The confusing thing is how can an "object" be zero dimensional?

assuming that nature is a smooth manifold,we can only create a model that is based on the smallest distance measured,therefore our model has a graininess.

 

You need extra input to define a notion of distence, but that is little problem. On a manifold equipped with a metric, what is the smallest distence you can define? I think it is zero, which is the mangnitude of a vector "joining a point to itself". The assumption here is that we do have a genuine Riemannian metric.

 

This may well be different to the actual physics, which may indeed have some fundamental graininess, but this would not be seen by modeling space-time by a smooth Riemannian manifold.

 

The confusing thing is how can an "object" be zero dimensional?

 

A point on a manifold can be thought of a just as an evaluation of the coordinates in some predefined coordinate system. For example, in three dimensions we have local coordinates (x,y,z), any particular point (assuming that it is coverted by the chart we are using) is just the specification of three actual numbers, say (0,0,0) or (1,0,1) etc.

 

However one should note that the points exists quite independently of the coordinates employed, that is we can think of manifolds as particular kinds of nice topological spaces.

More ontological enquiry regarding the nature of continuum split off to philosophy section

 

http://www.scienceforums.net/topic/73774-the-field-as-continuum-split-from-how-does-a-higgs-field/