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.

__Scenario A__

We add to 1 and 2 that Switch and Switcher have existed for exactly ten minutes.

__Scenario B__

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.

Mike,

You ask:

Your argument seems to assume that, if it was switched, it was switched at some finite time ago n. What requires that?

Condition 1 says that Switcher’s checks take place at one minute intervals. Each check in the series of checks occurs n minutes ago. As I was using “n,” it is any of the natural numbers. With that understood, each check by Switcher takes place “at some finite time ago.”

Condition 2 says that Switch is ON iff it was turned ON immediately after one of these checks. Each check takes place a finite number of minutes ago. Any “switching” takes place immediately after ones of those checks. So any “switching” must have taken place “at some finite time ago.”

Wes,

This seems to be the argument:

(1) Each check by Switcher takes place “at some finite time ago.

(2) Any “switching” takes place immediately after one of those checks.

(3) Therefore any “switching” must have taken place “at some finite time ago.”

My point is that, for any n, it is clearly true that the check & switch could have taken place earlier. And there is no earliest time at which the check & switch must have occurred. So, for all we know from (1) and (2), it is true at every n that the check & switch occurred earlier than n. If that’s true, then there is no earliest finite time at which the check & switch occurred. But then (3) is false: (3) entails that there is an earliest time at which the check & switch occurred.

Mike,

I quite agree that there can be no “earliest … time at which the check & switch occurred.”

I do not agree with this:

But then (3) is false: (3) entails that there is an earliest time at which the check & switch occurred.

(3) has no such entailment. It should be read this way:

(x) (if x is a “switching” then x took place “at some finite time ago.”)

Read this way (3) is obviously compatible with there never having been any switching.

I want to make sure I understand what you are trying to do here. You want to say that Scenario B (in my original post) does not lead to contradiction. You accept my argument (or at least some argument) for saying that Scenario B entails that Switch is ON. But you’re trying to show me that my argument for saying that Scenario B *also* entails that Switch is *not* ON is flawed. That is what we’re debating, isn’t it?

Yes, the debate concerns whether there is a contradiction here. I’m less sure than you are that there is. So, you concede this,

*I quite agree that there can be no “earliest … time at which the check & switch occurred.”*

I’m not sure what this remark (above) means. You might mean that for any time at which the c&s occurs, there is an earlier time at which it *might* have occurred. That is not what I asserted above. What I had in mind is this: *possibly, there is no earliest time t at which the c&s occurred.* To be clear, I’m claiming the following, *possibly, for each t it is true at t that the c&s occurred earlier than t.*

Now to my claim.

Claim: If there is no earliest time at which the c&s occurred, then there is no finite time at which the c&s occurred.

Proof:

(1) There is no earliest time at which the c&s occurred. Assumption

(2) For every time t it is true at t that the c&s has already occurred. From (1)

(3) If for every time t it is true at t that the c&s has already occurred, then there is no finite time at which the c&s occurred. Fact

(4) Therefore, there is no finite time at which the c&s occurred. From (1)-(3)

Ok, simpler version. For all we know (i) is true in the case you describe.

i. For each time t, it is true at t that the c&s has already occurred.

ii. If, for each time t, it is true at t that c&s has already occurred, then there is no finite time at which the c&s occurred.

iii. Therefore, there is no finite time at which the c&s occurred.

Wes,

It seems to me that my Grim Reaper based arguments against the infinite past use some sort of an intuitive rearrangement principle. I don’t know how to work out such a principle, and it may be that the problem is insuperable in which case the Grim Reaper arguments lose a significant part of their strength, but it is clear to me that the principle wouldn’t apply to your case. The reason for that is in the GR cases, the rearrangement principle is supposed to allow one to place copies of possible objects with specified dispositional natures at as many distinct spatiotemporal slots as one wishes.

But in your case, the specification involves more than a dispositional nature: you’ve given a historical condition for the Switch in condition (2), and there is no reason to think that a rearrangement principle will let one do that.

Here is an even simpler version of your arrangement:

(1) Every minute, the Switcher presses the Switch in the ON direction.

(2′) Switch is off at some past time.

This is compossible with any finite age of the Switch, but not with the Switch having infinite age.

But (2) and (2′) are historical conditions, and hence it is unsurprising that they are not compatible with some past histories. Imagine the Historian who never forgets any year, but who (2”) once knew nothing. This is compatible with a finite age, but not an infinite one. And no reasonable recombination principle will allow it.

