## Thursday, September 28, 2017

### Walking off to infinity

This is a simplified version of a paradox Josh Rasmussen sent me (“Rasmussen’s Rod”). Suppose that Laika is in a spaceship in a Euclidean non-relativistic space, and in one second she flies a kilometer, in the next half second another kilometer, and in the next quarter second another, and so on, all in exactly the same direction.

What will happen to Laika and the spaceship in two seconds?

1. Causal finitism: The story is impossible, as the outcome has infinitely many accelerations as causes.

2. Space is constituted by the relations between things in them rather than being a container. After two seconds, Laika will be infinitely far away from us. Where is that? It’s a place that didn’t exist until Laika got there, a place constituted by Laika’s being there and her distance from us.

3. Laika and the spaceship will leave space, and will exist as objects that aren’t externally spatial. (They might be internally spatial.)

4. Dogs and spaceships depend on space for their existence, and hence upon leaving they will cease to exist.

Joshua Rasmussen said...

A "rod" version: you place two blue blocks in front of you, then slide them apart and insert two red ones, then repeat the insertion of red blocks at increasing speed until, POOF: the infinitely many blocks all vanish out of existence, or the blue ends are infinitely far apart.

Which is more absurd: causal finitism or POOF? :)

Joshua Rasmussen said...
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Joshua Rasmussen said...

What if there are two spaceships headed in opposite directions. Each leaves a permanent, connected trail. What happens after two seconds then? Do the trails vanish with the ships?

(Compare: the rod with two end blocks grows with increasing speed until the end blocks are infinitely far apart.)

Alexander R Pruss said...

There is a trail and no ships on 4?

Joshua Rasmussen said...

So strange.

What happens if you record the distance traveled each iteration? Let's say you write the number of km in front of you after removing the number previously written (where the ship first began). What do you see written in front of you at the end of the process? I guess it must have been removed...

Hmmm. If a lamp goes on when you write a number and goes off when you remove that number, then the lamp must end up off -- because if it were on, then you'd see a number that is not a number. But suppose you don't do any writing. Then the lamp could end up on. But how can mere writing make a difference to the lamp? There is a logical connection here but not an ontological one. I think that's my same basic worry about the blue blocks (or ship) going out of existence. Maybe reason requires it. But what's the metaphysical explanation? ...

I'm probably confused.

IanS said...

On 1. Causal Finitism: How do you count causes?

Suppose the spaceship were accelerating continuously (e.g. x = 1 - (ln (2 - t)) / ln (2). This interpolates the trajectory in the O.P.). How many causes would there be? An infinite number, one for the acceleration at each time? Or a single one, for whatever was causing the acceleration?

If an infinite number, then Causal Finitism would rule out any continuous acceleration (e.g. that implied by Newtonian gravity). If just one, why couldn’t you say the same in the O.P.?

Alexander R Pruss said...

Ian:

Good question. It is possible to break up Newtonian causation into chunks, perhaps arbitrarily. Maybe the state of the universe at t0 causes the state of the universe for all the times in some small (or large?) interval (t0,t0+delta], and then the state at t0+delta causes the state at all times in (t0+delta,t0+2delta], and so on. That would let one accommodate Newtonian causation here.

But it wouldn't accommodate the spaceship case, I think.

Alexander R Pruss said...

Josh:

The lamp could still be on, with no text there. It could by like Thomson's lamp, which can end up on and which can end up off, and the final state is just not determined by the setup. Or at least that the most widely accepted solution to the lamp paradox.

johnsonav said...

If I'm remembering correctly, this kind of unbounded acceleration in a finite time can be caused just by (Newtonian) gravitational interactions. If this kind of acceleration is impossible, where does that leave Newtonian mechanics?

Joshua Rasmussen said...

Thanks, Alex.

Are familiar with Huemer's recent book on infinity? He argues that you can't get an indeterminate result from determinate causes (or something like that).

If only the writing of the number causes the lamp to go on while the removing of the number causes it to go off, then how could it be on with no number written?

Alexander R Pruss said...

I missed that in his book. Where does he say it?

Joshua Rasmussen said...

on Facebook somewhere. :) (I hope I haven't mis-represented or mis-remembered the exchange.)

Joshua Rasmussen said...

I assumed he unpacked it in the book (could be wrong).

Unknown said...

Couldn't one answer that the ship would simply stop at the speed of light since this speed cannot be exceeded?

Alexander R Pruss said...

Yeah, but relativity is not metaphysically necessary.