Talk:Fixed point (mathematics)

Pre- and postfixed points switched

Davey/Priestley are simply wrong™ on this. The earlier, better reference is Cousot&Cousot, 1979, http://www.di.ens.fr/~cousot/COUSOTpapers/publications.www/CousotCousot-PacJMath-82-1-1979.pdf , where the the prefixed and postfixed points are defined correctly™. Changed.

Pre- and postfixed points switched back

I think the Smyth-Plotkin 1982 usage is more common than the Cousot-Cousot 1979 usage. They seem equally natural to me. — Preceding unsigned comment added by 147.188.201.12 (talk) 21:44, 21 July 2022 (UTC)

Well there is also Shamir's 1976 thesis which says a prefixedpoint is p ≤ f(p), and accompanying 1977 paper "The convergence of functions to fixedpoints of recursive definitions". The Cousot 1979 paper cites Shamir's paper so this is probably why they agree. There are still people using Shamir's definition, e.g. [1] (cites Cousot) and [2] (probably copied from Wikipedia). In [3] Pitcher uses Shamir's definition and remarks "Warning: in [Gun92], [the post-fixed-point] is the definition of a pre-fixed-point." This implies that Pitcher considered both definitions and decided to use Shamir's definition, but unfortunately he doesn't explain why.

I guess I agree, overall prefixedpoint as f(p) ≤ p does indeed seem more common. But it would be nice to have a source which thoroughly compares the two definitions, however briefly, and comes to a conclusion. But Pitcher is not that source. --Mathnerd314159 (talk) 02:10, 22 July 2022 (UTC)

I should also mention Shamir's justification for his definition: a prefixedpoint is a function which is "almost" a fixedpoint, but is less defined. This "less defined" is similar to the meaning "before" of pre-. It is also similar to the use in preorder, which is almost a partial order but is not antisymmetric. In contrast the Smyth-Plotkin definition has no justification in that paper or in Davey-Priestley. Mathnerd314159 (talk) 03:31, 22 July 2022 (UTC)

Per [4] the justification for the "modern" definition is that the location of the symbol f is before the inequality sign in the term “f (x) ≤ x”. Mathnerd314159 (talk) 05:11, 22 July 2022 (UTC)

The analogy to "preorder" is not helpful, as it works equally well for each usage of "prefixpoint". 147.188.201.13 (talk) 20:51, 23 July 2022 (UTC)

But well done for finding all these sources. 147.188.201.13 (talk) 20:52, 23 July 2022 (UTC)

Requested move 4 October 2022

The following is a closed discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review after discussing it on the closer's talk page. No further edits should be made to this discussion.

The result of the move request was: no consensus. (closed by non-admin page mover) — Ceso femmuin mbolgaig mbung, mellohi! (投稿) 20:04, 23 October 2022 (UTC)

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• Fixed point (mathematics) → Fixed point
• Fixed point → Fixed point (disambiguation)

– After my edits this article is about the usage of fixed points in a range of areas, such as math, computer science, and logic, with a brief mention of the physics definition, so it is not just about mathematics anymore and is more suited to be at Fixed point. Looking at the dab stats it seems that it is 50-50 fixed point (mathematics) and fixed-point arithmetic, so no help there in determining a primary topic. As argued in the previous RM though, the primary topic of the term "fixed point" is the sense described on this page. "Fixed-point" uses a dash, for one thing - looking at a dictionary they are clearly separate. Also, in terms of long-term significance, the usage in computers is relatively recent and niche, and the usage as a point that is fixed has been around for centuries and is still going strong in every high school math classroom. Also the close last time as no consensus seems pretty suspicious by my count - the only actual oppositions were No such user and BarrelProof who preferred moving to invariant point, and they were rebutted by eviolite and also opposed by Felix QW. So if this page doesn't become the primary topic then Wikipedia's RM process is just broken. Mathnerd314159 (talk) 04:09, 4 October 2022 (UTC) — Relisted. P.I. Ellsworth , ed. ^put'r there 10:32, 14 October 2022 (UTC)

• Oppose: sorry, I still think of Fixed-point arithmetic. YorkshireExpat (talk) 19:33, 4 October 2022 (UTC)

