Wikipedia talk:WikiProject Mathematics/Archive/2025/Jul

Boundary parallel has a definition that appears to be garbled, and cites a source^[1] that has a different^[a] definition, also with an issue.^[b] See Talk:Boundary parallel#Unclear lead - wrong links. Can anybody suggest an alternative source with a better definition? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:21, 14 July 2025 (UTC)

Square root of 10 could use more eyes. For some context, after a related, protracted RFD debate about similar titles, bd2412 has unilaterally created a (bad) article on top of a redirect despite a reasonably strong consensus not to have such an article at Wikipedia:Articles for deletion/Square root of 10. There's some weird situation with revisions being undeleted (by BD), submitting and accepting his own draft, and I can't follow all of it. Moreover, I've been threatened with a block by a clearly WP:INVOLVED admin for edit warring, despite following WP:BRD, so I'm disengaging. 35.139.154.158 (talk) 23:54, 16 July 2025 (UTC)

I would probably support a second AfD of this article. The sourcing is quite dubious. For what it's worth, an IP editor can nominate for AfD, same as an autoconfirmed editor. Tito Omburo (talk) 00:01, 17 July 2025 (UTC)

Fwiw, I see you received a totally inappropriate warning on your userpage for this revert. You removed it, perhaps ill-advisedly, but it was bullshit. I haven't checked the other litany of complaints against you, but this one item seems very questionable and supports your assessment of WP:INVOLVED. Tito Omburo (talk) 00:09, 17 July 2025 (UTC)

For context, the content deleted in the previous AfD of this article is posted at Talk:Square root of 10, and constituted about 1/10 of the current amount of content, with no sources as compared to the dozen or so sources currently in the article. It is, of course, entirely appropriate to write a new article on a subject previously deleted for lack of sources if sources can in fact be found. BD2412 T 00:11, 17 July 2025 (UTC)

Why are you threatening an IP editor by trawling their prior contributions? Also, I do not think the "sources" you have given would survive another AfD. I suggest you nominate the article yourself and leave the matter to the community. Tito Omburo (talk) 00:12, 17 July 2025 (UTC)

Firstly, as this is an IP address it is impossible to know whose "contributions" are being reviewed. Secondly, the "contributions" are reverts of talk page warnings, which suggest an intent to conceal a pattern of conduct from scrutiny. This is in the context of the IP previously having been blocked twice for conduct issues, and warned numerous times. BD2412 T 00:15, 17 July 2025 (UTC)

Your own response contradicts itself. It is impossible to know whose "contributions" are being reviewed, and the same "person" was blocked on conduct issues? You are hereby recused. You should nominate this article for deletion and let the community decide. How are you an administrator? Go hang your head in shame and think about your future with the project. Tito Omburo (talk) 00:18, 17 July 2025 (UTC)

The activity of concern is emanating from the IP account, whether the edits themselves are being made by one person or a group of people with access to that IP address. When an IP address has been warned many times and blocked more than once for conduct emanating from that IP, then it does not matter whether it is one person or multiple people; we block the IP address. BD2412 T 00:30, 17 July 2025 (UTC)

Enough. I have, on my own authority, per WP:BRD restored the last WP:CONSENSUS version of the article, which was a redirect. If you want to re-create the article, please go through the process we all have to: WP:AfC. Also, ping this project, myself, and the IP, as per usual guidelines on article creation. Also please refrain from threatening editors, IP or otherwise, on manifestly manufactured pretext. In my opinion, you should be stripped of your administrative privileges on this project for this infraction, we don't need no stinking cops. But I am just one man. Tito Omburo (talk) 00:37, 17 July 2025 (UTC)

Can we all try to assume good faith and remain polite here? (Aside: This page just did apparently go through WP:AFC.) –jacobolus (t) 03:23, 17 July 2025 (UTC)

You're right, the article did go through AFC and no editor needs approval from a WikiProject to compose an article. And these insults are childish, if you want to take action on your complaints, there are noticeboards where that can happen but you don't need to belittle editors on talk pages. That usually doesn't end well for the insulter. I'd strike those remarks while you can. Liz ^{Read! Talk!} 06:32, 17 July 2025 (UTC)

My unsolicited four cents:

• BD2412's action in threatening to block 35.139.154.158 after only a second revert is not a good look. Without making any inference as to what was in BD2412's mind, admins need to be very careful to avoid the appearance that they might be using their admin status as leverage in a content dispute. (Well, any dispute, really)
• The article looks fine to me, and the subject seems "intuitively notable"; if it doesn't meet the immediate special notability guideline then maybe we should revisit that guideline.
• That said, are we going to just throw away the previous consensus at AfD? I'm not necessarily opposed to that; ten people six years ago shouldn't be able to ban a topic for all time, but I would like to see the point addressed.
• Tito's recent contributions sound emotional and a short voluntary absence from the dispute might be a good idea.

