0,t0+T1], it has to return the verdict “dead”.
    2b. If Fred is alive at all times in [t0,t0+T1], it has to return the verdict “alive”.
    (It doesn’t matter what happens if Fred is alive at some but not all times in [t0,t0+T1]. Any story you tell will let the paradox go through.)
    3. If the verdict is “alive”, then at t0+T1, the RGR executes a process which takes a random amount T2 of time (again, we suppose a fixed continuous distribution with non-zero density at 0) to kill Fred; if the verdict is “dead”, the RGR does nothing else.
    4. All the random variables, within and between the different RGRs, are independent.
    5. Fred is alive at all times prior to noon.
    6. Fred does not die except by the action of an RGR.
    Contradiction now ensues from the supposition that there are infinitely many RGRs. For if there are infinitely many RGRs, then:
    7. P(there is a time later than noon at which Fred is alive) = 0
    8. P(Fred is dead at every time after noon) = 0
    The argument for 7 is this: If Fred is alive at noon + 1/n seconds is smaller than every RGR that activated between noon and noon + 1/2n must have had T1+T2 > 1/2n. But the probability of this conjunction of claims about the RGRs is zero. So, P(Fred is alive at noon + 1/n seconds) = 0 for every n. By countable additivity: P(for some n, Fred is alive at noon + 1/n seconds) = 0, and (7) follows.
    The argument for 8 is this: If Fred is dead at every time after noon, then either an RGR acted to kill directly at noon, which event has zero probability, or else some RGR acted to kill after noon. If some RGR acted to kill after noon, then Fred must have been alive at some time prior to its action and after noon (by 2a), contradicting the claim that Fred is dead at every time after noon.
    But 7 and 8 contradict one another, since the two events in 7 and 8 are mutually exclusive and exhaustive of the allowed events, and hence their probabilities must add up to 1.
    I know some folks think only impossibilities have zero probability, so they’ll want to replace the zeros in 7 and 8 by infinitesimals. Contradiction still ensues since two infinitesimals can’t add up to 1.
    Now to get an argument against an infinite past, we add the following premises:
    9. Possibly there exists an RGR.
    10. If possibly there exists an RGR and there were infinitely many past events, then possibly there exist infinitely many RGRs.
    And from infinitely many RGRs we get 7 and 8, which are contradictory.

    October 12, 2009 — 11:05
  • Jeremy Pierce

    I’m not sure an infinite past is impossible, but I don’t think these arguments show why. Both of these arguments seem to me to have the following form:
    1. An infinite past does not rule out X.
    2. X is impossible.
    3. Therefore, an infinite past is impossible.
    Isn’t that an invalid inference? It reminds me of the similar argument that I’ve seen from time to time to the effect that time travel is impossible on a fixed view of time because it allows for close loops of causation with no explanation for why the closed loop would be impossible. But the Principle of Sufficient Reason explains why closed loops are impossible. So the fact that time travel on a fixed view of time doesn’t itself rule out closed causal loops doesn’t show time travel to be impossible.
    Similarly, the rabbit scenario is impossible because it violates PSR. The fact that an infinite past wouldn’t by itself explain why the rabbit scenario is impossible is irrelevant. PSR explains why it’s impossible. There’s no need for every metaphysical theory to explain every impossibility.
    Then in the bob case, the principle that explains why the bob scenario is impossible is Principle (*). That principle is all you need to explain the impossibility, so why do we need the infinite past to explain it?

    October 16, 2009 — 19:52
  • Jeremy:
    I think 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, though Josh has made a valiant effort in a comment that might do the job, and the argument in my October 12 2009 comment, above, makes clear in (10) what we need from the rearrangement principle.
    The same kind of thing is, I think, going on in the arguments against time travel.
    Here is a version of the rearrangement principle that isn’t quite strong enough for the argument. We need a notion of a “coherent bundle of causal powers located in spacetime region R” (The effects of the bundle might extend out of R). Then we say: if B is a coherent bundle of causal powers located in spacetime region R, and R1, R2, … are other spacetime regions of the same size and shape, then it is possible to have a world consisting of nothing but necessary beings, plus duplicates of B in R1, R2, …, plus whatever mereological sums, if any, are necessitated by this.

    October 17, 2009 — 10:30
  • Jeremy Pierce

    OK, I think I see what you’re doing. If I do, then I think there’s an easier way to make the point. The real philosophical work being done in the infinite bobs example can be done analogously in the infinite past case by pointing out what an infinite past of contingent beings would require (on libertarian conceptions of contingency anyway; some compatibilists would resist this). I’m not sure he right way to express it is that the possibility of an infinite past requires the possibility of the bobs scenario. It’s more that infinite past scenario involves an impossibility that is also present in the infinite bobs scenario.

    October 23, 2009 — 19:00