On the other hand, it seems that a reasonable recombination principle should allow one to place a new GR at every past January 1, and Fred at a couple of days before the interval over which the GRs are supposed to go off, where it is stipulated that objects persist unless there is a cause of their destruction.

Now, I am a little worried that the stipulation that objects persist unless destroyed is like the historical conditions (2), (2′) and (2”). But I think it’s only *somewhat* like them. For one, it might well be the case that it’s a necessary truth that objects persist unless there is a cause of their destruction, and so this isn’t an additional stipulation. For another, this stipulation seems to be a part of the dispositional natures of objects.

But this is of little help to me unless I work out the rearrangement principle.

Mike – Sorry, I still don’t understand why Scenario B isn’t obviously impossible. I should not have agreed that there “was no earliest time at which check & switch occurred” and left it at that. What I had in mind was that given my specification of the case there could have been *no time* at which a check was followed by a switch.

Alex – I think you must agree that my Scenario B is impossible, and also that this throws no light on the question whether the series of discrete past events has a beginning. But you think that with a yet-to-be-worked-out “recombination principle,” an argument involving GRs may do the job. It’ll be interesting to see how that plays out.

Thanks to both of you for helping me think this thing (part way) through!

I found your other examples of “historical condition” that can’t be combined with an infinite past to be helpful and illuminating.

I wonder if the Grim Reaper case might not simply serve as a reductio against any sort of rearrangement principle you formulate. It seems to me perfectly conceivable that some universes contain infinite pasts, and it’s difficult for me to intuit why the possibility of these universes should really be touched by such carefully concocted thought experiments. I’m reminded of the situation with time travel: clearly there are some time travel scenarios which are impossible involving inconsistent time loops, e.g., where I kill my distant ancestor; but many philosophers don’t take these to be conclusive proof that consistent time loops are impossible.

Or maybe a rearrangement principle, if one can be properly formulated, shouldn’t apply to such fanciful entities as Grim Reapers. It seems that they’re uniquely what’s causing the problem. Imagine you have just two Grim Reapers, each set to kill the other at time t if the other is alive just before t and not doing anything at t. What happens at time t? Or imagine that you have an infinite number of Grim Reapers stretched out in a line (or a ray, more accurately) labeled “0,” “1,” “2,” and so on *ad infinitum*. Suppose each one is set up to instantaneously kill/deactivate all the previous GR’s at time t. Is GR 0 alive at time t? It seems that it shouldn’t be, but none of the GR’s are alive at time t to kill him. Is this an argument against the possibility of the infinitude of space? But then you could imagine that each one gets smaller and smaller in geometrical progression, so that they only take up a finite amount of room. So maybe it’s just the infinitude of GR’s that’s impossible!

Alex –

I did have one more thought about this. Even if you come up with a plausible “recombination principle” that allows you to put in all the infinitely many GRs who “go off” at all the staggered times, and the existence of GRs (considered merely in terms of their “dispositional natures”) is possible in all the required spatiotemporal slots, it remains the case that you get a contradiction *only* because of the assumptions you have made about Fred – to wit, that he dies only at the hand (or whatever) of a GR and that he isn’t resurrected. It seems to me that this is a little like saying that I get a contradiction in my Scenario B only because of the extra assumption about Switch: viz., that it goes into the ON state iff Switcher turns Switch ON immediately after one of its checks.

I’m not at all sure about this, but maybe the general moral I was going for does apply to the GR scenario. Schematically, it works like this. We make up a scenario S that includes X, Y, and Z and that entails a contradiction. We then argue that it’s possible that X and conclude that Z is impossible.

So… Let X be the conjunction of all the Grim Reapers, let Y be the extra assumptions about Fred, and let Z be the infinite past. The easy (too easy?) rejoinder is: “Take out Fred.” No GRs “go off,” the past is infinite, and the contradiction disappears.

*Mike – Sorry, I still don’t understand why Scenario B isn’t obviously impossible. I should not have agreed that there “was no earliest time at which check & switch occurred” and left it at that.*

Wes,

It is not obviously impossible; my claim depends on the modal principle (P), which is very plausible given the linear structure of past time.

P. (Vt)M(F occurs before t) => M(Vt)(F occurs before t)

(P) reads intuitively that if for all times t it is possible that F occurs before t, then it is possible that for all t, F occurs before t. (P) holds in all finite cases. Take any finite sequence of times into the past t3, t2, t1, . . ., tn. If it is true for each of those times that it is possible that F occurs before that time, then it is possible that F occurs before tn. But if it is possible that F occurs before tn, then it is possible that F occurs before every time in the finite sequence (tn is the earliest time in this finite sequence).

