There are mathematicians who have rejected the idea of the real number line being made of points, perhaps for reasons like this. I don’t recall who, but pointless topology mght be relevant.
My understanding is that such a story relies on trying to define the area of a point when only areas of regions are well-defined; the probability of the point case is just the limit of the probability of the region case, in which case there is technically no zero probability involved.
Yes, it is relevant to algebraic geometry, which is important for the treatment of down-to-earth problems in number theory.
I think you’re confusing topos theory with pointless topology. The latter is a fragment of the former and a different fragment is used in algebraic geometry. As I understand it, the main point of pointless topology is to rephrase arguments to avoid the use of the axiom of choice (which is needed to choose points). That is certainly a noble goal and relevant to down-to-earth problems, but not so many in number theory.
There are mathematicians who have rejected the idea of the real number line being made of points, perhaps for reasons like this. I don’t recall who, but pointless topology mght be relevant.
My understanding is that such a story relies on trying to define the area of a point when only areas of regions are well-defined; the probability of the point case is just the limit of the probability of the region case, in which case there is technically no zero probability involved.
Is pointless topology ever relevant?
Yes, it is relevant to algebraic geometry, which is important for the treatment of down-to-earth problems in number theory.
I think you’re confusing topos theory with pointless topology. The latter is a fragment of the former and a different fragment is used in algebraic geometry. As I understand it, the main point of pointless topology is to rephrase arguments to avoid the use of the axiom of choice (which is needed to choose points). That is certainly a noble goal and relevant to down-to-earth problems, but not so many in number theory.