--Trovatore (talk) 08:00, 17 July 2025 (UTC)

@Trovatore: I threatened to block the IP when I saw that they had been warned for precisely this sort of conduct over, and over, and over, and over, and over again, and had recently been blocked for it. With respect to the previous AfD, that was for, functionally, a very different article. That version was a stub of just over 1,000k as compared to the 12,000k of the current article. Several of the participants complained of the article being unsourced, which it was. This version of the article has over a dozen sources. BD2412 T 12:29, 17 July 2025 (UTC)

Having the tools is not supposed to give you a leg up in disputes. Having the trust of the community, which got them to award you the tools, might, but the tools themselves never should. If it even looks like that could be happening, there's a problem. That's all I'm saying. --Trovatore (talk) 16:53, 17 July 2025 (UTC)

Blanking a page without discussion is not a "dispute", it is vandalism. There is no "leg up" to be had. BD2412 T 17:28, 17 July 2025 (UTC)

No, it is not vandalism: see WP:VAND, especially the sections WP:TPV and the subsections of WP:NOTV titled "Boldly editing" and "Disruptive editing or stubborness". Your behavior here with regards to administrative threats is grossly improper, much more so than anything the IP has done. (Have you noticed yet how many different people are saying versions of this to you?) Your failure to treat the IP with good faith, and your incorrect labeling of their edits as vandalism, are also improper. I don't think TO took you to ANI for precisely the correct reason, but if an administrative recall petition is started based on your behavior towards this IP (who is a long-term constructive contributor) and your failure to recognize the problems with it, I would strongly be inclined to support. --JBL (talk) 23:49, 17 July 2025 (UTC)

The IP has a lengthy history of warnings for incivility and edit warring and blocks for continuing such behavior after being warned. I grant that this history may have influenced my response. BD2412 T 01:08, 18 July 2025 (UTC)

That's not an answer. An uninvolved admin could possibly have made such an argument. You should not have. You were in fact in a dispute with 35.139.154.158, whose reverts were not "blanking" but the restoration of the prior consensus, and gave at least the potential appearance of using the tools as leverage in that dispute. --Trovatore (talk) 01:53, 18 July 2025 (UTC)

My response to the IP was prompted by my awareness of their history, not by the instant dispute. I do understand how matters could have appeared to those unfamiliar with that history. If the same actions had been undertaken by an IP who had not been recently blocked for edit warring, I would have responded differently. BD2412 T 02:08, 18 July 2025 (UTC)

You are still not engaging with the issue, which has nothing to do with the appropriateness of a potential block. The issue is the potential for admin privileges to put a thumb on the scale of a dispute. Admins are not supposed to have any advantages whatsoever in disputes, at least none deriving from their admin status. These sorts of warnings have potential to create such advantages. --Trovatore (talk) 18:08, 18 July 2025 (UTC)

Because you were in a content dispute, you should not have blocked! Paul August ☎ 02:42, 18 July 2025 (UTC)

@Paul August: I didn't. BD2412 T 03:21, 18 July 2025 (UTC)

Sorry, "threatened" to block. Paul August ☎ 03:29, 18 July 2025 (UTC)

I was very careful not to state that I would impose a block. If the behavior continued, I would have reported the IP for recidivism to the same conduct that led to their previous month-long block (which I had no involvement in). BD2412 T 03:33, 18 July 2025 (UTC)

I think the arguments for deletion in the old AFD discussion still hold up. None of the published sources are about ${\displaystyle {\sqrt {10}}}$ itself and some should not be cited here at all (for instance the Conway--Guy and the Wells books). As for the contents of the article, apart from the section on the historical use as an approximation to pi everything seems to be either random factoids, or things applicable to all square roots of integers. The stuff on pi could be shortened and included in https://en.wikipedia.org/wiki/Pi#Approximate_value_and_digits. jraimbau (talk) 14:12, 17 July 2025 (UTC)

Put it to AFD again then, if you want. Or remove sources you think don't meet WP:RS or claims you don't think stand up to scrutiny. I think the historical use as an approximation for π is likely enough to establish notability: if you hunt there are many sources about this. There is also plenty out there about the use for constructing logarithms or the relevance to slide rules (example). –jacobolus (t) 17:20, 17 July 2025 (UTC)

An entire article could exist on the historical use of the square root of 10 as an approximation to pi alone. This is reported to have occurred in multiple civilizations around the world independently, and certainly there is no other number for which such an association can be claimed. As for the Conway-Guy and the Wells books, those come straight from the OEIS, which itself lists them as references for this number; listing in the OEIS is itself a basis for notability per Wikipedia:Notability (numbers)#Irrational numbers, as is the number having a name other than "3.16227 etc.". I frankly don't see how the "trivia" about this number is any different than the "trivia" reported on any number typically covered in Wikipedia. BD2412 T 17:41, 17 July 2025 (UTC)

@BD2412: I don't understand the hostility to this article. I found the approximate of pi to be of interesting value, but an underdiscussed aspect is its role in order-of-magnitude calculations (see talk page). Expand on that, and you'll have a bulletproof article, IMO. Headbomb {t · c · p · b} 17:43, 17 July 2025 (UTC)

Re: Conway--Guy and Wells' books, these references contain each 1 line about ${\displaystyle {\sqrt {10}}}$, they should not be cited as providing support for notability.

