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Updates to "Math theorem" Templates: Improved style and new proof parameters

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Currently, there are two templates for inserting formatted theorems and proofs into articles: {{Math theorem}} and {{Math proof}} In many cases, however, the proof of a theorem directly follows the theorem. The formatting when juxtaposing these templates is not great, however:

Theorem — My theorem statement.

Proof

My proof statement.

I have written a modified version of {{Math theorem}} (see {{Math theorem/sandbox}}, at [this revision]) to improve the formatting by incorporating the proof as a new parameter for {{Math theorem}}, and also bringing the default formatting of theorems in line with typical math texts:

Theorem. My theorem statement.
Proof. My proof statement.∎

Please join the conversation at Template talk:Math theorem there if you have opinions about the proposed change. The-erinaceous-one (talk) 04:50, 2 September 2024 (UTC)[reply]

Can you give examples of articles where you think this template should be used? Personally I find that most of the time ordinary paragraphs are sufficient for theorem statements and proofs (in some articles where authors put theorems in flashy boxes, sometimes with color, etc., I have found the decorations more distracting than helpful). Proofs are helpful in particular in articles that are directly about a theorem or a few theorems, or occasionally in articles where a particular theorem is fundamentally important to the topic, but in cases I'm thinking of collapsing the proofs would defeat the point of including them. Most of the rest of the time I'd skip the proofs altogether (proofs in external resources can be linked from footnotes, or if a proof seems distracting but necessary, it could be put in a footnote in full). –jacobolus (t) 06:35, 2 September 2024 (UTC)[reply]
I envision the new version of the template replacing the current {{Math theorem}} template and also being used anywhere else that a formal theorem statement could be useful. As you noted, many pages currently display theorems and proofs in boxes, which I agree are undesirable. Editors might sometimes intentionally add boxes around theorems and proofs, but I think in many cases they simply use {{Math theorem}} because they assume it is the "standard" Wikipedia formatting of theorems. By updating the template, we would improve the formatting across all of those pages that use it and discourage editors from doing ad-hoc formatting of theorems (e.g., boxes and colors).
Regarding the formatting of proofs, I'm not married to the idea of making the proofs collapsed by default, or, in fact collapsible by default. We could choose the default to not make proofs collapsible and then allow editors to enable it using a parameter flag. I am also looking into adding another parameter that allows displaying the proof in a footnote, although I personally find this a worse option than a collapsible box since it requires readers to scroll up and down if they want to see the theorem while reading the proof.
One example of a page that would benefit from a nicely formatted Theorem template is Liouville's theorem (Hamiltonian). Despite the name of the page including "theorem", there is not a formal statement of the theorem. The closest thing it has is

The distribution function is constant along any trajectory in phase space.

but this doesn't state the formal assumptions of the theorem. The-erinaceous-one (talk) 22:58, 2 September 2024 (UTC)[reply]
I was looking around for an example of a proof in the footnote, but didn't find one quickly. Here's what I have tried for the {{Math theorem}} template.

Theorem. Mathy mathy math.[proof 1]