I’m using (P) in your case of an infinite sequence of times into the past. I instantiate (P) as follows,

PI. (Vt)M(c&s occurs before t) => M(Vt)(c&s occurs before t).

Can you give me some argument why (PI) fails in the infinite case? Certainly the antecedent of (PI) is true; why doesn’t the consequence follow given the structure of past time? It’d be good to have an argument that does not appeal to it being just obvious, since I don’t find it all that obvious.

Wes:

‘The easy (too easy?) rejoinder is: “Take out Fred.”‘

Well, let’s imagine it this way.

The condition on Fred and all the GRs is that they continue to exist unless they are caused to cease to exist by something outside them (they have no internal causes of death). The universe right now contains nothing but the GRs and a deity who is pretty powerful and can create ex nihilo. There is no Fred. The Grim Reapers will go off at t1, in ten years (more precisely: between ten years and ten years plus an hour from now). The deity is powerful enough to create Fred, and no contradiction results from the deity creating Fred, as long as the deity annihilates Fred within ten years. So, the deity has created Fred, and there is the deity, there is Fred, and there are the GRs. No absurdity!

But now here is something rather weird: the deity has to destroy Fred or destroy infinitely many of the GRs or resurrect Fred. But that’s really weird. There is the deity, there is Fred and there are the GRs–what forces the deity to destroy Fred or destroy infinitely many of the GRs or resurrect Fred?

Let’s vary the case a little (this is kind of like my widget stuff). Instead of the deity, we have a machine that exactly three minutes before t0 flips a coin, and if it’s heads, kills Fred. No incoherence results so far. But the machine has to flip that coin. What forces it to do so?

M.:

Very interesting that you brought up time travel. Exactly parallel worries come up there. There you are, with your gun, looking at your grandfather before your father was conceived. It’s impossible that you kill him. But why? Let’s say your marksmanship is such that you’ve got a 50% chance of hitting. But in this case your chance of hitting had better be 0. What stops you?

Suppose you shoot but are so nervous that you miss. Then your grandfather’s survival is explanatorily prior to your nerves. For were he not to have survived, you wouldn’t have existed, and hence you wouldn’t have been nervous. But your nerves are explanatorily prior to your grandfather’s survival. So we have an explanatory priority circle, which is absurd.

Considerations like this make me think that if time travel is possible, substantial causal interaction between the traveler and events in the traveler’s history is impossible. This places a severe constraint on the possibilities for time travel. Indeed, on a broad interpretation of “the traveler’s history” as the union of backwards light cones centered on the traveler’s past spatiotemporal locations, it makes time travel to the absolute past impossible–one cannot travel into one’s own backwards light cone, or else explanatory circularity results. One can, perhaps, travel to space-like separated locations, and in some reference frames this will count as time travel, but that’s all.

Now, an interesting question is whether a similar restriction can be made on the possibility of an infinite past: you can have an infinite past, but only in a context where you’re not going to be able to generate GRs. I don’t know. The restrictions had better not be ad hoc. I wonder if the restrictions might not be really severe like: you can only have an infinite past in a world without causal interaction. If so, then the Kalaam arguer still is OK, since our world doesn’t satisfy this restriction. But I don’t know.

The suggestion that GRs would be a counterexample to any plausible rearrangement principle is intriguing.

“Imagine you have just two Grim Reapers, each set to kill the other at time t if the other is alive just before t and not doing anything at t. What happens at time t?”

I am thinking, probably because of similar cases at the back of my mind, that the “dispositional natures” will need to require non-instantaneous action. If so, then it’ll take a bit of time for the GR to notice that the other is not doing anything at t, and so both will kill each other, a bit after t. (Yes, the standard GRs can be made to work non-instaneously. Hawthorne notes this.)

A different worry is this. The GR stories are all predicate on an “Aristotelian” notion of causation–causation being a relation between particulars with causal powers. If one sees causation as coming from global causal laws, the stories will have little plausibility, since the scenarios can be simply taken to show that the global causal laws presupposed by them are impossible. So the worry is that someone might simply take the stories to refute causal particularism.