I amso believe using (solely) OEIS for notability for a number is at best not completely accepted, in any case your link discusses an example which has many other more reliable sources for this.

Re:"there is no other number for which such an association can be claimed": 22/7 occurs in Chinese and Greek computing (and possibly Egyptian as well).

In general i don't have a strong opinion on whether the article should be removed or not but in its current form it looks filled with a distasteful mix of pedantry and triviality, which certainly explains part of the negativity towards it. jraimbau (talk) 18:31, 17 July 2025 (UTC)

@Jean Raimbault: The mix of content is fixable, and fixing it would be the most helpful thing to the reader interested in the subject. To my perhaps untrained eye, however, the mix of content just looks like how we write articles on numbers generally (e.g., 33 (number)). BD2412 T 18:43, 17 July 2025 (UTC)

A large fraction of our articles on individual numbers consist solely of "a distasteful mix of pedantry and triviality". Saying it's just like the others does not advance your case. —David Eppstein (talk) 00:17, 18 July 2025 (UTC)

Then delete all the numbers, David. Or, being such a proud mathematician, improve them. Other editors on this project have been doing that with the subject under discussion here. BD2412 T 00:48, 18 July 2025 (UTC)

Frankly Wikipedia / its readers wouldn't be affected too much one way or the other by the deletion of most of the current articles about numbers, though the process would be a waste of time, so I wouldn't recommend it.

On the other hand, most of them could have a decent article written about them if someone put in the effort. It takes significant amounts of effort to hunt for good sources and then summarize them with a clear and meaningful narrative, and the number articles don't seem to have generally attracted much interest from the type of folks who could write good articles about them. I don't think Wikipedia:WikiProject Numbers has overall been very successful as a project. –jacobolus (t) 00:55, 18 July 2025 (UTC)

Re "then delete all the numbers": I gave up on both trying to delete bad articles on numbers and trying to rescue rescuable articles on numbers after Wikipedia:Articles for deletion/198 (number) (2nd nomination). It is one thing to have an enormous collection of low-quality junk piles on Wikipedia to clean up; it is another thing entirely to have to fight off the many Wikipedia editors who appear to like keeping the junk piles junky. Somehow it reminds me of this. —David Eppstein (talk) 01:09, 18 July 2025 (UTC)

David, I do sympathize. I find bad content all the time and try to improve it, and am aware that there are millions of pages that I will never even see that could stand even more improvement. However, I have to have faith in the idea that eventually we will all pull together and make things right, fix all the errors, and cite all the propositions. BD2412 T 03:28, 18 July 2025 (UTC)

There is plenty enough bad content for me to improve, on topics that I am interested in, where my improvements are likely to stick. It's a waste of my time put efforts into areas where they are not appreciated and other editors will quickly make the content bad again. Number articles have become that to me. —David Eppstein (talk) 06:11, 18 July 2025 (UTC)

I would suggest picking one number (a less immediately prominent one, again like 33 (number)), and making a really good article of it, so there's something editors can look to for guidance on what a really good article on a number should look like. I did that with United States District Court for the Eastern District of Missouri back on '09, and although there is still a way to go with improving the district courts, at least there is now a model article. BD2412 T 17:22, 18 July 2025 (UTC)

You are free to make advice for your own behavior and then follow it. I am not lacking in projects here. —David Eppstein (talk) 18:00, 18 July 2025 (UTC)

Articles about numbers seem to inevitably attract heaps of miscellaneous trivia and not much meaningful connective prose. While they aren't generally particularly helpful to anyone, gathering trivia lists doesn't really do much harm, and for some readers trivia might be what they want, so I'm largely indifferent. Ideally we'd have nice readable and comprehensive articles about all of the small numbers, but currently even our articles about e.g. 2 (how is it possible that this doesn't mention parity (mathematics), division by two, binary logarithm, binary search, involution, duality (mathematics), mediation and duplication, octave, boolean function, sign (mathematics), even and odd functions, etc. etc.), 3, 4, and 5 are pretty bad. –jacobolus (t) 00:34, 18 July 2025 (UTC)

Frankly, I think the fact that we are still talking about this is a good thing. It is an indication that we are still building the encyclopedia, and that people are engaged in how that project will proceed. BD2412 T 01:10, 18 July 2025 (UTC)

Hi WikiProject Mathematics!

I am trying to shepherd an article about mathematician William Martin Boyce through the AfC process. Boyce disproved the common fixed point conjecture in 1967, and followed up with additional research on commuting functions, Baxter permutations, Steiner trees, and callable bond pricing. I'm hoping that someone better positioned at the intersection of mathematicians and Wikipedia editors can help me with this.

The first version of the article (which I admit was not very good) was declined. Thinking the article needed more context, I spent some time writing the Common fixed point problem article, which was accepted immediately and featured in DYK.

