Sunday, October 23, 2011

The deep question for the philosophy of spacetime

There is more than one way of putting this point, so the assumptions I will make are not at all essential, and I don't even endorse the assumptions. Assume absolutism about spacetime. On one reading of absolutism, there is then a location relation between objects and points or regions of spacetime (on another reading there is an object- or point-valued location determinable). Depending on the version of absolutism, the location relation may correspond to the predicate is wholly located at, is at least partly located at or is exactly located at (I may be leaving out some options).

Now the deep question is this: What is it that makes a relation between objects and points or regions of a topological space be a location relation? (The question can also be put on relationism. Then the question is what is it that makes a family of relations between objects be a family of spatial, or spatiotemporal, relations.)

There are two extreme answers.

Location monism: There is just one location relation. In a Newtonian and in an Einsteinian world and in a 12-dimensional discrete universe, one and the same relation relates objects to points or regions of a topological space, obviously a very different topological space in each case.

Location functionalism: Any natural (sufficiently natural? perfectly natural?) relation between objects and points in a topological space, where the topological space is either concrete and cosntituted as a topological space by natural relations, or abstract as in this post, is a location relation. What the axioms are will depend on which location relation one takes as fundamental as well as on difficult metaphysical issues. Supposing that the relation is being exactly located at, and the spatial relata are regions, then the axioms might be very lax. In fact they might be nothing but:

  • If xLR and x is a part of y, then there is a unique region R' that contains R such that yLR'.
  • If yLR and x is a part of y, then there is a unique region R' that is contained in R such that xLR'.
(If Thomistic part nihilism is true, do this with virtual parts.) On this view, a relation of being exactly located at any phase space with a topology will count as a location relation as long as the relation is in fact natural. One might additionally add some more axioms, such as that no object is exactly located at two distinct regions (though I myself am inclined to deny that as it's incompatible with my best account of transsubstantiation), but the result about phase spaces will remain true. If one wants to rule them, one can either insist that in fact there is no phase space location in which is natural or disallow abstract topological spaces as relata, despite the benefits of allowing them. One might also add an axiom that makes this be a location in spacetime, by using causation. For instance, we might require the topological space to have a partial ordering on its points (we might add something about how the ordering should play nice with the topology), which we will call "at least as late as", and then extend this to a relation between regions: R' is at least as late as R provided that every point y of R' is at least as late as some point x of R and every point x of R has some point y of R' such that y is at least as late as x. Then add:
  • Normally, if event E causes event E', and E and E' are exactly located at R and R' respectively, then R' is at least as late as R.

Monism and functionalism are extreme theories because functionalism classifies as locational as many relations as anybody could possibly reasonably want to do that to and monism classifies as locational as few as anybody who thinks location is real reasonably could.

I incline to functionalism here.


Heath White said...

Functionalism sounds terrible! :-)

If I understand it correctly, it would count all sorts of scientific or just measured data points as having spatial relations. E.g. consider a graph of a stock, with axes of time and price. People say things like "the stock has been in the region bounded by $10 and $20 for the last three years" and functionalism would call that a genuine spatial relation.

Alexander R Pruss said...

One question is going to be whether any of these involve a sufficiently natural relation to a topological space that's sufficiently natural. This depends on part on the nature of determinables like charge and mass.

It could be that further axioms could rule these out. For instance, if one added my causal axiom, that would go a long way to helping rule out many cases.

Alrenous said...

Space is what we call the property that allows objects to be separated. For an object that doesn't have a spatial relation, it is either identical to another arbitrary object, or unreachable by that object - it has no property it can change to come into interaction with it.

(So yeah, I'd say adding a causal axiom is necessary to complete the concept.)

Any property held by two separate objects that allows interaction and non-interaction will define what space is for those two objects.

The idea of location is derived from the relationship defined by this property.

(Result: I verified something about consciousness.)

It interests me that physics has only one spatial relation. (Handy - makes the calculations easier.) I can't think of any reason in principle some objects could participate in one subset of spaces and other objects could participate in an overlapping but separate subset of spaces.

Constantinos Miltiadis said...

May I ask what you mean by "xLR" and "if xLR"?

Alexander R Pruss said...

L was meant to be the name of the location relation. Sorry, I forgot to spell that out. "xLR" says that x stands in L to R.