*But your nerves are explanatorily prior to your grandfather’s survival. So we have an explanatory priority circle, which is absurd.*

Alex,

I’m not sure your nerves play any role in explaining why your grandfather survived. Notice that no one else can kill your grandfather either, though they might also be standing there ready to kill him and 99% accurate with a rifle. Your nerves play a role in explaining why you did not kill the person P who happens to be your grandfather. It is possible for you to kill P. But had you killed him, he would not have been so related to you.

Mike:

I believe in essentiality of origins. ðŸ™‚

But in any case, let’s call him P. Then my nerves are explanatorily prior to P surviving. But that P survives helps explain why I exist, and hence that P survives is explanatorily prior to my nerves. So the argument works with a rigid designator P, too.

Alex

Alex,

That sounds right, I think. But mutual explanations are not so troubling. Pressing my hands together, each hand explains why the other doesn’t move further, or each of two playing cards leaning against one another explains why the other does not fall, similarly with stones in archways, etc. There’s a lot of mutual causation, and so a lot of mutual explanation. None of these circles seem incoherent.

Thanks for your response!

I don’t quite see yet why the following very simple restriction isn’t reasonable: universes can have infinite pasts only if their global historical distribution of physical laws, events, objects and/or objects’ dispositions don’t result in a contradiction. Is this ad hoc? Or maybe we could try something weaker: a universe U can have an infinite past only if it’s impossible to describe a temporally paradoxical event *involving only objects and causal powers that can be found within U itself*. Since nothing like an infinite amount of Grim Reapers exists in this universe (nor does arbitrarily fast causation appear to be physically possible here, if I’m not misunderstanding the Planck time and all that; but I’m no physicist), we can still have an infinite past.

The position of card A at t0 explains why card B does not move at t1, where t1 is at least t0+x/c where x is the minimum distance between a molecule of A and a molecule of B. And the position of card B at t0 explains why card B does not move at t1. Suppose that card A were annihilated right at t0. Then, card B would still be standing at t0 just as it was before, until t0+x/c.

The same can be said for the other cases.

Even with Newtonian physics and solid objects, it’s not clear that the case works. The reason for that is that card A’s position at t0 does not explain why card B at t0 is where it is. Rather, card A’s position at t0 explains why card B is not beginning to move at t0 (i.e., why it’s not accelerating), and card B’s position at t0 explains why card A is not beginning to move at t0. But it is false that card A’s position at t0 explains card B’s position at t0.

And in any case, card A’s position at the first moment the cards are placed in position does not explain card B’s position at that moment.

Hi Wes,

I think the general lesson you draw is a good one: showing that p + an infinite past is impossible doesn’t show that an infinite past is impossible, even if p seems possible in its own right. What is needed is an argument for thinking that if p is possible and if an infinite past is possible, then p + the infinite past would be possible, which it is not. In your switcher scenario, I haven’t been able to think of a way to do that.

There, the problem as I see it is that we need general principles to help us out, yet no independently plausible principles guarantee the possibility of both conditions given an infinite past (at least I have yet to think of any). One principle that *might* help guarantee condition (2) is a causal principle that requires the button to be caused to be on. But this principle is in tension with the possibility that the switch had no beginning (when would it be caused?)… Thus, all I am able to conclude from your scenario is that it isn’t compatible with an infinite past; I agree that it would be a mistake to conclude that an infinite past is thereby impossible.

But in the case of the GRs, I think there may well be independently plausible principles that would guarantee the possibility of a GR scenario were an infinite past possible. In fact, I’ve recently been tentatively persuaded that an infinity of past events is impossible because of principles in the neighborhood of what I suggested here:

http://prosblogion.ektopos.com/archives/2009/10/two-more-argume.html#comment-107275.

Of course, it may be that an infinite past is possible while an infinity of past events is not.

Link without the period: http://prosblogion.ektopos.com/archives/2009/10/two-more-argume.html#comment-107275

*The position of card A at t0 explains why card B does not move at t1, where t1 is at least t0+x/c where x is the minimum distance between a molecule of A and a molecule of B.*

Ok, but it has nothing to do with what I said. I said this,

*. . . each of two playing cards leaning against one another explains why the other does not fall. . .*

I said nothing about the positions of the cards. And of course on any analysis of causation you please, it better turn out that card A explains why card B does not fall to the table.

