Wikipedia talk:WikiProject Mathematics/Archive/2024/Apr

How to deal with this example/proof heavy article? (Cauchy's theorem)

I was asked to edit the article on Cauchy's theorem (group theory). There are two example sections which are earmarked for lack of citations. But when I read them, they do not seem particularly appropriate for the encyclopedia format at all. Putting aside typoes and poor grammar for a moment, the examples are both presented like problem book exercises, and could be shortened to a one-sentence description, if they are even interesting enough to warrant inclusion at all.

I'm also surprised that the article includes not one, but two entire proofs of the theorem. There must surely be textbooks online with complete proofs of their own. The style manual suggests "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." - by this metric I would vote to remove proof 1 entirely, and heavily abbreviate proof 2.

I have ideas for content that could be added, but I thought before removing 50% of the article I should get a second opinion. Danielittlewood (talk) 21:02, 1 April 2024 (UTC)

I also am a fan of proof 2. As far as a citation for this proof goes: it is an exercise in Isaacs _Finite Group Theory_. The original proof is by J McKay in a 1959 article in the Amer. Math. Monthly. Russ Woodroofe (talk) 21:23, 1 April 2024 (UTC)

Thank you very much for that reference. I managed to find the exact reference (although I can't access it, I'll take your word for its content). I'll add that to the article.

https://www.jstor.org/stable/2310010 Danielittlewood (talk) 21:07, 2 April 2024 (UTC)

A few examples showing some basic consequences of a theorem generally seems like a good idea to include. It would be nice to show some picture with these. Any wikipedia article explicitly about a theorem should include at least one proof if it is not inordinately long, or a proof sketch if all of the proofs are extremely cumbersome. If there are multiple proofs with substantially different ideas, then including more than one proof is nice. The style guide you are quoting is discussing the use of proofs in articles that aren't explicitly about particular theorems. In an arbitrary mathematical article (something like Circle or Matrix (mathematics) or Differential calculus or Real number), including a proof of every statement would be a distraction. –jacobolus (t) 22:43, 1 April 2024 (UTC)

In this case i completely agree with removing (at least most of) the examples. The first one has nothing to do with Cauchy's theorem: it should be removed. The second one uses it (in a confusingly oblique way) while usually the result is deduced from Lagrange's theorem; it can conceivably remain if it is rewritten correctly. jraimbau (talk) 05:06, 2 April 2024 (UTC)

I'm certain there are better applications. I think the ones in the article currently can be abbreviated to a single sentence while losing nothing of value. I'll try to find some better examples, ideally that lead to proofs of deeper theorems. Danielittlewood (talk) 21:08, 2 April 2024 (UTC)

I think both proofs are good to include, since they're very short and the page is about the theorem. To my personal preference the second proof could be written a little condensed, something like this:

Given any group G, the cyclic group Z_p acts on the set of tuples (g₁,...,g_p) in G with g₁...g_p = e, by cyclic permutation of the elements. If p is prime and G is finite, it follows from the orbit-stabilizer theorem that each orbit of the action has size either 1 or p. Orbits of size one are in natural bijection with group elements g such that g^p = e. If there is no such element other than e, it follows that the cardinality of the set of tuples is not divisible by p (since it is equivalent to 1 modulo p); since it can be checked that the cardinality of the set is divisible by the order of G, it follows that the order of G is not divisible by p. So if the order of G is divisible by p, then there must exist a non-identity element g with g^p = e.

(with obvious room for improvement, and provided there's a reference given with details). If there's a similar way to slightly reduce the first proof, I think that would be ideal. But I also think that both as currently written are acceptable in terms of detail. Gumshoe2 (talk) 23:37, 1 April 2024 (UTC)

Inner measure

The inner measure article had an unsourced and incorrect definition from almost 14 years ago, which I have now removed. It leaves the article pretty spare. Thoughts on what to do are invited on the talk page. --Trovatore (talk) 05:50, 3 April 2024 (UTC)

Original research on Wikipedia

Hello, I am a mathematician from the German Wikipedia. There we had recently a user that basically "misused" the German Wikipedia to publish his own "research" (if you can call it even that...). Basically the user computed a LOT of things with Wolfram Alpha and published all his computations in the German Wikipedia to a point where the articles became unreadable. He even invented his own names for functions and the user - according to his own words - does not have a formal degree in mathematics. In my opinion most of the stuff was not even relevant for an encylopedia. In the end a lot of his entries were deleted and after a heated discussion the user got banned. Long story short the user was/is also active in the English Wikipedia (see Special:Contributions/Reformbenediktiner). I am not so familliar with the English Wikipedia policies but I know that original research is also not allowed, so I thought I should maybe notify people here and they could at least have a look at some of the affected articles like for example Theta function, Rogers–Ramanujan continued fraction, Fubini's theorem#Example Application, Jacobi elliptic functions, Rogers–Ramanujan identities etc. If you see some math in color, that was probably done by this user. In the German Wikipedia the user did not use any source material and just computed things with Wolfram Alpha. Whether it all was correct or not, I am not even sure. It would be good if people would have a look at the affected English articles as well and give their judgement.--Tensorproduct (talk) 19:42, 22 February 2024 (UTC)

FYI: User:Reformbenediktiner. PatrickR2 (talk) 19:48, 22 February 2024 (UTC)