The-erinaceous-one (talk) 23:27, 2 September 2024 (UTC)[reply]
I would recommend against replacing the previous template, since authors who used it were intending the behavior as provided at the time, not an entirely different appearance chosen by someone else later. I also disagree that Liouville's theorem (Hamiltonian) would benefit from having parts of it wrapped in boxes or reformatted. Ordinary paragraphs are working fine there. In my opinion you should make a new template under a new name if you want it, and then adopt it on pages you write yourself or do significant work on, but should leave other pages alone. Aside: your sandbox version probably has some kind of malformed HTML which causes it to render outside of a colon-indented talk page response (which uses a definition list element). –jacobolus (t) 23:44, 2 September 2024 (UTC)[reply]
It looks like the existing template does not work well in lists either. [Edit: I placed an example here, but it broke our ability to use the "reply" editor]. The-erinaceous-one (talk) 00:07, 3 September 2024 (UTC)[reply]
Regarding Liouville's theorem (Hamiltonian), I think a weakness of that page is that it is difficult to figure out what "the theorem" actually says. First you have to search through the page to find the quoted text I copied above (which is not clearly labeled as the theorem). Then, you have to reconstruct and/or guess what the assumptions of the theorem are from the rest of the article. The-erinaceous-one (talk) 00:13, 3 September 2024 (UTC)[reply]
In other words, it's a typical explanation of something that physicists call a theorem. XOR'easter (talk) 02:31, 3 September 2024 (UTC)[reply]
Skimming through links at Special:WhatLinksHere/Template:Math theorem and Special:WhatLinksHere/Template:Math proof, these templates aren't really all that widely used, and in my opinion most of the articles where they are used would be improved by avoiding the templates (and sometimes taking out the proofs). YMMV. –jacobolus (t) 06:51, 2 September 2024 (UTC)[reply]
Their usage might not be ubiquitous, but 400+ pages is not insignificant and improving the available template(s) would improve those pages and making nicely formatted theorems and proofs easier. Regardless, the {{Math theorem}} template already exists and is used, so the question is whether the proposed changes would be improvements---not whether we should completely stop using it. The-erinaceous-one (talk) 23:01, 2 September 2024 (UTC)[reply]
Also, many of the pages I've opened up use the templates multiple times, so the total number of uses is well over 400. The-erinaceous-one (talk) 00:34, 3 September 2024 (UTC)[reply]
Yeah, if it were up to me at least half of those would not have any such template. But it's disruptive to make changes like this at large scale. People should feel free to use this list as inspiration for finding articles which could be improved, including by removing the templates. –jacobolus (t) 01:45, 3 September 2024 (UTC)[reply]
I agree with @jacobolus that we don't want to put too much emphasis on proofs. Many articles would actually benefit from removing some of their proofs. This has been discussed multiple times on WPM. From the Proofs section of MOS:MATH: A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgment; as a rule of thumb, include proofs when they expose or illuminate the concept or idea; don't include them when they serve only to establish the correctness of a result. In many cases it would be more beneficial to work on replacing the proofs with some suitable references to reputable sources instead of incorporating them into some new template. PatrickR2 (talk) 03:19, 3 September 2024 (UTC)[reply]
Even without the issue of incorporating proofs or not, I often find the current template (and the new template) which wraps the result in a "template box" to be distracting and annoying. In articles that discuss multiple results, it gives undue weight to those that happen to have an official name of "Theorem of Such-and-such" compared with those results that don't. That unnecessarily breaks the flow of exposition. Better use something less intrusive like "Theorem of such-and-such: statement ..." in the text itself. One case where the template could be justified is an article or section dedicated to a single theorem. But most of the time, the use of the template seems misguided to me. PatrickR2 (talk) 03:28, 3 September 2024 (UTC)[reply]

References

  1. ^ Proof. This proof text should be placed in the footnote, but it is not yet working.∎
There are two different matters: how to update the current theorem template and whether its use is appropriate. I commented on the first in the talk page of the template and so here I comment on the second. As someone who actually likes using the template (and the one who actually imported the template from French Wikipedia), I think it depends on how it is used within an article. I agree in some instances, boxes can be jarring especially if there are too many of them. On the other hand, emphasizing a statement in some way is a good idea in some other instances. The axiom of choice articles gives a good example in my opinion: since the article is about a single statement. (As the proof template, I have never used it personally but apparently some people like it) —- Taku (talk) 08:26, 3 September 2024 (UTC)[reply]

Square bipyramid proposal to split

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Discussion on splitting article the square bipyramid is ongoing. See Talk:Octahedron#Create a square bipyramid or regular octahedron article. More opinions are welcome. Dedhert.Jr (talk) 11:08, 4 September 2024 (UTC)[reply]

Help needed on several elementary articles

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Farkle Griffen made many edits on several elementary articles including Variable (mathematics), Mathematical object, Indeterminate (variable) and several others. Generally, these article are of low quality, but IMO, most of their edits are disimprovements, as consisting generally of misinterpretations of randomly chosen sources. One of their typical edit is to change the first sentence of Variable (mathematics) from "In mathematics, a variable is a symbol, typically a letter that is used for naming a mathematical object, often a number" to "In mathematics, a variable is a symbol, typically a letter, that holds a place for constants, often numbers".

