Thanks Jonathan for the journal reference, although my mathematical credentials are not excellent.
You say, "I think Joy's topological approach may absolve us of the need to quantize space, in order to reconcile things,...".
I beg to disagree, while a topological approach may help, Joy's approach MUST NOT shy away from its clear responsibility having started his work with a statement like... "The Point Bell Missed" and "Although lines and planes contain the same number of points...".
Firstly, this statement admits that lines and planes consist of points.
Secondly, by saying the points are numbered is to say they are discrete. Only discrete things can be numbered.
Thirdly, a line is claimed to be 1-dimensional and a plane 2-dimensional. How is a 0-dimensional object contained in a 1-dimensional one? Are the points in a line and those in a plane of same dimension?
Fourthly, the claim that lines and planes contain the SAME number of points needs to be clarified. Is it supportive of what Tom said that "...There are as many points in this line: ___, as there are in the entire universe" and "I am not going to discuss the point-line thing in this forum. It's well understood geometry". What Tom is saying is that number of points on the line is infinite and the number in the universe is infinite, so both the line and the universe contain the SAME number.
The question I would have asked Tom, but can't since he says for him the case is closed is, whether his line, ___ and a segment of it also contain the SAME number of points. If so, whether points can then be said to be capable of being counted as to make assertions like two geometrical objects having the same number as Joy started with.
Jonathan, henceforth we must demand strict definition of what anyone, particularly the more mathematically inclined are asserting. It is from not being as demanding that mathematicians were allowed to introduce 'a line having length but with a breadth of absolute zero' and a 'surface of absolute zero thickness' into our physics.
Akinbo