It is indeed not so simple. If I continue my thought experiments about sets not being universal or foundational, I run into a myriad of problems.
For one, how can one reason about anything when rejecting concepts? How can one conclude anything when rejecting logic? How can one infer anything when rejecting time?
These problems seem to point to some recursive or symmetric (or circular as the article suggests) dependency between the realist and non-realist perspectives.
I don't yet fully understand why there would have to be three worlds -- I'd intuitively say that two (e.g. physical and mental) suffice. The platonic world might simply follow from the mental one, or vice versa. I'll put in several hours of thought and report back in the next post that touches upon this subject.
I concluded for myself that logic, science, nor philosophy are going to be of much help with this. I therefore turned to contemporary art, where such thought still has some kind of validity. Let's see where that leads me :)
Edit: It seems that the "three world" idea is originally an idea by Karl Popper. Wikipedia [1] explains this in some detail, from which it becomes clear why the thought experiment has three, not two worlds.
Part of the reason that you're going to run into trouble with sets being universal, or foundational, rightly found in a theorem that was birthed from set theory, Gödel's Incompleteness Theorem[0].
I will confess that when I took discrete in college I was seduced by set theory. I literally had the, naive, thought to myself "you could prove all of math with just sets!". As my education continued I found out, much to my chagrin, and Hiblert's[1], that I was very, very, mistaken.
This hasn't stopped everyone from trying to continue Hilbert's program, though with a bit more limited scope[2][3].
By and large, all math is undergirded by a set of axioms that have to be taken as true with no proof. Even as far back as Euclid's Elements basically starts with a set of axioms and then proceeds from there. Strange that something that is so real must have a bunch of rules given as true with no proof of their validity beyond "well, everyone can see that it's true".
In a final, ish, dig, I'll just say leave it up to Penrose to take Popper's sensible cosmology and turn it into a quasi-religious one.
It is indeed not so simple. If I continue my thought experiments about sets not being universal or foundational, I run into a myriad of problems.
For one, how can one reason about anything when rejecting concepts? How can one conclude anything when rejecting logic? How can one infer anything when rejecting time?
These problems seem to point to some recursive or symmetric (or circular as the article suggests) dependency between the realist and non-realist perspectives.
I don't yet fully understand why there would have to be three worlds -- I'd intuitively say that two (e.g. physical and mental) suffice. The platonic world might simply follow from the mental one, or vice versa. I'll put in several hours of thought and report back in the next post that touches upon this subject.
I concluded for myself that logic, science, nor philosophy are going to be of much help with this. I therefore turned to contemporary art, where such thought still has some kind of validity. Let's see where that leads me :)
Edit: It seems that the "three world" idea is originally an idea by Karl Popper. Wikipedia [1] explains this in some detail, from which it becomes clear why the thought experiment has three, not two worlds.
[1] https://en.m.wikipedia.org/wiki/Popper%27s_three_worlds