I re-wrote the Boyce article with many more sources and references to the other article. But the article was again declined, this time raising concerns about whether or not Boyce was notable. I specifically wrote the article to meet the criteria in Wikipedia:Notability (academics), so I felt stuck.

Hoping to avoid another 3-month AfC review cycle, I posted a couple of questions on the AfC help desk page, but I wasn't able to get anyone to reconsider the article. The previous reviewer suggested detailing the argument in favor of notability on the talk page and resubmitting it, which I did. The reviewer also suggested posting a message here.

Would someone from WikiProject Mathematics be willing to look at the draft article and talk page and either help with the notability argument, or let me know how I can improve the article to better establish notability? (Any other advice is also welcome.)

Thank you! WillisBlackburn (talk) 02:37, 19 July 2025 (UTC)

Is it familiar to say regular quadrilateral and regular hexahedron instead of square and cube, respectively? Recently, I have some dispute with the user regarding to include "3-cube" as the alternative way to say cube. For some reason, n-cube is for the hypercube in ${\displaystyle n}$-dimensional, something kind of analogy between those two figures I have mentioned above. But at the same time, n-cube can also mean for hypercube graph. So I removed the hypercube terminology, but the user replies with the same thing . Maybe someone can have an opinion? Dedhert.Jr (talk) 01:20, 19 July 2025 (UTC)

Google scholar: «"regular tetragon"» = 216 results, «"regular quadrilateral"» ≈ 3,000 results, «square quadrilateral» (papers including both words, not necessarily together) ≈ 145,000 results, "regular hexahedron" ≈ 2,300 results, "3-cube" ≈ 18,100 results (not all relevant).

So these terms are occasionally used, but much rarer than square or cube. I don't think there's a problem with mentioning these in some contexts (I would recommend including the name "regular hexahedron" in the lead of cube, as the name "hexahedron" is apparently the common name for the shape in crystallography), but I wouldn't belabor it. –jacobolus (t) 01:31, 19 July 2025 (UTC)

"Regular hexahedron" is definitely not first-sentence material. Inserting it there throws a stumbling block of needless complexity into the very beginning of an article about a basic concept. It has a certain feel, intended or not, of putting on airs. (We don't wedge Ludolph's number into the first sentence of Pi.) It might be introduction material, if the body text follows up with sufficient detail on where that terminology is used (e.g., crystallography). Stepwise Continuous Dysfunction (talk) 02:18, 19 July 2025 (UTC)

That a cube is a type of hexahedron, and that it is "regular" are both essential facts which must be mentioned in the lead section. They don't need to be turned into a bold term (and perhaps should not, since hexahedron should be wikilinked). In just the same way that square leads with "In geometry, a square is a regular quadrilateral." it might be reasonable to lead with something like "In geometry, a cube is a regular hexahedron." –jacobolus (t) 05:20, 19 July 2025 (UTC)

I'm not convinced that's the best way to begin Square, but anyway, hexahedron is a more obscure word than quadrilateral. The latter appears in middle-school geometry texts, whereas the former belongs to the professionals, and may be more obscure than Platonic solid. I'd rather stick with language like "shape with six square sides" to begin with, and then bring out "regular hexahedron" somewhat later. Stepwise Continuous Dysfunction (talk) 07:24, 19 July 2025 (UTC)

It's only more obscure because solid geometry is a more obscure topic than plane geometry, (sadly) not taught to middle school students, not because it is dramatically more complicated or less useful. (Although... a Google scholar search for «hexahedron "middle school"» turns up hundreds of papers where the word is used in the context of instruction for middle school students, and an Internet Archive search turns it up in dozens if not hundreds of middle school geometry textbooks, commonly as part of a list of Platonic solids where the cube is described as "hexahedron (cube)", so I'm not sure your impression of those is accurate.) –jacobolus (t) 07:49, 19 July 2025 (UTC)

Whatever the reason for it being a more obscure word, it is a more obscure word. (Compare the 142 GS results for «hexahedron geometry "middle school"» to the ~10,500 for «cube geometry "middle school"». The corresponding numbers for the Internet Archive are 1,256 versus 31,224. Or, for a very broad-strokes picture, consider the ngrams.) I don't see the point of having readers trip over it at the very beginning. Stepwise Continuous Dysfunction (talk) 15:08, 19 July 2025 (UTC)

In the article Modular representation theory the phrase "the maximal order R of the field K" is used; I'm not familiar with that concept. Does anybody know how to explain or fix it? Thanks, AxelBoldt (talk) 13:53, 19 July 2025 (UTC)

The definition of an order is somewhat context-dependent. For a number field, an order is a subring that is a lattice containing a ${\displaystyle \mathbb {Q} }$-basis of the field. For a local field, it is a compact-open subring. Looking at the article, it seems likely that it is the latter case, a local field (of characteristic zero), that is intended, but it is far from clear. In that case, the more usual term would be the "valuation ring" of the field. Tito Omburo (talk) 14:07, 19 July 2025 (UTC)

I have linked maximal order. This answers your question. However, this is not sufficient to clarify the section, which requires a complete rewrite. IMO, this is also true for large parts of Modular representation theory and Maximal order. D.Lazard (talk) 15:01, 19 July 2025 (UTC)

Thank you, D. Lazard, for your constructive feedback. I agree that simply linking to the Maximal order page only partially addresses the conceptual gaps in the section. Given your observation, I will begin drafting a revised version that improves clarity and structure. If you're open to collaboration, I would appreciate your input as I refine the content—especially concerning key aspects of modular representation theory where precision is essential.

