Talk:Involution (mathematics)

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)

Is there any reason not to move this to Involution (esoterism)? --DavidCary (talk) 18:35, 26 November 2013 (UTC)

Transpositions

This statement is false:

A permutation is an involution precisely if it is a transposition.

(1 2)(3 4) is an involution, but it is not a transposition (at least according to the definition on the Wikipedia transposition page). I've changed it. -- Walt Pohl 21:20, 7 Dec 2004 (UTC)

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 [1]. 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).

If I were familiar with the templates, I'd fix it. Michael Hardy (talk) 12:48, 4 September 2008 (UTC)

Boolean exclusive-or

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)

Ridiculous reference

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 128.84.79.145 (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. Zero^talk 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)

Babbage

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)

Transpose in linear algebra

Transpose was cited as a key involution in linear algebra since matrix (mathematics) is an essential part of that study. An editor reverted this link, saying matrices are no longer central to linear algebra. No evidence was given for this assertion. Books on linear algebra are largely concerned with matrices. Mystified by this revert, a discussion is initiated here. — Rgdboer (talk) 19:51, 4 October 2018 (UTC)

The material on transposes was largely unchanged except in positioning by this sequence of edits. The actual addition that was reverted was the editorial, unsourced, and debatable opinion that "The matrix is an essential element of linear algebra." —David Eppstein (talk) 20:47, 4 October 2018 (UTC)

Thank you. The following sentence was inserted lower in the section: "For a specific basis, any linear operator can be represented by a matrix T. Every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices." The contribution was not obviated. The variable representing the matrix may be reconsidered. — Rgdboer (talk) 02:00, 5 October 2018 (UTC)

I apologize for my edit summary "matrices, while certainly important, are not (anymore) in the center of LA", which may be perceived as somewhat rude, especially to editors in our age (a parenthesized anymore, yuck!). My only exoneration is pleading "while certainly important". :)

I wanted to preserve the transpose, I used undo not to deprecate this edit, but to make utmost explicit that I changed it, and I also pondered the variable name, swaying between none at all and keeping up the connection to the name of the linear operator $T.$ I cannot judge if spending more effort on naming pays the rent. Purgy (talk) 07:35, 5 October 2018 (UTC)

No harm, Purgy. Have moved on to § Mathematical logic with a contribution on relations. Always appreciate your edits. — Rgdboer (talk) 22:21, 5 October 2018 (UTC)

Please give the reader the general case

I think the examples should be the general cases of -x, and 1/x (to be C-x and C/x). I also think that it is a bit misleading to show that the composition of the above functions is also an involution, cause the reader might think that any composition of two involutions is also involution. besides, composition of the above function, can be considered as the mere one case of the general case, which is C/x (where C=-1) 2A02:6680:1102:385C:1841:9A2D:D966:A308 (talk) 20:04, 7 June 2022 (UTC)

Generel Properties

All given examples for involutions in the reel numbers can be created with the composition of a bijection g with an involution f with the formula $g\circ f\circ g^{-1}$: e.g.: $f_{3}(x)=f_{2}(x-1)+1$ and $f_{4}(x)={\sqrt {2}}f_{3}(x/{\sqrt {2}})$. A special case is a shift along the line y=x.

Is it possible to reduce all involutions to a composition starting with the inverse operations of the reell numbers $-x,{\frac {1}{x}}$? Conscious 'n' curious (talk) 11:34, 18 November 2022 (UTC)

Complex numbers

Is there a reason that there are sections on real numbers and quaternions but not on complex numbers? Blitzer99 (talk) 07:21, 18 October 2024 (UTC)

These are not involutions

The article incorrectly lists $ln(x), e^x$ and $x^2$ as involutions. 2001:BB6:316E:C400:6806:27CA:6BE3:B660 (talk) 12:26, 12 November 2024 (UTC)

No, it doesn't. It lists then as functions that can be combined with their inverses and another involution to make involutions. Some IP editor recently took out the explanation, making that part more confusing. —David Eppstein (talk) 17:48, 12 November 2024 (UTC)