Dr. Pruss, one more question. In order to make the standard GR case work without using instantaneous causation, we have to make the GR’s kill the subject faster and faster the closer the GR activates to time t. For instance, suppose GR #n, the GR that activates at time t + 1/n seconds, takes an additional 1/n seconds to kill the subject (assuming he was alive at t + 1/n to begin with). But it appears on the surface of this description that the GR’s have different causal dispositions: some are disposed to kill the subject much faster than others. Therefore I have difficulty seeing how a rearrangement principle could be brought to bear here – the GR’s simply aren’t perfect duplicates of one another!

Mike:

First of all, cards don’t do any explaining. What does the explaining is propositions. So, the claim is presumably something like: That card A is in such-and-such a position explains that card B is in such-and-such a position. But this isn’t true at all times, but only on an interval of times. And so, when we make this precise, the temporal stuff becomes relevant.

M.:

This is a nice point to pick up on. And that’s part of why it’s nice to do this probabilistically. On a probabilistic setup, the GRs are indistinguishable until, say, ten minutes before the interval J over which they activate. (So we apply the rearrangement principle prior to that moment.) Make sure J does not contain its lower bound–i.e., that J is open at the lower end. Then at that moment each independently and randomly picks both an activation time, within J, and a duration of the detection and kill process. The activation time is chosen uniformly randomly within J. The duration can be chosen uniformly over some interval or via an exponential decay probability distribution or whatever.

Then we have a Theorem: With probability 1, it is the case that for every time t in J, there is a GR that (a) activates at some time t’ < t, and (b) whose duration of activity d satisfies t’ + d < t.

In other words, in this arrangement, the probability of paradox is 1. But a paradox is an impossibility, and the probability of an impossibility is 0. So 1=0, which is absurd. ðŸ™‚

*Mike: First of all, cards don’t do any explaining.*

Yes, right, I’m aware of that. I talk of cards doing the explaining deriviatively. It’s common and natural to speak of objects as standing in causal relations, even if they are not directly so. On my view, events stand in causal relations, not propositions. So if anything at all turns on being precise right at this particular point (and surely nothing does) I’d say that the leaning of card A against card B explains the failing to fall of card B or the remaining standing of card B, and vice versa. Something along those lines. But just to repeat, I said nothing about explaining the position of the cards. The position of the cards has a much more involved explanation.

But aside from that, I wasn’t posting a thesis about the true analysis of causation. Before the sidetrack into technicalities in the analysis of causation, the point was simply that there are benign circularities in causal explanation that virtually everyone recognizes. Maybe you don’t, who knows. Analyses of causation generally accomodate those circularities. The point is that not every circularity in explanation is somehow logically devastating or incoherent, as you seemed to be suggesting.

Mike:

“I’d say that the leaning of card A against card B explains the failing to fall of card B or the remaining standing of card B, and vice versa.”

Let’s fill out the vice versa: “the leaning of card A against card B explains the failing to fall of card B or the remaining standing of card B, and the leaning of card B against card A explains the failing to fall of card A or the remaining standing of card A.”

But this is not circular, because you don’t have EAB explaining FBA and FBA explaining EAB, but rather you have EAB explaining FBA and EBA explaining FAB.

Moreover, for precision, you need to specify the time. I am taking your assertion to be in the present tense. But it can’t just be talking about the present moment, because A’s leaning against B in the present moment does not explain B’s failing to fall in the present moment. So, the assertion has to be about leanings and failings to fall that are temporally extended. So the explanation has to be over an interval. And then there are issues about the beginning. The interval had better be open at its lower end. And the explanation then is regressive in nature. This stuff is tricksy.

*But this is not circular, because you don’t have EAB explaining FBA and FBA explaining EAB, but rather you have EAB explaining FBA and EBA explaining FAB.*

That might be an artifact of the particular case and description. Take a simpler case. You might have two trains, at some throttle otherwise sufficient to move them, face to face, so to speak, on a train track. The failing to move of T1 explains the failing to move of T2 and vice versa. Over a suitable interval, we seem to have a case of benign circular explanation.

Wes (or anyone, for that matter) I wanted to move this comment down front. It would be great to learn where this goes wrong. Wes writes,

*Mike – Sorry, I still don’t understand why Scenario B isn’t obviously impossible. I should not have agreed that there “was no earliest time at which check & switch occurred” and left it at that.*

Wes,

It is not obviously impossible; my claim depends on the modal principle (P), which is very plausible given the linear structure of past time.

