Alice has lived forever in a universe with an infinite road that has a beginning and no end, and is marked every mile. Every day of her life, by an irresistable longing, she has followed these rules:
If somehow she’s not on the road, she goes to mile zero on the road.
If she is on the road, she walks a mile in the endless direction of the road.
Besides the movement required by 1 and 2, she stays in place.
Where is Alice now? Nowhere! There is no possible present scenario compatible with the rules. She can’t either be at mile zero or off the road, because if two days ago she was on the road, she would now be on the road now be past mile zero, and if two days ago she wasn’t on the road, she would now be on the road past mile zero. She can’t be at mile n, because then she would have to have been off the road some time back, which violates the previous argument.
The rules are all coherent. A person who has to follow these rules seems to be possible. Yet the story is impossible. What went wrong? The neatest explanation seems to be that a (causally connected) infinite past is impossible.
(This is inspired by a recent infinite past Grim Reaper story I read from Rob Koons.)
This is an interesting variation.
ReplyDelete> A person who has to follow these rules seems to be possible.
Alice is, essentially in this story, an unstoppable force who can find and get to mile zero in one day, from anywhere in the universe. If she and the road both exist, and she is not on the road, then nothing can possibly stand between her and mile zero. It seems easier (to me, at least) to reject the possibility of physical stuff with that power, than to reject the possibility of an infinite past.
I'm not sure how to plausibly relax 1, while leaving the impossibility in place.
That's a good point, but it depends also on which kind of possibility we're talking about, right? Even if "Alice" is physically impossible, she might still be metaphysically possible, if she is, as Dr. Pruss adduces, genuinely coherent (and presumably doesn't violate essential natures). Dr. Pruss' point, I think, is that such a scenario is not just physically, but metaphysically impossible.
DeleteIt is the metaphysical possibility of Alice that I am considering. It seems less costly to reject the metaphysical possibility of things like Alice (unstoppable forces, immovable objects, impassible barriers, etc) which have unbounded physical capacities, than the possibility of an infinite past.
Deletejohnsonav,
ReplyDeleteI think it's more with not that nothing can stop Alice, but that it's merely possible that nothing happens to stop Alice. Alice doesn't need to be unstoppable. It just merely needs to be possible that nothing happens to stop her.
Heavenly,
DeleteSuppose that one of the ways Alice is not unboundedly capable is that she has a maximum acceleration, or top speed. If that were the case, there's a perfectly coherent story involving Alice, the road, and an infinite past, it seems like, and Alice can be somewhere at every time.
It's easy to modify the story to work with Alice having a maximum speed and acceleration. Forget about the road. Replace "mile 0" with a point Z on an infinite plane (or in 3D space), and suppose there is a "north" direction somehow specified. Now specify Alice follows these rules each day:
ReplyDelete1. If she starts the day at Z, she goes one mile north.
2. If she starts the day anywhere else, she goes one mile in the direction diametrically opposed to Z.
3. She makes no other changes of position.
Observe: She can't be at Z at the start of any day by the rules (if the day before she was at Z, she would be one mile north of Z, and if the day before she was other than at Z, she would be one mile further away from Z). Every day her distance from Z increases. So she must be an infinite distance away from Z. But there is no point in space infinitely far away from Z.
Okay, I like this modified story.
DeleteImagine Betty moves in the opposite direction from Alice. Betty follows one rule: every day she moves one mile closer to Z. Betty is going to follow this rule, starting today, for every day into the infinite future. So where is Betty now? Nowhere! She'd have to be infinitely far away from Z today in order to follow the rule into the future forever.
Betty can't go on following her rule forever into the future. And Alice can't have been following her rule forever in the past. The situations seem pretty symmetrical. Why not take this as a fact about rules and their repeated application, how some cannot go on forever, and others cannot have been going on forever, rather than saying something about the possibility of an infinite past or future?
Notice that while Alice's rule gives a well-defined action no matter where and when Alice is, Betty's rule doesn't say what to do when she is at Z.
ReplyDeleteAh, I was unclear with my use of "rule", which may differ from how you were using it. Sorry. I'm talking about relations between the pairs of locations that are visited, rather than a guide for action. On my usage, a thing was following Betty-rule for a trip between p and q, iff q is 1 mile in the direction of Z from p, regardless of why it moved between those two locations.
DeleteIf an object is at Z today, there is no location where that object could be tomorrow where Betty-rule isn't being violated (no place is one mile closer to Z than Z). So if an object follows Betty-rule every day it exists, it must have a last day of existence.
The opposite holds for Alice-rule. If an object has traveled to Z today, it must have violated Alice-rule to get there (Z isn't one mile farther from Z than anywhere). Anything which follows Alice-rule every day it exists must have a first day of existence.
Both Betty (a person who follows Betty-rule every day into the infinite future), and Alice (a person who has followed Alice-rule from infinity past) are impossible. How does the impossibility of Alice relate to the possibility of an infinite past in such a way that the impossibility of Betty does not relate to the possibility of an infinite future in the same way?
I am thinking somthing like this. Alice has a set of coherent causal powers, coherently reactive to any input possible in this universe, all encapsulated by the rules. If an infinite past is possible, it should be possible for Alice with these causal powers to have existed forever. But the Betty-rule is not one that can be defined by a set of coherent causal powers that are coherently reactive to any input possible in this universe.
ReplyDeleteThat said, I am starting to get a bit of a sense for why I've never been that convinced by these kinds of stories, while I have been convinced by Grim Reaper types of stories.
Ah yeah, I do see the distinction you're drawing between Alice and Betty.
ReplyDelete> If an infinite past is possible, it should be possible for Alice with these causal powers to have existed forever.
That’s where I don’t feel the force, I think. If Alice exists, she has a location. That location is some distance from Z. So Alice can be no more days old than she is miles distant from Z. Every location is a finite distance from Z. So no matter where Alice is, she had a beginning some days ago. Having a beginning a finite number of days ago seems essential to Alice, whether or not the past itself is infinite.
But maybe I’m just tollensing your ponens.