Best regards,

Saadat 31.7.98.24 (talk) 16:59, 19 July 2025 (UTC)

I think it should be replaced by a link to valuation ring, because is this not an order in the sense of the article I think, but agree that this article needs a rewrite. Tito Omburo (talk) 17:00, 19 July 2025 (UTC)

Title. Does anyone here have access to the source "Geometry and its Applications; In Honor of Morio Obata"? It's under paywall. MathKeduor7 (talk) 05:35, 18 July 2025 (UTC)

No one? :( MathKeduor7 (talk) 17:05, 19 July 2025 (UTC)

If you have access to Wikipedia:The Wikipedia Library, then you do, but only as an "EPDF", with no ability to download. –jacobolus (t) 18:08, 19 July 2025 (UTC)

Thank you very much! MathKeduor7 (talk) 20:29, 19 July 2025 (UTC)

I've started stubs about them. There are many great theorems named after them. Any geometer willing to help improving the articles with details about their mathematical contributions!? Thank you very much! MathKeduor7 (talk) 05:31, 20 July 2025 (UTC)

See Talk:Tetrahedron#Proposal: split regular tetrahedron to have its own article. Dedhert.Jr (talk) 02:13, 22 July 2025 (UTC)

This seems to have been copied unilaterally from a declined draft article, with sources almost entirely linked to the original draft author. I'm pretty sure any mention of it is undue but wanted to double check to see if I'm missing something. At the very least, the Spanish-language sources, if they're actually reliable, should probably get evaluated by a Spanish-speaker. Sesquilinear (talk) 21:13, 22 July 2025 (UTC)

It is sourced to the springer encyclopedia, so I would presume appropriate. I have not checked the exact contents, but this seems formally appropriate under a first pass. Tito Omburo (talk) 23:32, 22 July 2025 (UTC)

The Springer encyclopedia page is written by Cullinane himself; I think it may be reliable as a mathematical source, but I'm really worried about due weight in a notability sense, so I'm really asking about the Spanish sources and the IEEE one. Sesquilinear (talk) 00:09, 23 July 2025 (UTC)

It's probably not notable given the lack of references I can find when searching, but it is interesting. I'm glad Cullinane published it on his website. It would be better still if he tried to unpack the explanation a bit further. I don't think the mention at https://www.revistaunion.org/index.php/UNION/article/view/1608 is enough to establish notability of this topic. –jacobolus (t) 00:26, 23 July 2025 (UTC)

There are four equations in the logo of the Georgian Mathematical Union. One of them is the Helmholtz equation, and I'd be interested to know if any of the other three are well-known and/or named. Perhaps one of you has an idea? GanzKnusper (talk) 15:25, 23 July 2025 (UTC)

The equation ${\displaystyle \partial _{\bar {z}}w+Aw=F-Bz}$ is a propagation equation for sections of a holomorphic vector bundle equipped with a connection, with an external potential B, and forcing term F. This seems like the obvious interpretation, but it's not necessarily uniquely determined. For instance, it could plausibly be the equation of a Higgs field (with an external coupling). Tito Omburo (talk) 17:22, 23 July 2025 (UTC)

Since Nov 2005, Closed formula redirects to Closed-form expression, which shows a corresponding hat note. However, the target article doesn't mention "closed formula" anywhere in the body, so there isn't even a chance to place a {{citation needed}} tag for it. For this reason, and since I only know that notion from logic, I'd like to retarget Closed formula to Sentence (mathematical logic). Is there any objection? Merging Sentence (mathematical logic) into Closed-form expression came to my mind, but seems inappropriate, since the latter doesn't mention bound variables (are they allowed in a closed-form solution?). - Jochen Burghardt (talk) 15:13, 26 July 2025 (UTC)

Sentences and expressions are completely different things. Sentences encode propositions, whereas expressions define functions. --Trovatore (talk) 02:18, 27 July 2025 (UTC)

To me, "closed formula" is just a nominal form of the adjectival phrase "closed-form"; I would not expect it to appear necessarily in the text. Personally I think the current situation (most efficient for a non-specialist reader who uses a variant phrasing, slightly less efficient for people well up on logical jargon) is mildly preferable to the alternative, but I wouldn't pick a fight about it. --JBL (talk) 23:51, 26 July 2025 (UTC)

