I find this post very interesting, but I disagree with your examples about God. This comment is rather lengthy, and rather off-topic, so I apologize, but I wanted to bring it up because your post features these questions so prominently as examples.
Specifically, I don’t think that the answer to the questions about God can be written off so easily as “no”. It seems to me that the questions
“Could God draw a square circle?”
“Could God create a stone so large that even He could not lift it?”
are asking about the bounds on omnipotence.
Suppose an omnipotent being exists in a universe, and that universe operates under some fundamental laws that among other things define what can exist vs. what is a contradiction. It seems fairly obvious, based on the standard definition of omnipotence, that the omnipotent being should be able to do all things that do not violate these fundamental laws and cause a contradiction. I’ll call this Level 0 Power.
“Could God draw a square circle?” is asking about what I’ll call God’s Level 1 Power. It is asking “Does an omnipotent being have the power to change the fundamental laws of the universe?”, or, if you like, “Can God reprogram the universe?” If God is in charge of the rules, then presumably he could rewrite them such that things which are currently a contradiction are no longer a contradiction. In terms of the circle/square question, this seems kind of silly, since circles and squares are not contained in the universe but are products of formal systems invented by humans. Alternatively we could ask “Can God make 1+1 add up to something other than 2?” and the answer is “Of course; even mathematicians can do this, by redefining the axioms or working in the integers (mod 2) or something.” In terms of this example, then, Level 1 Power is asking “If the universe is a formal system of sorts, can God change the axioms?”
Suppose God has Level 1 Power and can change the axioms of the system (or systems) he creates. This isn’t so hard to imagine; it’s like a human programmer rewriting a piece of the code in the middle of running a simulation. But the question about whether God, if he existed, actually had such a power, seems like it would be a reasonable subject of discussion for theologians.
“Could God create a stone so large that even He could not lift it?” is yet a more difficult question. Is it asking about God’s Level 1 Power? I think it depends on where omnipotence comes from. If God is omnipotent within the system because that’s the way he coded it, then it is asking about God’s Level 1 Power: all he has to do is go into the code for the universe and change the part that says he should be able to lift all stones. But if God’s omnipotence is something that exists independent of the system, then this question is asking whether God can change the rules which define himself.
Anyway, your answer of “no” to these questions indicates not that the questions are worthless but that you assume an omnipotent being could only have Level 0 Power.
Suppose an omnipotent being exists in a universe, and that universe operates under some fundamental laws that among other things define what can exist vs. what is a contradiction.
Contradictions are feature of a language (or some more formal system used to describe the universe), not of the universe. What we call physical laws are regularities which allow us to compress the observed data a bit—e.g. instead of keeping a list of planet positions at each moment, it is enough to have the initial positions, velocities and few equations of motion. Absence of contradictions is not such a law. (It is easy to imagine what violation of a particular physical law would look like, but try to imagine how a contradiction would look like. What would you observe if there was a lizard on your table and simultaneously there was no lizard on your table?)
Alternatively we could ask “Can God make 1+1 add up to something other than 2?” and the answer is “Of course; even mathematicians can do this, by redefining the axioms or working in the integers (mod 2) or something.” In terms of this example, then, Level 1 Power is asking “If the universe is a formal system of sorts, can God change the axioms?”
This is exactly changing the language, and very uninteresting to theologians when, as you correctly note, mere mathematicians can do it. “1+1=2” is a string in some formal system which acquires its meaning by isomorphism with real world situations. You can redefine your alphabet to exchange the symbols “2“ and “4”, which would make “1+1=4” true, but its meaning would be absolutely the same as the meaning of “1+1=2” before the redefinition. It has nothing to do with fundamental laws of the universe, whatever they are.
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.
Perhaps I shall spell this out better, but the impossibility is linguistic. A cleaner example I mention is:
Where “bachelor” means “man who is not married,” God could not create a married bachelor. A married bachelor is not a thing. If you break down the definitions of circle and square, you’ll see that a “square circle” is not a thing. A heavy stone that has no mass (or a heavy stone that is not heavy), or a circle that is not circular, or any other number of direct contradictions seem impossible, not as limits on power, but mostly as limits on language. That’s the point I’m getting at.
I think this post is right on. I think we are IN this universe with a brain to match it, with 3-d, separation of time and space appropriate to non-relativistic speeds, and so on strongly coded in.
In terms of any powerful god, she either “lives” in a much larger universe than ours, which kicks the can down the road (is there an omnipotent god who created that universe?) or she essentially comprises the entire universe. What other way to have an entity which “knows” how every particle moves at every instant other than having that entity be the universe? The most powerful, most accurate “simulation” of any system is the system itself. Obviously, the system itself can’t get a wrong answer from a bad approximation somewhere, every other simulation can. I’m talking of simulations because omniscience presumably means the model god carries in her mind is complete and completely accurate. The model in god’s mind is as big and complex and at least as fast as the system itself.
But be that as it may, the only way you can play the linguistic trick of saying square circles are not real so god can still be omnipotent without being able to make them, you have, it seems to me, created a higher level physics which constrains all the universes that might be created. But where can the higher level physics come from? Is it just there, in which case our god is not the creator of the UNIVERSE universe, just of a very constrained universe that follows a bunch of atheistically determined rules. The can is kicked down the road.
So if you are interested in a god-the-creator which has had no cans kicked down the road, I don’t see how you can rule out ANYTHING. Things we can’t concieve of are not ruled out, certainly things we can sorta concieve of like square circles and married bachelors can’t be ruled out.