I must stop to revert them, because WP:3RR, and because of the lack of third party input, I cannot open an ANI thread for disruptive editing (this appears as content dispure).

So, I need some help. D.Lazard (talk) 18:36, 5 September 2024 (UTC)[reply]

Mathematical object seems broadly improved. Variable and indeterminate seem tricky to me. There's at least one way these are different, in that variables usually have a domain, while indeterminates are purely formal symbols. (E.g., random variable, real variable, etc.) But many people in casual discourse make no such distinction. Tito Omburo (talk) 18:53, 5 September 2024 (UTC)[reply]

Eigenmode vs. eigenmodes (redirects)

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Currently,

This is obviously confusing, because eigenmodes ([[eigenmode]]s) and eigenmodes ([[eigenmodes]]) are different link destinations.

Suggestion: the plural redirect [[eigenmodes]] should point to Eigenvalues and eigenvectors, just like the singular version. Thoughts?  — sbb (talk) 03:30, 5 September 2024 (UTC)[reply]

The word “mode” (with or with prefix) does not appear at Eigenvalues and eigenvectors; should it? 100.36.106.199 (talk) 10:36, 5 September 2024 (UTC)[reply]
I believe that the best way forwards is to look at all of the articles on eigenfunction, eigenmode, eigenstate, eigenvector and normal mode collectively, then discuss what to merge and what to redirect. As part of this, the redirects for singular and plural should be brought into alignment with each other after looking at what links to what.
My first take is that eigenfunctions and eigenvectors should be in the same article, but I could make a case for keeping them separate. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:52, 5 September 2024 (UTC)[reply]
Clearly the target of any of these links should not depend on whether it is plural or not. That's easy to fix.
I think Eigenfunction should be kept separate from Eigenvalues and eigenvectors, since the latter article is necessarily focused on linear endomorphisms in general, whereas the former should be focused on the specific application to function spaces. It's fine to leave a summary in Eigenvalues and eigenvectors § Eigenvalues and eigenfunctions of differential operators pointing to Eigenfunction as a main article. –jacobolus (t) 17:25, 5 September 2024 (UTC)[reply]
A mode is a standing wave. See eg https://www.feynmanlectures.caltech.edu/I_49.html#Ch49-S5
In mathematical models of wave systems these standing waves appear as "eigenfunctions", also called "eigenvectors" in some representations. A mode, a wave concept, is not a synonym for "eigenfunction", a mathematical concept.
Absent a significant reliable reference, "eigenmode" should be deleted. The word is redundant by repeating itself. Similarly "eigenmodes". Without a reference having these pages is misinformation. Johnjbarton (talk) 15:34, 5 September 2024 (UTC)[reply]
According to Eigenmode expansion, eigenmode is a specific term-of-art for solutions of the Maxwell equation along a waveguide. This usage seems to be consistent with many of the hits in a cursory Google scholar search. Tito Omburo (talk) 17:19, 5 September 2024 (UTC)[reply]
Thanks, but I disagree with your characterization based on the three sources in that page. None of these references define "eigenmode" and as far as I can tell they only use the word as a modifier as in "eigenmode propagation algorithm".
The solutions to Maxwell's equations along a waveguide seems to be just "modes". Jackson discusses "wave guides" in section 8.3 and says:
  • "There will be a spectrum of eigenvalues and corresponding solutions which form an orthonormal set. These different solutions are called the modes of the waveguide.
Here is a review of quantum optics that uses the word 'mode' many times, but 'eigenmode' rarely and inconsistently.
  • Fabre, Claude, and Nicolas Treps. "Modes and states in quantum optics." Reviews of Modern Physics 92.3 (2020): 035005.
My guess is that "eigenmode" has a specific technical meaning like you say. But what meaning?
Based on what we know so far, "eigenmode" should redirect to Eigenmode expansion since at least the word is used there. Johnjbarton (talk) 19:38, 5 September 2024 (UTC)[reply]
Ok I think I found a review that sorts this out at least for radio waves:
  • Huang, Shaode, Jin Pan, and Yuyue Luo. "Study on the relationships between eigenmodes, natural modes, and characteristic modes of perfectly electric conducting bodies." International Journal of Antennas and Propagation 2018.1 (2018): 8735635.
    • Eigenmode expansion method (EEM) [3], singularity expansion method (SEM) [4], and characteristic mode analysis (CMA) [5] are three common modal analysis methods in electromagnetic engineering. The three modal analysis methods result in three different kinds of modes generally, that is, eigenmodes, natural modes, and characteristic modes, respectively.
Based on this reference (and thus restricted to the corresponding field), "eigenmode" is not a synonym for "eigenvector" or "normal mode", but a specialized term related to the "eigenmode expansion method". Johnjbarton (talk) 19:57, 5 September 2024 (UTC)[reply]
The edit history for the eigenmode redirect shows that it used to point to Normal mode, but the latest (Oct 2022) edit by Constant314 (talk · contribs) states "Eigenmode is much more general that normal mode. When modes are mapped onto a vector space, a mode becomes a vector. Hence, eigenmode and eigenvector are nearly the same thing", and was changed to point to Eigenvalues and eigenvectors.
I can't comment on whether or not the edit comment is correct, but that was the rationale/statement.  — sbb (talk) 23:22, 5 September 2024 (UTC)[reply]