P. (Vt)M(F occurs before t) => M(Vt)(F occurs before t)

(P) reads intuitively that if for all times t it is possible that F occurs before t, then it is possible that, for all t, F occurs before t. (P) holds in all finite cases. Take any finite sequence of times into the past t3, t2, t1, . . ., tn. If it is true for each of those times that it is possible that F occurs before that time, then it is possible that F occurs before tn. But if it is possible that F occurs before tn, then it is possible that F occurs before every time in the finite sequence (tn is the earliest time in this finite sequence).

I’m using (P) in Wes’s case of an infinite sequence of times into the past. I instantiate (P) as follows,(with c&s for check and switch),

PI. (Vt)M(c&s occurs before t) => M(Vt)(c&s occurs before t).

Can you give me some argument why (PI) fails in the infinite case? The biggest obstacle to doing so, I think, is that certainly the antecedent of (PI) is true. How to explain then why the consequence does not follow given the structure of past time? It looks like the consequent has to follow if the antecedent in (PI) is true. But if that’s so then Wes’s Scenario B involves no contradiction.

Mike:

We’ll still run into trouble at the beginning of the interval.

Suppose the interval is closed. Then it has a first moment, t0. At t0, the first train’s not-moving is explained by the second train’s not-moving *prior* to t0, i.e., before the interval. So it’s false that the train’s not moving during the interval is explained by the other train’s not moving during the interval.

Suppose the interval is open, and of the form: I = { t : t0 < t < t2 }. Let d be the minimum distance between particles of the two trains during the interval. Then d > 0. Let t1 = t0+d/(2c). Then, why the first train is not moving at t1 has nothing to do with anything the second train is doing during the interval I, because no particle in the second train during I is in the past light cone of any particle of the first train at t1.

*We’ll still run into trouble at the beginning of the interval.*

All I need is *some* interval during which there is benign circular explanation. You’re not suggesting (are you?) that the non-movement of T1 has nothing to do, during any interval, with the non-movement of T0? That’s all I need to illustrate the fact that such events are not logically worrisome.

Question: if the argument presented by Wes (and possibly similar ones offered in earlier threads) demonstrates that there is no infinite past does it not then prove that there is an infinite future? If one asserts that the goal of the switcher is to turn on the switch then once it is on the switcher cannot turn it off. So once it is on there will never be time when it is off. This seems to be consistent with 1 and 2.

Dr. Pruss,

That’s ingenious! I also see now that you had discussed it in the comments section of a past posting of this blog, so apologies for the redundancy my comment engendered.

I’m wondering now if the paradoxical nature of the example just comes out of the weirdness of infinite stochastic processes in general. Suppose a switch is set to off (and will stay that way unless some individual affects it), but an infinite number of people P1, P2, P3, … all want to turn it on. They simultaneously flip a coin at time t. If everyone’s head comes up tails, the switch stays off. Otherwise, the person with the largest index whose coin comes up heads gets to flip the switch. Then the probability that the switch gets flipped is 1, but the probability that anyone flips it is 0. Paradox!

It seems intuitive here that the problem simply lies in a quirk of the setup. We’re not guaranteed the existence of “the person with the largest index whose coin comes up heads,” even though it’s possible (with probability zero) that one exists. Similarly, all of the Grim Reapers in the probabilistic GR thought experiment are intuitively compossible, since there’s nothing preventing them all from simultaneously existing if, say, their victim dies before or at the moment they activate. But as soon as we add the stipulation that the victim is alive when they activate, things go haywire. So perhaps we should simply conclude that it’s impossible for the victim to be alive by the time of the thought experiment, not that the existence of infinitely many past times is impossible.

Whoops! Bad typo there. I meant to write, “If everyone’s coin comes up tails,” not “if everyone’s head comes up tails!”

Here’s another probabilistic oddity involving space. Suppose Tim is alive and standing somewhere at time t. There is a countably infinite sequence GR1, GR2, GR3, … of Grim Reapers who do the following: at time t, they all activate. Thereafter, the nth GR flips a coin. If the coin lands on tails, it does nothing. If the coin lands on heads, the GR magically creates a bullet out of thin air precisely 2 – 1/n feet in front of Tim (we may have to assume the bullets get smaller and smaller so they don’t overlap) travelling at a velocity of 1 foot per second toward Tim. This process takes exactly one second for each GR. Assume Tim, if struck by a bullet, dies too quickly for the next bullet to hit him while still alive (this doesn’t require Tim be able to die arbitrarily quickly, given the constant velocity of the bullets).