  Per WP:NWFCTM it doesn't matter what you first think of. Mathnerd314159 (talk) 02:02, 6 October 2022 (UTC)

  Well WP:5P5 to that. As someone with a fairly general technical education including maths, computer science, and physics, I regard myself as a well read layman in this topic, and don't think it's unreasonable that others would think the same way as me. YorkshireExpat (talk) 17:27, 6 October 2022 (UTC)

• Oppose: The computing topic is not any more niche than topic in the analysis of mathematical functions. Computing may be a more recent topic, but computing seems unlikely to fade away in importance anytime soon. Any simple search will show that the computing terms "fixed point" and "floating point" are often not hyphenated in practice. If the dab stats are showing a 50-50 split in the selection between the two disambiguated topics, that indicates there is WP:NOPRIMARY topic and the dab page is doing its job of helping people find what they want when what they want is statistically unclear. —⁠ ⁠BarrelProof (talk) 23:02, 4 October 2022 (UTC)

  Honestly I'm not sure what's up with the stats. Outside of WP, a simple Google Books search shows 52 out of 430 results with the computing usage. (At page 43 Google stopped matching text in books and just started recommending based on topic), confirming that the computing usage is quite niche. The standard definition was used in diverse areas such as fiction, astronomy, railways, and patents. It was even used in a computing magazine to describe computing square roots via iteration. Meanwhile the computing usage was used with.. well... computers. Similarly if we look at Ngrams and charitably assume that fixed point in the computing sense and floating point were used equally, subtracting that out the standard sense was only briefly below the computing sense around 1990 and since then it has rebounded to 3x the usage.

  As well the hyphenation there is also the adjective-noun distinction - in the computing usage, fixed-point is always applied to a noun, so you get fixed-point signal processing, fixed-point arithmetic, fixed-point numbers, etc. In fact users use various names to get to the page such as fixed-precision arithmetic and fixed-point number. So FPA can't be a primary topic because it would be an incomplete title. (WP:TITLEPTM) Mathnerd314159 (talk) 03:08, 6 October 2022 (UTC)

• Support As mentioned above and in the last Requested Move discussion, the computing use is differentiated by a hyphen and is a partial title match, since it is rarely used as an adjective outside of set phrases such as fixed-point arithmetic. Felix QW (talk) 19:17, 6 October 2022 (UTC)

  This is the first non wiki link I get when I Google 'fixed point'. No hyphen. YorkshireExpat (talk) 21:12, 6 October 2022 (UTC)

  Yeah, but that article also has obvious grammatical issues. "The use of fixed point data type is used widely in ..." Mathnerd314159 (talk) 21:32, 6 October 2022 (UTC)

  Here is a real example: [5]. This was on page 8 of Google Scholar results for "fixed point", quite buried. Most results were of course for the mathematical meaning, but all the fixed-point arithmetic results before then used the dash. Mathnerd314159 (talk) 21:42, 6 October 2022 (UTC)

• Comment: I linked this discussion in Wikipedia talk:WikiProject Mathematics#Fixed point requested move and Wikipedia talk:WikiProject Computing#Fixed point requested move to hopefully get a wider variety of views. Mathnerd314159 (talk) 21:24, 6 October 2022 (UTC)
• Support: Per WP:NOUN: "Fixed point" [of a function] is a noun, whereas in the computing sense, it is an incorrectly dehyphenated adjective with the real noun omitted (I see I am independently making the same point as Felix QW above). We should not be using titles in the sense of an adjective simply because it brings to mind a topic by suggestion. There is no other topic that competes as a possible primary topic. We should not be supporting what amounts to illiteracy on WP. —Quondum 22:01, 6 October 2022 (UTC)

  The grammatical accuracy of the page title(s) is not what's in dispute. It's what topic people expect to get when they type 'fixed point' into the search bar, and whether or not the extra click from the dab page is wasted or not. YorkshireExpat (talk) 16:58, 7 October 2022 (UTC)

  That is what disambiguation hatnotes in articles are for, but rather what the primary topic is (and my argument here is that the arithmetic sense is disqualified). We should not use search bar behaviour to decide article titles. Besides,the person looking for Fixed-point arithmetic gets an extra click either way. The person looking for the fixed point of a function gets one less click if not sent via a dab page. —Quondum 17:53, 7 October 2022 (UTC)

