In this post, I will copy-paste some definitions about some concepts, that I believe they somehow relate to each others.
- Self-reference (Recursion): occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding (ex. the word “I”).
- Fractals: A fractal is a natural phenomenon or a mathematical set that exhibits a repeating pattern that displays at every scale. If the replication is exactly the same at every scale, it is called a self-similar pattern. Fractals can also be nearly the same at different levels.
- Strange loop: A strange loop arises when, by moving only upwards or downwards through a hierarchical system, one finds oneself back where one started. Strange loops may involve self-reference and paradox. The concept of a strange loop was proposed and extensively discussed by Douglas Hofstadter in Gödel, Escher, Bach, and is further elaborated in Hofstadter’s book I Am a Strange Loop, published in 2007.
And yet when I say “strange loop”, I have something else in mind — a less concrete, more elusive notion. What I mean by “strange loop” is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive “upward” shifts turn out to give rise to a closed cycle. That is, despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop. (pp. 101-102)
- Gödel’s incompleteness theorems: The self-referential structure at the core of Gödel’s incompleteness theorems. Gödel’s theorem shows that, a complete and consistent finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. Each time a new statement is added as an axiom, there are other true self-reference statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent.
- Consciousness: We are conscious that we are conscious. Consciousness in itself is an example of a strange loop. Atoms form neuron, that form brain, that think about concepts, that include strange loops, atoms, neuron, brain, self and consciousness itself.
- Intelligence: has been defined in many different ways such as in terms of one’s capacity for logic, abstract thought, understanding, self-awareness, communication, learning, emotional knowledge, memory, planning, creativity and problem solving. Yet, the ultimate quest of intelligence is to understand/recreate intelligence itself.
- Superrationality: Is a way of thinking, assuming that everyone else is taking into calculation that everyone else is assuming that everyone else is taking into consideration [to any n level of recursion …] the decisions of the others.
“Supperational thinkers, by recursive definition, include in their calculations the fact that they are in a group of superrational thinkers.”
- Common knowledge: A knowledge is said to be a common knowledge, when every one knows this knowledge, and everyone knows that everyone else know this knowledge, and everyone knows that everyone else knows … [to any n level of recursion]. For ex.: Superrationality can be achieved when it is a common knowledge. (aka: when everyone is superrational, and when everyone knows that everyone is superrational, etc to any n-level).