Can we ever understand the totality of our world? What are the limits of knowledge? One way to approach this is to consider our system of understanding as a formal and consistent system (a theory), which has some axioms and theorems. Can we prove all the theorems from within our system or cognition?
Generally, the answer is no (Gödel 1931). Loosely, I think it is because our system cannot simulate itself. There are more truths in the world than computable ways to reach them. Trying to add ways/axioms creates new Platonic truths, truths which cannot be proven from within the enhanced system. It is like trying to lift yourself by pulling one’s hair. What if one considers all possible axioms? Then we do not have a theory, we have the world. The unity of things might understand itself, but this is a theological perspective even.
I think there is a sort of a mental space of possibilities a priori, which is generated by the complexity of the axioms of the system. Think of a system consisting of some algorithms. If we for example enumerate all simple functions in a system, we can always take that list and create a simple variation of this list. So our definitions in language (natural or math) allows us to cheat in a way. The natural language and formal systems are therefore generally incomplete. There is no way to be completely exhaustive in our system. With rules of computation, it is impossible to consider exhaustive and enumerable lists and so on. One can always push the envelope a bit further.
Think of the sentence declaring an object:
”The shortest Berry paradox not definable under ten words”
I came up/invented this version of the original Berry paradox today, thinking of how to interpret Gregory Chaitin’s incompleteness theorem. I in a way declare an object, which is inherently paradoxical. Of course I cannot define complex objects with simple rules, that is the essence of the paradox here. More formally it maps to the fact that the complexity of the axioms cannot prove the Kolmogorov complexity of any given truth. So we cannot in a way recognize complex objects with simple rules. The paradox was put forward by Bertrand Russell, but it is attributed to a librarian named Berry in Oxford.
The original paradox goes:
https://en.wikipedia.org/wiki/Berry_paradox
I guess the interpretation could go something like this: We cannot state or prove complicated things on particular entities with simple rules or axioms –> our language (natural laws or natural language) is too small compared to the totality of the world. In physics, we might say that a physical law is a statement about the state of affairs (a rule on how to replicate reality), but as the particular reality can be complex, we cannot recognize a new physical law easily, which would deduce the observed complex universe, if we have simple axioms. And if our physics becomes more complex (a universe with more rules, or more axioms), the truths remain elusive. The larger the sea of knowledge, the longer the coastline of our ignorance.
If there are indeed always entities with high complexity, beyond our axioms in the sense that we could determine their high complexity, perhaps then the totality is essentially random in the sense that causal laws are not sufficient to describe them. This in turn would suggest that instrumentalism makes sense. Good theories are as simple as possible, and predict out of the sample as well as possible. This is the principle of sufficient reason by Leibniz. The thing is, that we cannot create a complete system of laws using low complexity/information, as the world is complex. Therefore induction and inference is needed. The best model of the totality of things is the totality of things itself. If we cannot in principle prove the facts of this world from our language and logic, we have essentially the illusion of a free will, which is a grace.
Vastaa