  But there are more pages being dabbed. YorkshireExpat (talk) 18:17, 7 October 2022 (UTC)

  The additional dab links seem pretty tangential, and can go through an extra click to get to the dab page. —Quondum 18:51, 7 October 2022 (UTC)

  Maybe to you, but remember WP:NWFCTM ;) YorkshireExpat (talk) 08:32, 9 October 2022 (UTC)

• Oppose. There is no primary topic, but I'd support some move since the current title can easily be confused with fixed-point arithmetic. Vpab15 (talk) 15:23, 12 October 2022 (UTC)

  I'd propose fixed point (invariant) or also invariant point which was suggested in previous discussion. Vpab15 (talk) 15:25, 12 October 2022 (UTC)

  Invariant point was also clearly rejected last time. Quoting: '"Invariant point" is far too rare for fixed points to be "commonly called" that for WP:NATURAL purposes.' Similarly invariant is too unusual a word to be used as a subject specifier for fixed point (invariant); it will just read like fixed point (*gibberish*). Mathnerd314159 (talk) 23:23, 15 October 2022 (UTC)

• Support, though not very strongly. I can see there's good competition for the name but there is no stronger one for the precise name. In fact I'm a bit surprised a couple of other uses like in sci-fi aren't mentioned as well in the disambiguation.. I would not move it to a title like invariant point - that is simple not used anywhere nearly as much. NadVolum (talk) 11:47, 19 October 2022 (UTC)

• Oppose - There is no clear WP:PRIMARYTOPIC here so Fixed point needs to remain a disambiguation page. Renaming Fixed point (mathematics) may be appropriate but I don't have any useful input to share on that.

~Kvng (talk) 16:09, 22 October 2022 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

"Transformation" inconsistency in current article

The current article defines a fixed point in the first sentence as an element sent to itself by a transformation function. The article on transformation functions declares that these are functions sending a set to itself, i.e., that the domain and codomain are the same set, i.e., the picture of transformations relevant to the transformation monoid. The text of this article and recent edits, however, stated that the domain and codomain may be different. This has been an inconsistency that seems to need correction one way or another.

It is absolutely true that the image of a transformation function need not be identical with its domain, and also that there are mathematical conventions defining fixed points for functions that are not transformations. But there are also mathematical conventions for fixed points of non-functions! Partial functions (very simple by restriction) and morphisms (less trivial), for example. I edited to over-restrict the notion of fixed point in my last copyedit for consistency with the lead sentence, but I see that didn't go over well with other active editors of the page; I'll edit to expand the notion next time, which will mean redefining mathematical fixed points as something broader than specifically fixed points of transformation functions. RowanElder (talk) 13:52, 31 August 2024 (UTC)

The transformations article also has the more general definition: "When such a narrow notion of transformation is generalized to partial functions, then a partial transformation is a function f: A → B, where both A and B are subsets of some set X." There is no need to change anything. Mathnerd314159 (talk) 15:51, 31 August 2024 (UTC)

(1) That more general definition does not address all of my concerns, which were not about partial functions alone and specifically.

(2) The generalized claim in the transformation (function) article about partial functions is a contentious simplification that probably needs editing itself; it is not the case that a partial function "just is" a function f: A → B, where both A and B are subsets even though that's good enough to build shared intuition in informal use and conversation. There are exact correspondences between the partial functions and those functions of subsets but they have differences.

(3) Both this and the transformation article are start class articles and have been for a long time. They both need a lot of work that's not getting done. There is a lot to change. RowanElder (talk) 19:32, 31 August 2024 (UTC)

Regarding the rating, for a long time this article was C-class. AFAICT it should still be C-class, Lazard's demotion of it was because too many topics were in "see also", which I addressed. It could even be B-class with a few general sources. In contrast I would say transformation really is Start-class. Mathnerd314159 (talk) 23:07, 31 August 2024 (UTC)

This isn't convincing to me. It seems to be a skeletal article for an important topic. To answer the commit message, I'd just foolishly assumed Wikipedia's "point (mathematics)" would be more than a disambiguation page when tentatively proposing a new point/mapping wording for the lead rather than value/function. I'm new to Wikipedia and still adjusting to the low quality of the mathematical reference information on the encyclopedia; I'm an expert mathematician but I only decided to pick up editing Wikipedia as a hobby recently. Not the most rewarding hobby so far with interactions like these! RowanElder (talk) 00:25, 1 September 2024 (UTC)

