I suppose I am assuming that the universe operates under some set of formal rules (though they might not be deterministic) independently of our ability to describe the universe using formal rules. I would also say that our inability to comprehend a given contradiction is related to the fact that we are inside the system. If God were outside the system he would not necessarily have this problem.
I disagree with your second point, though. Sure, 1 and 2 are labels for concepts that exist within a formal system we’ve developed, and sure, we can create an isomorphism to different labels. But I would consider this to be the same formal system. The example I gave (working in the integers mod 2) involves switching to a formal structure that is decidedly not isomorphic to the integers under addition.
Also, sorry if I was unclear—I did not mean to imply that mathematical formalisms as we’ve developed them are related to the fundamental laws of the universe. I only meant to say that if the universe is a formal system of some sort, and God operates outside that formal system, then it is conceivable that God could switch to a different formal system where things that we consider impossible are not, just like we can switch to a different formal system where 0 and 2. Maybe God could do something analogous and put me in the universe (mod 10 feet) so that if I walk ten feet straight across the room I’ll end up where I started; this seems like a contradiction in our universe but is definitely imaginable.
[Quick edit for clarity: maybe it doesn’t seem like a contradiction that I could walk ten feet away and end up back where I started, but it does seem like a contradiction that I could walk ten feet and both be ten feet away, and also be exactly where I started. This is what I imagine happening in the universe (mod 10 feet).]
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.
I suppose I am assuming that the universe operates under some set of formal rules (though they might not be deterministic) independently of our ability to describe the universe using formal rules. I would also say that our inability to comprehend a given contradiction is related to the fact that we are inside the system. If God were outside the system he would not necessarily have this problem.
I disagree with your second point, though. Sure, 1 and 2 are labels for concepts that exist within a formal system we’ve developed, and sure, we can create an isomorphism to different labels. But I would consider this to be the same formal system. The example I gave (working in the integers mod 2) involves switching to a formal structure that is decidedly not isomorphic to the integers under addition.
Also, sorry if I was unclear—I did not mean to imply that mathematical formalisms as we’ve developed them are related to the fundamental laws of the universe. I only meant to say that if the universe is a formal system of some sort, and God operates outside that formal system, then it is conceivable that God could switch to a different formal system where things that we consider impossible are not, just like we can switch to a different formal system where 0 and 2. Maybe God could do something analogous and put me in the universe (mod 10 feet) so that if I walk ten feet straight across the room I’ll end up where I started; this seems like a contradiction in our universe but is definitely imaginable.
[Quick edit for clarity: maybe it doesn’t seem like a contradiction that I could walk ten feet away and end up back where I started, but it does seem like a contradiction that I could walk ten feet and both be ten feet away, and also be exactly where I started. This is what I imagine happening in the universe (mod 10 feet).]
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.