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« on: February 12, 2022, 08:36:46 PM »
@Troolon; can we go back around 5 pages to our carpet-fitting business?
It turns out that we are quite the success story, and we have expanded into carpet production and installation; our factory in the West Midlands of England receives orders from far and wide, including from the Lord Mayor of Hobart, Tasmania, who wants to carpet his office. We send out a technician to measure-up and (conveniently) he finds the office to be exactly 3m x 4m, with diagonals of 5m, so it's precisely a right-angle rectangle. Our representative makes a drawing of the office floor and e-mails it to Company HQ.
Question 1; Hobart is around 42 deg S latitude. At what point, if at all, should our technician be using bendy rulers or variable-scales to draw his map, because the drawing is, after all, a map?
The federal Australian government gets to hear about the Lord Mayor's carpet and they are well impressed with the quality, precise manufacture to specified dimensions, and its absolute resistance to stretch and shrinkage. It can be rolled up, but it is incompatible with curvature in more than one dimension.
In an effort to brighten up Australia, they decide to cover the complete country in a nice floral pattern axminster. A rectangular remnant precisely 3000km x 4000km should do it, and extend a little over the coast. So we manufacture a carpet loom 3000km wide, and use it to produce a carpet 4000km long. We know it is a precise right-angle rectangle, because the loom is precisely 3000km, and with our top-notch quality control (the technology of which which we can leave someone else to work out) we used it to produce a carpet exactly 4000km long, without any warpage whatever. We ship it off to Canberra. (Again, we can leave the logistics to FedEx or whatever).
So now the Aussies roll it out from Brisbane towards the West.
Question 2; How does it look? Is it wrinkled at the edges? Is there a ridge over Melbourne? If we push it down over Victoria, does it leave a gap over Sydney?
Perhaps it lies perfectly flat; no gaps. We have proven that the Earth is flat, or cylindrical, or a cone, or a pyramid or some-such. We possibly don't know which, but we have completely refuted the idea that the world is spherical, or shaped like Angela Merkel. If we continue to identify what shapes it isn't, eventually we will know what shape it is.
Confession time; unbelievably, the above story is not true. (We went bust in the first year; the clever money, apparently, is in laminate). The point though, is that as a thought experiment it is true. If we did it, either the carpet would lie flat, or it wouldn't. It can't do both. It would falsify one part of the argument.
Your contention that we cannot know the shape of anything is a nonsense. Its not just about the maths. The Earth, like our fictional carpet, is an entity with fixed dimensions. If we measure it enough ways, using a constant metric against entities who's size we know, we can know its shape or, at the very least, identify what shape it isn't.