The universe with the 10-feet torus topology would certainly be a different universe governed by different laws. Still, one could conceive of a formal system of addition which would be exactly same as our present one, only it would not apply to distances (in a straightforward way). The same way as we can conceive the addition mod 2 arithmetics.
As for the seeming contradiction, if you define “p being x feet away from q” as “there is a geodetic of length x connecting p and q”, then obviously “I am ~40,000 km far from Istanbul while I am in Istanbul” isn’t a contradiction, although it may look like one on the first sight. If you define distance as the length of the shortest geodetic, then it is a contradiction. Once again, this is a feature of language, not of the world.
I have no problem with the idea that God could switch to a different formal system governing the world, perhaps even one we cannot describe now formally and consider it impossible, but that would only mean that certain formal systems, such as standard arithmetic, would have less practical applications, while others, maybe the mod 2 arithmetics, or something entirely exotic, would have more. It wouldn’t make “1+1=0” a theorem of standard arithmetics. In the same way, we have rules which attach adjectives “round” and “square” to objects and these rules (implicitly) specify that these categories are exclusive. Perhaps, in the new world, there would be objects which may lead us to generalise the notions of “square” and “round” to have some overlap; but then, we will not be speaking about “square” and “round”, as we understand the terms today.
The universe with the 10-feet torus topology would certainly be a different universe governed by different laws. Still, one could conceive of a formal system of addition which would be exactly same as our present one, only it would not apply to distances (in a straightforward way). The same way as we can conceive the addition mod 2 arithmetics.
As for the seeming contradiction, if you define “p being x feet away from q” as “there is a geodetic of length x connecting p and q”, then obviously “I am ~40,000 km far from Istanbul while I am in Istanbul” isn’t a contradiction, although it may look like one on the first sight. If you define distance as the length of the shortest geodetic, then it is a contradiction. Once again, this is a feature of language, not of the world.
I have no problem with the idea that God could switch to a different formal system governing the world, perhaps even one we cannot describe now formally and consider it impossible, but that would only mean that certain formal systems, such as standard arithmetic, would have less practical applications, while others, maybe the mod 2 arithmetics, or something entirely exotic, would have more. It wouldn’t make “1+1=0” a theorem of standard arithmetics. In the same way, we have rules which attach adjectives “round” and “square” to objects and these rules (implicitly) specify that these categories are exclusive. Perhaps, in the new world, there would be objects which may lead us to generalise the notions of “square” and “round” to have some overlap; but then, we will not be speaking about “square” and “round”, as we understand the terms today.