Just some random , unauthoritative thoughts.

  • An eigenmode is a mode (whatever that is) that can be mapped onto an eigenvector. More specifically, the numbers that describe the mode can be the components of an eigenvector. Doing that allows the machinery of linear transformations to be used to analyze the mode. Casually speaking, we may say that an eigenmode is a type of eigenvector. Of course, speaking more formally, we mean that the numbers that describe the eigenmode are treated as components of eigenvectors. Again, speaking casually, we say that an eigenmode is an eigenvector, but we do not say that (all) eigenvectors are eigenmodes.
  • An electromagnetic field mode is any configuration of the electromagnetic field that satisfies Maxwell’s equations, the constitutive equations, and the boundary values. Solution and mode are used interchangeably. I have not heard the term eigen-solution.
  • Mode is not restricted to mean an electromagnetic field mode.
  • The voltages and currents of multi-conductor transmission lines are analyzed by the use of eigenmodes.
  • It is not an eigenvector unless it is associated with a linear transformation which has an input vector and an output vector.
  • The Eigenmode expansion article seems underdeveloped and focused on waveguides. There are no in-line citations in the body of the article. Of the three citations, two are used to establish the name and the other establishes that the method is useful. The external link leads to a not found page. The term eigenmode has been in use since the 1950’s and predats any of the references. I hesitate to redirect anything to this article.

Constant314 (talk) 17:00, 6 September 2024 (UTC)[reply]