To me, "closed form" is about a format that is described as "closed", which makes it not at all synonymous with "closed formula", whatever that is. @Jochen Burghardt if you have a reasonable definition for the latter, feel free to change the redirect. —Quantling (talk | contribs) 00:52, 27 July 2025 (UTC)

A merger has been proposed for Plethystic logarithm and Plethystic exponential, but it's been up for a few months and received no input. The discussion is at Talk:Plethystic logarithm#Feedback from New Page Review process if anyone wants to weigh in. Thebiguglyalien (talk) 🛸 01:36, 26 July 2025 (UTC)

Be bold! Johnjbarton (talk) 02:16, 26 July 2025 (UTC)

I don't know whether these should be merged or how the content would be arranged. Or what plethystic means, for that matter. Thebiguglyalien (talk) 🛸 02:20, 26 July 2025 (UTC)

I've carried out the merge. --JBL (talk) 18:11, 27 July 2025 (UTC)

As far as I can tell. There are two relatively common meanings of the term expression used in mathematics, something like:

1. Any syntactically valid chunk of mathematical notation, possibly including examples like ⁠${\displaystyle x^{n}}$⁠, ⁠${\displaystyle {\tfrac {3+2}{7}}}$⁠, ⁠${\displaystyle \textstyle \int _{0}^{2}x^{2}\,dx}$⁠, ⁠${\displaystyle 3\nmid 5}$⁠, ⁠${\displaystyle x=4y+2}$⁠, ⁠${\displaystyle f:X\to Y}$⁠, and ⁠${\displaystyle \textstyle \{z\in \mathbb {C} \mid z^{2}=1\}}$⁠.
2. A chunk of mathematical notation of a specific type – consisting of number symbols, variable symbols, symbols for operations, and the like, such that it can be evaluated and represents a single mathematical object. Under this meaning, the examples ⁠${\displaystyle x^{n}}$⁠ and ⁠${\displaystyle {\tfrac {3+2}{7}}}$⁠, and ⁠${\displaystyle \textstyle \int _{0}^{2}x^{2}\,dx}$⁠ from above are included, but mathematical statements or propositions are excluded, meaning that examples like ⁠${\displaystyle x=4y+2}$⁠ and ⁠${\displaystyle 3\nmid 5}$⁠ are not considered expressions. I'm not entirely clear whether ⁠${\displaystyle f:X\to Y}$⁠ or ⁠${\displaystyle \textstyle \{z\in \mathbb {C} \mid z^{2}=1\}}$⁠ would count as expressions under this definition, but I suspect generally not.

@Farkle Griffen wants to focus the article Expression (mathematics) on #2, and recently made a hatnote pointing to Formula § In mathematics in case people are thinking of #1. I found the text of the hatnote to be confusing, and don't really think it helped readers understand the difference, so I removed the hatnote and started a talk page discussion. But I'm not sure we're making much progress without broader feedback, and I'm not sure quite what the best approach is, so I wanted to ask folks here before trying to make any concrete proposals.

Some possible approaches might be: (a) focus on topic #2, but try to mention very clearly in the lead section that #1 can be routinely found in use; (b) make topic #1 the main definition in the article, but mention topic #2, and discuss material pertaining to #2 in one or more subsections of the article; (c) cover both topics in the article on relatively equal footing; (d) split the article into two, with prominent cross-links in the lead of each; (e) merge the content of the current article into mathematical notation and try to describe the possible meanings of this and related terminology there in a unified way; (f) ???

I don't have a super strong preference, and am not trying to advocate for any of those possibilities, but I want to make sure we try to give readers a clear and accurate idea of what these terms commonly mean. What do other folks think?

(Aside: @Farkle Griffen: hopefully the above seems like a fair characterization?) –jacobolus (t) 22:40, 27 July 2025 (UTC)

In my view, statements involving "=" or "${\displaystyle \implies }$" are also expressions. For example ${\displaystyle 1=0}$ is an expression. I don't think anyone could disagree with this. Tito Omburo (talk) 22:54, 27 July 2025 (UTC)

I'm not disagreeing that they are "expressions". The english term "expression" means essentially "any syntactically valid chunk of language". E.g:

The Berry paradox is a self-referential paradox arising from an expression like "The smallest positive integer not definable in under sixty letters" (a phrase with fifty-seven letters).

But the mathematical term "expression" has taken on a very specific meaning. E.g:

An equation is written as two expressions, connected by an equals sign ("=").

If things like ${\displaystyle (3\geq 2)}$ are expressions, then ${\displaystyle (x\geq 2)=(f:X\mapsto Y)}$ is an equation.

