Religious stuff (not math)
I removed this from the main page as it has not context. We need to say who believes this, what religion (if any) these beliefs are a part of, etc etc. Probably beter if it lives in its own article.
(As developed by Sri Aurobindo:)
The process by which the Divine manifest the cosmos is called the Involution. The process by which that which was created rises to higher states and states of consciousness is the evolution. The Involution is essentially up to the point of the Big Bang; the Evolution is from that point forward.
After the creation, the Divine (i.e. the Absolute, Brahman, God; all these essentially mean the same thing) is both the One (the Creator) and the Many (that which was created).
The process by which the Many is created from out the Absolute is called the Involution. Once that process ends, the process of Evolution begins. In essence the evolution begins with creation (the Big Bang being a stage) and continues with all that follows.
The involution is the process by which the Absolute manifest the creation, the universe. It is the process by which the Many emerged from the One as a universe of divided, ignorant forms. The involution is that which occurred that enabled the creation, the universe, the cosmos to manifest from out of the Original Principle, the Divine, God, the Absolute. Involution is the process of self-limitation, of densification, by which the Absolute, Brahman veils itself by stages until it assumes the appearance in the cosmos, the universe we know of. It wishes to create the universe to objectify itself and its spiritual properties in infinite possibilities, for the purpose of delight of discovery which it will achieve thereafter.
The evolution is the movement forward by which the created universe evolves from its initial state of divided, ignorant forms, emerges as Life and Mind, and in that process rediscovers its Source. The evolution occurs after the involution. It is the development and progressive movement of all in the cosmos, including humans, to attain its fulfillment, including rediscovery in delight of the spiritual aspect, that Consciousness-Force, that was the source of the creation. The evolution is the progressive development from the first inconscience in matter into life (movement, sensation, etc. and living physical beings), to mind (in conscious being, animals, including the human, the self-conscious thinking animal), to spiritualized mind, culminating in Supermind, Truth Consciousness (as supramental individuals, leading to a supramental, i.e. a divine life on earth.)
DJ Clayworth 15:28, 26 Jan 2004 (UTC)
This statement is false:
Old meaning of involution
In the edit summary of 09:58, 28 June 2007 (diff), JRSpriggs wrote: ... involution is an old name for exponentiation; evolution is the old name for extracting a root. The web page on Involution (How to find square roots by hand), linked from the article, on the other hand says: ... Involution, or extracting a square root, ... Can this be resolved? — If involution is an old name for exponentiation (rather than root extraction), the external link should be removed. In either case, could the old meaning be referenced and mentioned in the article? -- LBehounek 23:25, 1 July 2007 (UTC)
- Look "involution" and "evolution" up in a good dictionary. Also see Exponentiation, where it says:
- Another historical synonym, involution,< ref >This definition of "involution" appears in the OED second edition, 1989, and Merriam-Webster online dictionary . The most recent usage in this sense cited by the OED is from 1806.< /ref > is now rare and should not be confused with its more common meaning.
- OK? JRSpriggs 05:34, 2 July 2007 (UTC)
Incorrect dab notice
This disambiguation notice is obviously wrong:
- This article is about involution in mathematics. For other uses, see Involution (mathematics) (disambiguation).
Should the exclusive-or operator be mentioned? Given an arbitrary value x, the operation f(a) = a xor x is its own inverse. In particular, g(a) = a xor 0 is the identity function. | Loadmaster (talk) 19:45, 4 February 2009 (UTC)
This page has an absurd reference. The paper referenced in the sources section, "Quaternion involutions and anti-involutions", regurgitates mathematics that has been known since the time of William Hamilton. That paper should not have even been published. Use a serious reference, such as "The book of involutions" (http://www.math.uni-bielefeld.de/~rost/BoI.html). This is a comprehensive text written by leaders in algebra. Or if you're looking for something more accessible, why not go for a paper of John Voight (of U. Vermont)? He's the current world expert on the study of quaternion algebras. I could think of dozens of more appropriate references. — Preceding unsigned comment added by 22.214.171.124 (talk) 16:15, 16 September 2011 (UTC)
Involution in group theory
The group theory section is different from all the other sections, and from the general definition at the beginning, in that the identity is defined to not be an involution. I looked in several mathematical dictionaries, several group theory books, and a random selection of group theory papers, and I never found the identity explicitly excluded. Instead, every group element g with g2=1 is called an involution. Hundreds of papers use the phrase "non-trivial involution" when they want to not allow the identity. I think our present text is a mistake, but if there is an authoritative source defining it like now we can give both options. Zerotalk 08:31, 7 November 2012 (UTC)
- I am also surprised at this definition.
- I added a reference that supports the current definition -- unless I am misunderstanding that reference?
- I agree that this article should list both definitions, as per WP:YESPOV. --DavidCary (talk) 18:35, 26 November 2013 (UTC)
The specific kind of involution used in most reciprocal ciphers
Is there a name for the subset of involution functions used in most reciprocal ciphers? i.e.,
- The function pairs up each element x with some other element y, such that
- f(x) = y and x != y and f(y) = x, for each and every x.
Or in other words,
- f(x) != x for each and every x, and
- f(f(x)) == x for each and every x
(In particular, the identity function is not a member of this subset). My understanding is that the Enigma machine and nearly all other reciprocal ciphers always use such a function to transform each plaintext letter to the corresponding ciphertext letter. (I suspect because of the misunderstanding that "when we encrypt a plaintext letter, obviously the encrypted ciphertext should not be the same letter"). I've been calling such a function a pairing, but I've recently discovered that term is usually used to mean something quite different -- what term should I use instead? --DavidCary (talk) 18:35, 26 November 2013 (UTC)
The following text was removed:
- For x in ℝ, this is often called Babbage's functional equation (1820). Ref: Ritt, J. F. (1916). "On Certain Real Solutions of Babbage's Functional Equation". The Annals of Mathematics 17 (3): 113. doi:10.2307/2007270. JSTOR 2007270.
The function f(x) = 1 – x is indeed an involution, but the article cited is about more general periodic functions and is inappropriate here. Perhaps another editor can find a place for it elsewhere.Rgdboer (talk) 23:10, 28 June 2015 (UTC)