As far as I know, the level of development of an article is not a criteria for redirects nor are unauthoritative thoughts. My unauthoratative take is that 'eigenmode' is used in different ways in different subfields and mostly as an informal synonym for 'mode' because that does not sound fancy enough.
I have provided a reference that identifies "eigenmode" as a type of "mode". This particular type of mode is discussed in sources listed in eigenmode expansion. So far these are the only source we have that discusses "eigenmode" directly. (Many sources use 'eigenmode' in the sense of 'eigenmode expansion'.) Asserting that 'eigenmode' is a much broader subject and predates the 1950s doesn't really help us here. We can't verify you ;-).
Even with a source that shows 'eigenmode' is an 'eigenvector' representation of a mode, I still do not agree that this redirect to eigenvalues and eigenvectors makes sense. A specialized, modified noun should redirect to the noun, not the adjective. Moreover, all the sources indicate that 'eigenmode' is associated with physics and engineering, not mathematics as topic.
A reasonable alternative to eigenmode expansion could be a redirect to normal mode. Johnjbarton (talk) 17:32, 6 September 2024 (UTC)[reply]
I don't think normal mode is a candidate. It is clearly talking about resonances at a fixed frequency. The eigenmodes of a waveguide have a continuous frequency spectrum. Further, it describes a mode as a standing wave. The eigenmodes of waveguides are traveling waves. Constant314 (talk) 21:37, 6 September 2024 (UTC)[reply]
Yes, I agree. I think normal mode should be renamed "Mechanical mode" or similar. What if we merge transverse mode and longitudinal mode into eg "Waveguide modes" and add a short section on "eigenmodes"? Johnjbarton (talk) 23:14, 6 September 2024 (UTC)[reply]
Of course! We also have Mode (electromagnetism). Bah. Johnjbarton (talk) 23:32, 6 September 2024 (UTC)[reply]
I share your frustration. Wikipedia has never been accused of being overly organized. I did not take part in the seminal conversations that established Wikipedia culture, but it seems to have settled to this: verifiability takes priority over completeness which takes priority over avoiding redundancy. You are welcome to reorganize, but don't lose anything. Sound also has longitudinal modes, transverse modes (in solids), and waveguides. You cannot just absorb those into Mode (electromagnetism) or Waveguide modes. However, both of those could use an expanded section on transverse and longitudinal modes. Constant314 (talk) 03:22, 7 September 2024 (UTC)[reply]
Redundancy is totally fine in my opinion (and by Wikipedia convention), as long as (1) each subject is encyclopedic ("notable"), (2) the scope of each article is reasonably clear and not entirely overlapping, and the article is reasonably complete and balanced within that scope without putting undue weight on minor aspects of the topic or fringe viewpoints, (3) each article is moderately self-contained and accessible, not dependent on text in other articles, (4) related articles are each correct, don't contradict each-other, reflect the consensus of reliable sources, (5) related articles link to each-other so that readers can find the information they are looking for, and maybe some others I'm not thinking of. With that said though, also feel free to reorganize material by moving it from one article to another, merging articles together, splitting them apart, etc. if a different high-level inter-article organization seems clearer. –jacobolus (t) 04:07, 7 September 2024 (UTC)[reply]
I updated Mode (electromagnetism) and included a sentence about eigenmode with the two refs I found. Johnjbarton (talk) 00:51, 8 September 2024 (UTC)[reply]
I just made an enquiry at the Teahouse about subpages. Seems that it is not allowed. I was referred to Wikipedia:Splitting. Noit sure if that helps. It looks like your main interest is the eigenmode redirect. Perhaps he should create an eigenmode stub which could point the reader to all the appropriate targets along with some commentary. Does that sound like a good idea? Or, even simple, create an eigenmode dab page. I may go ahead and do that.Constant314 (talk) 18:32, 8 September 2024 (UTC)[reply]
I essentially used the page Mode (electromagnetism) for the disambiguation purpose. I think that should be the target for the redirect unless we find a lot more sources and content. We could add an anchor to the paragraph. Johnjbarton (talk) 18:55, 8 September 2024 (UTC)[reply]
I am good with that. Eigenmodes is probably a little more general than that, but Mode (electromagnetism) is a probably a good redirect target. You have my support to make the change. I presume that include both eigenmode and eigenmodes. Constant314 (talk) 19:18, 8 September 2024 (UTC)[reply]
Thanks,  Done Johnjbarton (talk) 19:39, 8 September 2024 (UTC)[reply]
 Courtesy link: Wikipedia:Teahouse § Adding a subpage to an existing article—  jlwoodwa (talk) 20:46, 8 September 2024 (UTC)[reply]
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