All I'm saying is the article Expression (mathematics) should be about the mathematical term, not the english one. – Farkle Griffen (talk) 23:40, 27 July 2025 (UTC)

There are definitely some folks who define things otherwise, along the lines of this simplified definition for high school students: "An expression is a mathematical phrase that does not contain an equals or inequality sign", or this definition from the VNR Concise Encyclopedia of Mathematics:

"An expression is a combination of numerical symbols, numerical variables, and (meaningful) juxtapositions of such symbols with operational symbols and brackets, for example: ⁠${\displaystyle 1/4}$⁠; ⁠${\displaystyle 12-5}$⁠; ⁠${\displaystyle 3\cdot a}$⁠; ⁠${\displaystyle 5\cdot (17+60)}$⁠; ⁠${\displaystyle (2z-13):(5z+10)}$⁠. However, the combinations ⁠${\displaystyle 5:0}$⁠ or ⁠${\displaystyle 7+\cdot a(}$⁠ are not expressions; and ⁠${\displaystyle 7+8=15}$⁠ or ⁠${\displaystyle a-3<3a}$⁠ are not expressions but propositions.

–jacobolus (t) 02:04, 28 July 2025 (UTC)

This seems fair.

My biggest issue is there needs to exist an article about "a sequence of symbols that represents a mathematical object" (which is what the article curretly is). I'd rather change the title of the article (a possible option f) than options (b)-(e). – Farkle Griffen (talk) 23:27, 27 July 2025 (UTC)

Are you working from the point of view of formal languages or mathematical practice? Because in practice, expressions are often two-dimensional arrangements of markings (some of which might not be called symbols) rather than sequences of symbols. You could linearize them using TeX but the TeX markup is not the expression in the same way that the expression is not the mathematical object that it represents. On the other hand, from the formal language point of view, determining whether a formula represents a valid object is problematic and could easily become undecidable depending on which formalisms you allow. —David Eppstein (talk) 23:41, 27 July 2025 (UTC)

Either? Both? Neither? You could say a "natural extension" of the formal language definition, where syntax doesn't matter quite as much as the semantics. But also, whether a computer can determine if a given expression is valid isn't that relevant to the definition... It's an interesting question, but the definition itself is well-defined and easy to use. – Farkle Griffen (talk) 00:20, 28 July 2025 (UTC)

[citation needed]. Instead of undecidable, I could equally well have said independent of ZFC (or your favorite axiomatization, or of any axiomatization). For instance, if BB denotes the busy beaver function, is BB(600) well-defined? Its value cannot be determined in ZFC . —David Eppstein (talk) 00:54, 28 July 2025 (UTC)

I don't really see how it's relevant? The discussion is about whether equations/formulas should be part of the article. It's not hard to show that BB(600) is well-formed by the definition in the article. How to know if a well-formed expression is well-defined isn't really part of the discussion. – Farkle Griffen (talk) 01:09, 28 July 2025 (UTC)

It's syntactically well formed, but you appear to want the article to be about expressions that have a well-defined semantics. Is that true for BB(600)?

And even if you're concerned purely with syntax, if your syntax incorporates some reasonable level of type-checking (to disallow invalid expressions like 1.5 + {{},{{}}}), in a sufficiently expressive type theory, you're likely to run into the same problem of undecidability/independence. See e.g. [hdl.handle.net/1721.1/149683]. —David Eppstein (talk) 01:34, 28 July 2025 (UTC)

No, as below, I'm advocating for a definition of expression that is, essentially, just the usual definition of well-formed expression. Well-formed, ill-defined expressions would still be expressions, like ${\displaystyle 0\div 0}$ or ${\displaystyle 0^{0}}$ even if left undefined.

Type checking is an even stricter definition than what I'm advocating for. Similar to ${\displaystyle 0\div 0}$ I have no problem saying 1.5 + {{},{{}}} is a well-formed, ill-defined expression. – Farkle Griffen (talk) 02:00, 28 July 2025 (UTC)

I just loosened the language at Expression (mathematics) § Formal definition slightly. Essentially what's there currently is what I'm advocating for. – Farkle Griffen (talk) 00:30, 28 July 2025 (UTC)

Does that mean that propositions are expressions? I would expect, e.g., ${\displaystyle \{(x,y):x<y\},}$ to qualify as an expression. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 09:29, 28 July 2025 (UTC)

That would, but propositions would not. Essentially any symbol (or sequence of symbols) that "represents an object" is an expression. Only formulas/propositions would not. – Farkle Griffen (talk) 14:33, 28 July 2025 (UTC)

I don't think set-builder notation usually qualifies as an "expression" under the more restrictive kind of definition, because it cannot be combined with other expressions (but such "expressions", in general, can be so combined). For instance, something like ⁠${\displaystyle \textstyle (2,5z)\cdot {\bigl \{}(x,y)\mid x<y{\bigr \}}}$⁠ is not ordinarily considered syntactically valid. –jacobolus (t) 23:49, 28 July 2025 (UTC)

Maybe I'm misunderstanding your comment, but you would use set-theoretic operations on sets. So ${\displaystyle \{(x,y)|x<y\}\cup \{(x,y)|x=y\}}$ is both well-formed and well-defined.

