The nature of number

[Categories+Sheaves]

Thanks for your help, but before we move into other dimensional use’s of this form. I think it would be nice to examine a possibility if this graph stands.



We seem to have a range of zeros or a stretched zero which is not a point, but that has length.
And although the value does not change we can see a definite progression along the line.

Only in the sense that the preimage of 0 is a set with those properties. Or that you've taken something with these properties and called each of its constituents "0". It seems like we could call each point on the line "fish" or "paisley" and achieve the same result. Is there some concrete sense in which these points actually walk or talk like 0?

My thinking is this way; I have a tours. I press the middle until it is almost connected, but is not actually connected. The central hole exists, but it occupies Zero space. If I categorise the shape by the flow over it, it will fulfil all the criteria of being a sphere, however if I categorise the same shape by the internal side of this zero space skin, it is in all effect a torus. We can do the same the other way round.
Now how could you examine a shape to find holes of zero size and get any meaningful result? So is it not so that Grigori Perelman’s theorem is only proof to a given resolution and not an absolute?

1. This mapping of the torus i) fails to be differentiable ii) fails to be an embedding (even in the weakest topological sense) iii) does not seem to be well-defined (holes of zero width are o.k. if we're just deleting a point from a plane, but have you thought about what it means for a cylinder to have zero width?)
2. The type of vector fields/flows topologists love to put on manifolds are nonvanishing ones. If I understand what you're saying, the flow on the torus in your example has to vanish on the "inside" of your torus. So this doesn't conflict with the established math.
3. Even if this was problematic, Perelman's proof applies to 3-folds, not 2-folds. And if I understand what you mean when you say 'proof to a given resolution'.... yes: Perelman's proof is a technical result about a class of objects that operate under a very rigorous and particular set of rules.
4. A reprise of #1: Topological properties aren't invariant under the stuff you're doing to torii/spheres here, so you and Perelman are playing different games in yet another sense...

Geometric shapes are made up of points, which effectively have no dimension.

This is absolutely correct, but it's still worth noting that the "Geometric object A is a collection of points in space" perspective is surprisingly modern. It isn't too hard to learn/imagine what a straight line is, and we can intuitively understand what "a point on a line" is. But do these points themselves constitute the line, or does the line just happen to have points lying along it? That issue isn't quite so clear. While the former line of thinking is far superior when you have a robust theory of sets + calculus/analysis lying around, Euclid (and a whole bunch of other dead mathematicians) originally approached geometry from the other angle.

Something to think about. (Because: why should the heterodox guy have all the fun here?)