On Logical Stability and Non-Euclidean Geometries
"To know what would have happened, child?" said Aslan, "No. Nobody is ever told that."
I received a question about my last post, which deserves a full response:
Does Aquinas think that one could conceive of a world/universe in which there was no necessity for God, but that it’s just not the world in which we find ourselves in? What does it mean to have a logically stable universe without God? -A Stochastic Chanter
Dear Stochastic Chanter,
Aquinas does not directly discuss this question to my knowledge, but that is unsurprising. The distinction between logical possibility and observed actuality was fuzzy until quite recently, at least in the sense I am referring to,1 and this is because the first clear example of it only emerged in the early 1800s. I speak of non-euclidean geometries.
Geometry was formalized by Euclid in his treatise The Elements in Ptolemaic Greece. The Elements starts with a number of definitions, and then five postulates, as follows:
Let the following be postulated:
[The ability] To draw a straight line from any point to any point.
[The ability] To produce a finite straight line continuously in a straight line.
[The ability] To describe a circle with any center and radius.
That all right angles equal one another.
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
I hope that you can tell that one of these is not like the others. The fifth postulate, the “parallel postulate,” bothered mathematicians ever since Euclid wrote it down. It seems like it should be a theorem, not a postulate, but no one ever managed to prove it from the other five. Between 1810 and 1832, however, a number of mathematicians2 all independently discovered something: If you negate the parallel postulate, nothing bad happens. More technically, no matter how far they derived assuming the parallel postulate was not true (i.e. that two non-parallel lines would never meet, or that two parallel lines would meet eventually), they never found a contradiction. A bunch of things that were highly counterintuitive, sure, but never something that negated something they had already proved or assumed. They finally concluded that there was more than one logically consistent geometry.
This made a bunch of people very upset, and may have started the ball rolling for the destruction of enlightenment rationalism, but that is a story for another day.
Obviously, our own world can only be one geometry, but on the level of logic, there is no way for us to tell which one it is. We need to go and observe it, which means admitting the validity of our senses. Of course, this is fine for most well adjusted people, but it lays an axe to the root of our ability to trust our own reason above all else, bringing back the need to obey an authority.
Returning to the question of God, we can proceed in the same way. As an exercise, assume there is no God. Then you need to go through all arguments for the existence of God and see what they assume, and deny those assumptions as well.
This is where most well adjusted people will stop, because that means denying a ton of things that seem horribly obvious, like the existence of free will, the existence of any sort of reason to do anything, even seek out pleasure, our natural feeling of being a bit discontented with things, and any number of other very natural human emotions.
But we are madmen, and we cry “Out, damned spot” to the shadow of God on the world, and care not what we destroy, if only we can get rid of His omniscient gaze. So we continue, denying anything that can be used to prove God’s existence. It is a sad life, no life at all really, but nothing logically contradictory will arise, so long as we follow it to the end, wiping out any trace of the supernatural, any trace of the divine, any trace of the non-material. What will be left? What does it look like?
I don’t know about you, but as far as I can see, it looks like the stuff of nightmares.
The scholastic distinction between act and potential is related, but not to what I am referring.
Lobachevsky and Bolyai published it, but Gauss, Schweikart, and Taurinus also had work on it.
Imagining a world where God doesn’t exist is to imagine nothing. Non-reality, not an alternate reality, but non-reality, where no thoughts can make sense, so there is no possibility of beginning a syllogism, in any sense. We cannot reason to God‘s existence, we can merely reason from it in order to make sense of anything. Since he is who he says he is, and has revealed himself in such a way that we know in the absolute sense he exists, we can therefore make sense of all other things, even if we haven’t yet done so in however many particular instances.