How could this god create a square circle? Of course, I don’t know. But I’d imagine that when you saw the square circle you would know it, even if you couldn’t reconcile it with everything else you see and know about circles and squares. Indeed a square circle is trivial enough, god merely has to trigger your brain beyond the traditional sensory pathways limitation.
I find this post very interesting, but I disagree with your examples about God. This comment is rather lengthy, and rather off-topic, so I apologize, but I wanted to bring it up because your post features these questions so prominently as examples.
Specifically, I don’t think that the answer to the questions about God can be written off so easily as “no”. It seems to me that the questions “Could God draw a square circle?” “Could God create a stone so large that even He could not lift it?” are asking about the bounds on omnipotence.
Suppose an omnipotent being exists in a universe, and that universe operates under some fundamental laws that among other things define what can exist vs. what is a contradiction. It seems fairly obvious, based on the standard definition of omnipotence, that the omnipotent being should be able to do all things that do not violate these fundamental laws and cause a contradiction. I’ll call this Level 0 Power.
“Could God draw a square circle?” is asking about what I’ll call God’s Level 1 Power. It is asking “Does an omnipotent being have the power to change the fundamental laws of the universe?”, or, if you like, “Can God reprogram the universe?” If God is in charge of the rules, then presumably he could rewrite them such that things which are currently a contradiction are no longer a contradiction. In terms of the circle/square question, this seems kind of silly, since circles and squares are not contained in the universe but are products of formal systems invented by humans. Alternatively we could ask “Can God make 1+1 add up to something other than 2?” and the answer is “Of course; even mathematicians can do this, by redefining the axioms or working in the integers (mod 2) or something.” In terms of this example, then, Level 1 Power is asking “If the universe is a formal system of sorts, can God change the axioms?”
Suppose God has Level 1 Power and can change the axioms of the system (or systems) he creates. This isn’t so hard to imagine; it’s like a human programmer rewriting a piece of the code in the middle of running a simulation. But the question about whether God, if he existed, actually had such a power, seems like it would be a reasonable subject of discussion for theologians.
“Could God create a stone so large that even He could not lift it?” is yet a more difficult question. Is it asking about God’s Level 1 Power? I think it depends on where omnipotence comes from. If God is omnipotent within the system because that’s the way he coded it, then it is asking about God’s Level 1 Power: all he has to do is go into the code for the universe and change the part that says he should be able to lift all stones. But if God’s omnipotence is something that exists independent of the system, then this question is asking whether God can change the rules which define himself.
Anyway, your answer of “no” to these questions indicates not that the questions are worthless but that you assume an omnipotent being could only have Level 0 Power.
Contradictions are feature of a language (or some more formal system used to describe the universe), not of the universe. What we call physical laws are regularities which allow us to compress the observed data a bit—e.g. instead of keeping a list of planet positions at each moment, it is enough to have the initial positions, velocities and few equations of motion. Absence of contradictions is not such a law. (It is easy to imagine what violation of a particular physical law would look like, but try to imagine how a contradiction would look like. What would you observe if there was a lizard on your table and simultaneously there was no lizard on your table?)
This is exactly changing the language, and very uninteresting to theologians when, as you correctly note, mere mathematicians can do it. “1+1=2” is a string in some formal system which acquires its meaning by isomorphism with real world situations. You can redefine your alphabet to exchange the symbols “2“ and “4”, which would make “1+1=4” true, but its meaning would be absolutely the same as the meaning of “1+1=2” before the redefinition. It has nothing to do with fundamental laws of the universe, whatever they are.
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.
Perhaps I shall spell this out better, but the impossibility is linguistic. A cleaner example I mention is:
Where “bachelor” means “man who is not married,” God could not create a married bachelor. A married bachelor is not a thing. If you break down the definitions of circle and square, you’ll see that a “square circle” is not a thing. A heavy stone that has no mass (or a heavy stone that is not heavy), or a circle that is not circular, or any other number of direct contradictions seem impossible, not as limits on power, but mostly as limits on language. That’s the point I’m getting at.
I think this post is right on. I think we are IN this universe with a brain to match it, with 3-d, separation of time and space appropriate to non-relativistic speeds, and so on strongly coded in.
In terms of any powerful god, she either “lives” in a much larger universe than ours, which kicks the can down the road (is there an omnipotent god who created that universe?) or she essentially comprises the entire universe. What other way to have an entity which “knows” how every particle moves at every instant other than having that entity be the universe? The most powerful, most accurate “simulation” of any system is the system itself. Obviously, the system itself can’t get a wrong answer from a bad approximation somewhere, every other simulation can. I’m talking of simulations because omniscience presumably means the model god carries in her mind is complete and completely accurate. The model in god’s mind is as big and complex and at least as fast as the system itself.
But be that as it may, the only way you can play the linguistic trick of saying square circles are not real so god can still be omnipotent without being able to make them, you have, it seems to me, created a higher level physics which constrains all the universes that might be created. But where can the higher level physics come from? Is it just there, in which case our god is not the creator of the UNIVERSE universe, just of a very constrained universe that follows a bunch of atheistically determined rules. The can is kicked down the road.
So if you are interested in a god-the-creator which has had no cans kicked down the road, I don’t see how you can rule out ANYTHING. Things we can’t concieve of are not ruled out, certainly things we can sorta concieve of like square circles and married bachelors can’t be ruled out.
How could this god create a square circle? Of course, I don’t know. But I’d imagine that when you saw the square circle you would know it, even if you couldn’t reconcile it with everything else you see and know about circles and squares. Indeed a square circle is trivial enough, god merely has to trigger your brain beyond the traditional sensory pathways limitation.
I think for this to be meaningful I’d need to know what your working definition of “omnipotent” was.