Well, it is pretty tricky editing articles with vague titles, just because people argue over what it should mean. IMO this page is more of a disambiguation page than anything else, sort of an overview of the topic. And naturally when you edit an article with a vague title you end up spending most of your time trying to figure out what it means. IMO, an expert mathematician like you would probably have more fun finding stub articles or even red links, like Category:Mathematics stubs or Wikipedia:Requested articles/Mathematics. Since those articles are undeveloped, you can write pretty much whatever you want and it will stick - the only requirement is verifiability and citations. As opposed to here, where "ease of understanding" and so on are considerations, and the page has already been through a few rounds of debate. Mathnerd314159 (talk) 03:58, 1 September 2024 (UTC)

I am doing mostly copyediting and cleanup on Wikipedia, trying it out as a more prosocial hobby alternative to crosswords. If I want to have the sort of fun of "writing what I want" I write a paper or a blog post, where I'll have a bigger, friendlier, and more expert audience.

In this case I really don't care which of a few decent definitions of fixed point is used in the lead paragraphs so long as the article refers to it clearly and consistently and develops it clearly and consistently through the subsequent sections. I was not trying to argue the merits of one definition or another. I was solely trying to make the article more consistent and clearer. RowanElder (talk) 14:32, 1 September 2024 (UTC)

Thanks for your observations - I wasn't aware of the different definition in transformation (function). Apologies for any confusion that I may have caused.

We could start by defining fixed points only for functions with identical domain and codomain, which is easiest to understand and which (I guess) covers the vast majority of all applications. Lateron, we could mention the more general definition as a generalization. I thought of a sentence like As a generalization, the fixed points of an arbitrary function f:X → Y can be defined as those of f|_X∩Y: X∩Y → Y, but this restriction needn't map into X. My second thought was to restrict as f|_X∩f^-1[X]: X∩f^-1[X] → X , but this needn't map into f^-1[X]. We could restrict to $\bigcap _{i=0}^{\infty }\underbrace {f^{-1}[\ldots f^{-1}[X]\ldots ]} _{i{\text{ times}}}$, but this is hard to read, let alone understand. So, it is best not to define by restriction, but just reuse the current definition instead: c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.

Anyway, starting with a narrow definition, and mentioning the more general definition(s) lateron would nicely match the structure of transformation (function), too.

I suggest to avoid mathematical point (which is a disambiguation page, anyway). Just "value" (without a link) should be sufficient in the lead, or, maybe, mathematical object. As for partial functions, I'd move them down to the generalization section. - Jochen Burghardt (talk) 16:24, 31 August 2024 (UTC)

Thank you, too, and sounds good. Introducing the most essential prototype definition in the lead and later discussing generalizations makes good sense to me. RowanElder (talk) 19:35, 31 August 2024 (UTC)

I don't really think the notion of fixed point is well defined, e.g. fixed point iteration is not actually a generalization of the notion of a fixed point but more like a related concept. It sounds like you want to write a long-ish article on the "essential prototype definition", the fixed point of a function. I would suggest doing that in a new article. The title Fixed point of a function has been suggested before and then the current section could have a "main article" hatnote like the others. Mathnerd314159 (talk) 04:01, 1 September 2024 (UTC)

I didn't want to write that article, no. I just wanted to clean up inconsistencies and unclarities in this one to clear the way for others with more ideas. The article initially struck me as an intimidatingly tangled-up mess of conflicting conventions. It still does.

There are a few well-defined notions of fixed point that should be perfectly adequate for the lead, any of them could be great, and fixed point iteration deserves some discussion on an encyclopedic page about fixed points. I would agree that ideally the page would be reworked so that it's not a skeletal disambiguation page where the section on "fixed point iteration" could be mistaken for "introducing another type of fixed point" if you wanted to argue that. However, I have no plan to do larger rework now or in the short term. I'm currently busy with the awful mess that has been the geometric series page, and I'm still generally getting my bearings on Wikipedia. RowanElder (talk) 14:15, 1 September 2024 (UTC)