18 Matching Annotations
  1. Apr 2021
  2. Mar 2021
    1. In computer science, a tree is a widely used abstract data type that simulates a hierarchical tree structure

      a tree (data structure) is the computer science analogue/dual to tree structure in mathematics

  3. Jul 2020
    1. In logic, functions or relations A and B are considered dual if A(¬x) = ¬B(x), where ¬ is logical negation. The basic duality of this type is the duality of the ∃ and ∀ quantifiers in classical logic. These are dual because ∃x.¬P(x) and ¬∀x.P(x) are equivalent for all predicates P in classical logic
    2. the ∧ and ∨ operators are dual in this sense, because (¬x ∧ ¬y) and ¬(x ∨ y) are equivalent. This means that for every theorem of classical logic there is an equivalent dual theorem. De Morgan's laws are examples
  4. May 2020
    1. Related concepts in other fields are: In natural language, the coordinating conjunction "and". In programming languages, the short-circuit and control structure. In set theory, intersection. In predicate logic, universal quantification.

      Strictly speaking, are these examples of dualities (https://en.wikipedia.org/wiki/Duality_(mathematics))? Or can I only, at strongest, say they are analogous (a looser coonection)?

    1. Mathematically speaking, necessity and sufficiency are dual to one another. For any statements S and N, the assertion that "N is necessary for S" is equivalent to the assertion that "S is sufficient for N".
    1. In mathematical contexts, duality has numerous meanings[1] although it is "a very pervasive and important concept in (modern) mathematics"[2] and "an important general theme that has manifestations in almost every area of mathematics".[3]
  5. Oct 2016
    1. Tiresias, though blind, throbbing between two lives, Old man with wrinkled female breasts, can see

      Tiresias is a hermaphrodite; possession of female/male body parts gives him more insight and is the "true prophet".

  6. Nov 2015
    1. You have been concerned in the last few days that you are not making the connection between what I have been discussing with you and your daily experience. The simple fact is that you have been trying to relate infinite ideas to finite beliefs. There is no connection. There is no connection between Mind and Its manifestation. They are one. It is only from the standpoint of ego—of finite, three-dimensional misperception—that there can seem to be a “connection.” Since ego separates itself from what it sees, it divides things into subject and object, “in here” and “out there,” me and thee, and all the rest of the dualism. This prompts the attempt to manipulate all of these disconnected aspects, so that they will harmonize in such a way as to not destroy the basic illusion of the existence of “ego.” Thus, from the three-dimensional standpoint, it seems obvious that the misidentification of Identity called “ego” must become, do, initiate, manipulate, and achieve. One cannot bring these endeavors into that Place where Self is properly identified and experienced as Conscious Being or Fourth-dimensional Man, since the basic illusion of separateness has no existence there.

      Quote: Only from the standpoint of ego can there seem to be a "connection".

      Quote: There is no connection between Mind and its manifestation. They are one.

      The ego must become, do, initiate, manipulate, and achieve in order to find a connection between disconnected parts. These endeavors are not aspects of Conscious Being because the illusion of separation does not exist there.

  7. Oct 2015
    1. PAUL: Raj, something is blocking me from being able to be comfortable with what you are saying. RAJ: I would think by now that “comfortable” and “uncomfortable” would no longer be valid guideposts for you, since those are three-dimensional points of view. You know that when you are in that Place, you are experiencing Absolute Comfort, undivided in any way. If you want to bounce back out onto the surface of your finite consciousness, you will, indeed, find comfort and discomfort, pain and pleasure, gain and loss. I am afraid, Paul, that this is the one essential step which you must make yourself. I can encourage you to take it. I can explain to you why you can take it. But it has to come from within your Being. What is blocking it is simply fear. There is no actual opposition or roadblock. PAUL: In the past, I have experienced healings without taking this radical step. I have experienced benefits of prayer and metaphysical work. I did not have to be under the gun, so to speak. Isn’t unfoldment supposed to be natural?
  8. Jan 2014
    1. How we meet this challenge depends on how we address the following fundamental question about teaching our digital-age children: Should we teach our children as though they have two lives, or one?

      two lives or one? Also, what about two names? A public name and a private name, as some cultures already have where only your friends call you by a certain name that others might not know.