Monday, April 21, 2014

From relationalism about times to infinitesimal lengths of time

Assume that simultaneity is a reflexive and symmetric relation between events. I will, however, not think of it as transitive. This lets me say that an event that goes from 2 pm to 3 pm is simultaneous with one that goes from 2:30 pm to 3:30 pm. (This is important if there is to be any hope of the thesis that all causation is simultaneous being true.)

Can one construct times out of the simultaneity relation between events? Well, a natural attempt is to say that any maximal set T of pairwise simultaneous events is a time (we can use the Axiom of Choice to show that every event is contained in such a maximal set), and an event E happens at a time T if and only if E is a member of T.

This account, however, has a curious consequence. Consider some event En that starts right after noon, and ends right at noon plus 1/n hours. Thus, En takes place on the time interval (12,12+1/n] (non-inclusive at 12, inclusive at 12+1/n). Let T be any maximal set of pairwise simultaneous events that contains the En. (By the Axiom of Choice, T exists.) By the above account of times, T is a time, and all the events En occur at T. But when is T? It's not noon: none of the events En occur at noon. But for any positive real number u, most of the events En occur before 12+u, so T is not 12+u.

In other words, T is a time between 12 and 12+u for every positive real u>0. It is, thus, a time that is infinitesimally after noon. Thus, curiously, the natural construction of times out of the simultaneity relation very naturally leads to times that are infinitesimally close together, as long as there are events like En.

This is quite interesting, because it suggests that a hyperreal timeline may not be such an outlandish hypothesis (Rosinger has also suggested this hypothesis in a number of preprints, e.g., this one). It is a hypothesis that one is led to quite naturally from a relationalist picture, a hypothesis that given such a picture and such an account of times might very well be true.

Of course, the above depended on one particular way to construct times out of simultaneity. And it depended on a simultaneity, a somewhat fishy relation. But still, it's suggestive.

I think there is a way of seeing the above remarks as a reductio of the relationalist program. That's how I saw the observation when I started writing this post. And maybe that's right, but it's not clear to me that that's right.

4 comments:

Drew said...

Most of these demonstrations seem like arguments for finitism. The more I read this blog, the more I am convinced it is true, even in the mathematical realm.

elliottroland said...

The construction of events E_n (or at least something similar to it) is already precluded by Grim reaper-type arguments, is it not?

Alexander R Pruss said...

Elliott:

Depending on what conclusion on draws from the reaper arguments, yes.

Drew:

The hitch for me with that is that the future is infinite.

Alexander R Pruss said...

I've been thinking some more about this post. The link to hyperreals is a little less clear to me now, though you do get some sorts of infinitesimal durations.