3 upvotes, 1 direct replies (showing 1)
View submission: Logic has no foundation - except in metaphysics. Hegel explains why.
The conjecture I see here, about the very foundation upon which logic rests, reminds me so much of the work of Alfred Tarski around language paradoxes (This sentence is false). He essentially solved this by creating a metalanguage. However, an equivalent paradox could be created in his metalanguage, solvable only by the creation of a newer meta-metalanguage. This chain continues infinitely. This work led to Kurt Gödel's incompleteness theorem, which in a sense, is saying that our axiomatic rules are true only because we assign truth to them with nothing else holding up the turtle.
Am I off target here? (I'm not a scientist, let alone a logician).
Comment by Sabotaber at 31/01/2025 at 19:44 UTC
2 upvotes, 2 direct replies
You should take a look at Turing's Halting theorem. It's much easier to digest than Godel's work, but touches on a lot of the same ideas. An interesting exercise is to look at Machine X and figure out if it will halt, which should give you some insight into the whole metalanguage thing.
Also relevant is that information requires context to have meaning. If the universe we inhabit is informational, then it is very easy to assume(but not prove) a cosmology where it's turtles all the way down. It goes like this:
Information requires context to have meaning.
My context is made of information.
Repeat.
And this is bizarre because there are lots of reasons to think the universe might be purely informational, which also brings up the notions of philosophical ideas like Plato's world of forms and whether or not numbers exist abstractly. If they do exist abstractly and the universe is informational, then you can even argue that their existence explains our existence. But the context problem of information also lets every set argue it is uniquely represented by the number 1, which breaks any sense of mapping that you could use to describe the relationship between the evolving state of the universe and the set of all integers.
Another interesting quandary is that Graham's number is expressible in this universe. To write it out long-form would supposedly require the space and mass of a ridiculous number of universes the size of our observable universe, which means there MUST be gaps in the list of integers we can express in this universe. This is because of how the pigeon hole principle applies to the combinatorics of the arrangement of all the mass in the universe: A ridiculous number of combinations are taken up by all the possible arrangements that represent Graham's number, so there are also going to be a ridiculous number of smaller numbers that cannot have unique represensations. The big question I have is this:
What determines which integers are actually expressible in this universe?