This answer is about: randomness and chaos; and how they relate to logic and computability.
"What is randomness and where does it come from?
This is one scary place to venture in. We take for granted the randomness in our reality. We compensate for that randomness with probability theory. However, is randomness even real or is it just a figment of our lack of intelligence? That is, does what we describe as randomness just a substitute for our uncertainty about reality? Is randomness just a manifestation of something else?"
— "Medium." Medium, < medium.com/intuitionmachine/there-is-no-randomness-only-chaos-and-complexity-c92f6 >.
"Many natural intensional properties in artificial and natural languages are hard to compute. We show that randomized algorithms are often necessary to have good estimators of natural properties and to verify some specific relations. We concentrate on the reliability of queries to show the advantage of randomized algorithms in uncertain cognitive worlds." de Rougemont, Michel. "Logic, randomness and cognition." — Logic, Thought and Action. Springer, Dordrecht, 2005. 497-506.
Unfortunately, quantum chaos in relation to randomness, is a profoundly technical topic. I have managed to track down sources that relatively aren't overly technical.
As a starting point, this medium article is worth reading:--
For profoundly technical topics, I recommend this book series, as they are written by experts in basic technical terms, for the laypersons wanting to study technical topics:--
I recommend reading:--
- Chaos: A Very Short Introduction
- Probability: A Very Short Introduction
- Fractals: A Very Short Introduction
Other References for the Layperson:--
Some broader implications of chaos [Link to Stanford Encyclopedia of Philosophy].
When I think of randomness, I'm inclined to think in cosmological terms. Is randomness is a structural property of the universe? Is anything in the universe truly random?
"In mathematical logic, independence is the unprovability of a sentence from other sentences." Wikipedia contributors. — "Independence (mathematical logic)." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 3 Feb. 2019. Web. 29 Aug. 2019.
"We propose a link between logical independence and quantum physics. We demonstrate that quantum systems in the eigenstates of Pauli group operators are capable of encoding mathematical axioms and show that Pauli group quantum measurements are capable of revealing whether or not a given proposition is logically dependent on the axiomatic system. Whenever a mathematical proposition is logically independent of the axioms encoded in the measured state, the measurement associated with the proposition gives random outcomes. This allows for an experimental test of logical independence. Conversely, it also allows for an explanation of the probabilities of random outcomes observed in Pauli group measurements from logical independence without invoking quantum theory. The axiomatic systems we study can be completed and are therefore not subject to Gödel's incompleteness theorem." Paterek, Tomasz, et al. "Logical independence and quantum randomness." — New Journal of Physics 12.1 (2010): 013019.
- I usually do not use Medium or Quora as a source, with some exceptions. I have chosen to do so here.
- I've decided to place Stanford Encyclopedia of Philosophy sources in the layperson's section.