What you wrote would be well-formed (i.e. , one could define a product between a point in ${\displaystyle \mathbb {R} ^{2}}$ with an arbitrary subset in ${\displaystyle \mathbb {R} ^{2}}$)

but semantically meaningless (undefined). – Farkle Griffen (talk) 00:08, 29 July 2025 (UTC)

Maybe you are right. Is there a reliable source, about mathematical notation in general (rather than a more niche definition used on a narrow context), which (either explicitly or implicitly) describes the concept of "expression" as allowing arbitrary sets specified by whatever method but disallows equations? The sources I've been able to find which exclude equations are all aimed at a very elementary audience and list off some possible notation that can be used in "expressions" which represent single numbers, and don't consider more complicated types of objects or situations. –jacobolus (t) 00:23, 29 July 2025 (UTC)

The word "expression" appears in articles about elementary mathematics. These include equation, factorization, cancelling out and completing the square, all of which link to Expression (mathematics). They need an article which is reasonably easy to understand. It would be helpful if it included an explanation of what is meant by an "equivalent expression". "${\displaystyle 8x-5\geq 3}$" can be an expression because it can be evaluated as true or false, but I think it may confuse some readers and so it should not be in the article lead. The article should be focussed on topic #2. JonH (talk) 10:41, 28 July 2025 (UTC)

They need an article which is reasonably easy to understand. A different article, or an article that covers both usages, but it would be wrong to have an article with the title Expression (mathematics) exclude, e.g., ⁠${\displaystyle \textstyle \{z\in \mathbb {C} \mid z^{2}=1\}}$⁠. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 11:31, 28 July 2025 (UTC)

The lead and first part of the article should be reasonably easy to understand. It could go on to say that the value of an expression is not limited to being a number, and could be a logical value, a set, etc, with links to relevant articles. I agree it would be wrong to exclude these. JonH (talk) 12:18, 28 July 2025 (UTC)

Afaik, mathematical logicians distinguish an expression (having the type of a mathematical object) from a formula (having the type Boolean), e.g. Hermes 1973 "Introduction to Mathematical Logic". That is, "8x-5" is an expression as it evaluates (after x is appropriately instantiated) to some number, while "8x-5 > 3 and x > 0" then evaluates to true or false. Thus, they use notion #2.

There is a smooth transition between expressions and formulas when e.g. Boolean algebra is considered (e.g. "for all p,q: p and q iff q and p" would still be considered a formula, but its part "p and q" would be considered an expression); I'd suggest to mention that in a footnote. (The expression / formula distinction is even more blurred when considering boolean rings as a part of ring theory.)

When we consider operators that take a formula as an input and result in an object, like set comprehension, we may have a formula, like "x<y", as part of an expression, like "{(x,y): x<y}". This is not allowed by Hermes, but doesn't cause extra problems, afaik.

As for syntatic vs. semantic well-definedness, there is a smooth transition as well. Expressions like "${\displaystyle 1.5+\{\}}$" can be excluded by many-sorted logic, declaring ${\displaystyle (+):\mathbb {R} \to \mathbb {R} }$. As another example, "${\displaystyle 0\div 0}$" may be excluded using order-sorted logic, declaring ${\displaystyle \mathbb {R} _{\text{nonzero}}\subset \mathbb {R} }$ and ${\displaystyle (\div ):\mathbb {R} \times \mathbb {R} _{\text{nonzero}}\to \mathbb {R} }$. Type systems come in various degrees of sophistication, with the highest-degree versions (like Martin-Löf type theory) being undecidable. Imo, "syntatic well-definedness" should be something that is decidable within reasonable complexity. I'm not sure whether well-definedness should be discussed at all in the article. - Jochen Burghardt (talk) 12:10, 28 July 2025 (UTC)

Perhaps we can leverage the expression (computer science) article. There's a definition of expression there. Also, I might add to that the common use of "expression" in a computer language's parse tree (which I have abstracted and simplified here):

E = T | E + T | E − T

T = F | T * F | T / F

F = variable | number | '(' E ')'

...

where E = expression, T = term, F = factor and the '|' symbol separates alternatives.

—Quantling (talk | contribs) 15:03, 28 July 2025 (UTC)

The page on Infinite groups is currently a stub, and only lists a few examples of infinite groups. Is there any broader coverage on the subject that could be included, like properties that hold generally for infinite groups, or comparing different kinds of infinite groups? ALittleClass (talk) 02:21, 29 July 2025 (UTC)

The article Group (mathematics) actually says more about infinite groups than Infinite group does. The Numbers and Lie groups sections have a lot of prose about them, for starters. Stepwise Continuous Dysfunction (talk) 02:37, 29 July 2025 (UTC)

Based on that, a better solution would be to delete that article altogether. PatrickR2 (talk) 18:55, 29 July 2025 (UTC)

I think the existence of Gromov's "Asymptotic invariants of infinite groups" may caution against this. His basic idea is to study infinite groups metrically, using the word metric. Whereas, "locally" an infinite group has no interesting structure, asymptotically they have non-trivial structure, including a Carnot-Caratheodory metric on the asymptotic cone. Tito Omburo (talk) 20:10, 29 July 2025 (UTC)