Then with probability 1, Tim dies from a bullet wound; but the probability that any bullet kills him is 0.

Actually, to make that last example more in accord with the spirit of your probabilistic GR case, we should imagine each Grim Reaper being designed to, at time t, choose a positive integer n according to some discrete probability distribution and place a bullet 2 – 1/n feet away from Tim, etc. Also stipulate that if two Grim Reapers choose the same n, then no additional bullet is placed. Then the GR’s are identically disposed up till time t, a la your example.

M.:

My intuition is that in my case this is not just the weirdness of infinite stochastic processes. (I have reason for this intuition: namely, that long years of work in probability theory have inured me to the weirdness of infinite stochastic processes, but I still find my case weird.)

“If everyone’s [coin] comes up tails, the switch stays off. Otherwise, the person with the largest index whose coin comes up heads gets to flip the switch.”

This says nothing about the case where somebody’s coin comes up heads, but there are infinitely many other coins that come up heads. So it is not a complete specification of a case.

Here is a variant which might work better for you. There is a balloon. You have uncountably infinitely many people, and each one holds a different real number. All the real numbers between 0 and 1 are represented. You choose a random number between 0 and 1 uniformly, and the person who chooses that number pops the balloon. With probability 1, the balloon is popped. For any particular person x, the probability that x pops the balloon is zero. However, this is not paradoxical. Quantifiers and probability operators don’t commute.

In your second case, I think the probabilities make no difference. Just remove the stochastic aspect, and hence suppose that everybody makes a bullet materialize, and you get the standard GR story.

Getting back to my case, your suggestion is that we’ve just shown that Fred has to die before the time they all start activating. But that’s really weird. Suppose there is only one other thing that can kill him. What forces it to do so? (Logic never forces, only persuades. ðŸ˜‰ )

Nothing seems to force Fred to die, but it also seems like nothing forces me not to shoot my own grandfather once I’ve traveled back in time and have the crosshairs pointed at his head. But if I’ve gotten that far in the first place (assuming I *could* get that far, which you seem to deny above), clearly something about history had to have conspired to make me miss, or refrain from shooting. Maybe I should say that if Fred has lived long enough to be alive at t, something about history simply has to have conspired to ensure that no infinitude of Fred-killing GR’s exist at that point (at least, which can operate over arbitrarily small times as detailed above). Or something about history simply has got to have ensured that Fred died by then. That seems arbitrary and unintuitive, but for my part I find it a bit less unintuitive than the impossibility of an infinite past. History and physics necessarily conspire to prevent impossibilies!

In fact, I wonder (and I know this sounds silly, but in my defense I’m a complete amateur) if this intuition could actually be turned around and used as an anti-theistic argument as follows:

1. An infinite past is possible by conceivability arguments.

2. If an infinite past is possible, then if a necessary and omnipotent God is possible, there’s nothing about history that could stop such a being from possibly setting up and running the probabilistic GR scenario if he wanted to.

3. Running the probabilistic GR scenario is impossible.

4. Therefore, necessarily no being is both necessary and omnipotent.

I want to respond to your other comments (which I have to concede are all correct!), but presently need to run.

Oh, and regarding the case where only one other thing can kill Fred, but is (say) a thousand lightyears away: I’d have to say that’s impossible, too! On my tentative suggestion, a universe’s timeline couldn’t progress in such a way that something is capable of preventing a paradox, but fails to do so. So it’s not that Fred magically dies at the hand of the only other thing that can kill him (which is, after all, too far away to do so) just before the Grim Reapers kick in; it’s that Fred always had to be vulnerable to something else all along, something that was close enough to affect him.

I’ve been replying to myself way too much on this thread, so I’m just going to shut up for a while now!

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.Wait. Why couldn’t it be true that, for any m in the sequence, there is an earlier time n at which it was already switched? So, for any m in the sequence there is some earlier n at which the switcher turned the switch on. That is consistent with there being no particular n at which the switcher turned on the switch and consistent with the switch being on at every finite time ago. There is no finite time ago at which the switch was off but that presents no problem given an infinite past. That is, it presents no problem that,

. . .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.Each of the checks occurs at some finite time ago, and for any one of those, we’re assuming, it is true that the Switcher switched it earlier. That he did not switch it at some finite time does not entail that he did not switch it, since we have a past that is not finite. Your argument seems to assume that, if it was switched, it was switched at some finite time ago n. What requires that? So we can’t conclude that the switch is not on.