Thanks for this report. I just removed a lot of this from Poisson summation formula (two long and almost entirely unsourced sections). Probably the others listed above and the contributions of this user need similar scrutiny. —David Eppstein (talk) 20:28, 22 February 2024 (UTC)

Yes, unfortunately every post by him needs scrutiny. In the German Wikipedia eventually almost all of his math edits were removed. Many users asked him many times to provide sources but he kept on editing without providing any source. It seems to be the same here as Jacobolus' example below shows--Tensorproduct (talk) 21:18, 22 February 2024 (UTC)

Example discussion: Talk:Lemniscate_elliptic_functions#Sources? –jacobolus (t) 20:49, 22 February 2024 (UTC)

Thanks for letting us know. Bubba73 ^{You talkin' to me?} 21:03, 22 February 2024 (UTC)

Here was another recent discussion: Wikipedia_talk:WikiProject_Mathematics/Archive/2023/Jul#Theta_function. --JBL (talk) 18:58, 2 March 2024 (UTC)

Huh. I spotted the stuff at theta function, and scratched my head a bit about it. I would be happier if much or most of this was removed, or maybe moved to a distinct article. Many of the relations are cool-looking! Yes, it is not uncommon for stuff similar to this to be published in journals. However, the cutting edge academic journals & books will say things similar this in the intro: In 1837, Kummer listed three identities for hypergeometric functions; this was extended to 50 by 1880, and 240 in 1920 and a general algorithm to generate a countable number of such identities was given in 1960. However, it did not list all of them, and neither did algorithms x,y,z proposed in 1980 and in this paper we explore the structure of algorithmic generators ... and so you realize these guys are talking about a kind-of fractal splattered all through this landscape of inter-related identities, and how to best understand/describe that fractal. (As far as I know, there aren't any articles on WP that even scratch the surface of this topic, and it would be cool if there were... but, whatever.) The problem is that the enthusiastic amateur is unaware that he's dong the algebraic equivalent of publishing cool-looking zooms of the Mandelbrot set. Yes, its still cool looking. But is not where the action is, and it is a clutter and distracting, if you were reading the article to find something else, e.g. look up some factoid about riemann surfaces, and that factoid is now buried in reams of wild identities. 67.198.37.16 (talk) 08:13, 4 March 2024 (UTC)

It's up to you guys if you want to check every edit of him, whether it is legit or not (like we did in the German Wikipedia), or you want to save time and just remove them. For me is "computing stuff with Wolfram Alpha and not adding to the mathematical theory" not mathematics and hence not relevant for an encylopedia.--Tensorproduct (talk) 22:05, 8 March 2024 (UTC)

I literally just came across a few edits by that user at the article Jacobi elliptic functions while looking for articles to translate into Spanish. Unfortunately that user's article edits go back to 2022, and I can't justify reverting that far back since I can't integrate other people's edits while deleting that user's edits. I can confirm that a good chunk of their vocabulary is gibberish from at least 19 March 2024, but because the topic is somewhat beyond me, I can't even confirm the tremendous amount of notational modifications that user also made since 2022. I do suspect that reverting all of that user's edits is the right move. JuanTutors (talk) 00:21, 26 March 2024 (UTC)

Wow, yeah. Vast stretches of Jacobi elliptic functions and Theta function are garish, under- or unreliably sourced, and basically impenetrable. There's no way to tell what is important and what is just a formula included for the sake of having a formula. XOR'easter (talk) 01:35, 2 April 2024 (UTC)

I'd go so far as to say that Theta function should be reverted to the version of 14 April 2022. XOR'easter (talk) 22:34, 3 April 2024 (UTC)

I agree with this. It seems that there were some intervening good edits, but it seems like the simpler approach would be to merge those in manually. Tito Omburo (talk) 10:45, 4 April 2024 (UTC)

I can hit revert myself, but I'd have to clear time/gather energy to do manual merging of any intervening good edits. Perhaps some of the additions to these various pages could be saved in a List of identities for the such-and-such function kind of article (like Exact trigonometric values and List of trigonometric identities). But we'd need better grounds for preserving such material than "Wolfram Alpha says so" or "one random preprint on ResearchGate includes it". XOR'easter (talk) 18:11, 4 April 2024 (UTC)

I've removed a section of geometric series by this editor that was obvious offtopic original research. I looked at their edits to Fubini's theorem, which I consolidated into Fubini's theorem#Calculation examples and I don't feel very strongly about it (although it badly needs edits for style). Tito Omburo (talk) 10:39, 4 April 2024 (UTC)

Note: It seems like @A1E6: has had a lot of interactions with this editor in the past, and presumably could weigh in. Tito Omburo (talk) 10:55, 4 April 2024 (UTC)

Some of his work exceeds the limits of what WolframAlpha can do. But what matters is that he often harms the Wikipedia project by adding his original research that is clearly beyond WP:CALC and not only that; the research is sometimes hardly notable/interesting from a mathematical standpoint. I've had a conversation with him on several occasions; you can check out my Talk page if you want.

When I first started editing Wikipedia, I had a similar mindset like Reformbenediktiner – but that changed (a long time ago) when I understood what this project is about.

I'm not active on Wikipedia anymore though; otherwise I would have already done all the "dirty work" myself – like deleting some of his contributions (or adding warnings about original reseach for readers) and discussing on the Talk pages – I'm familiar with all the mathematical articles that he edits.

But don't get me wrong – some of his contributions are good... A1E6 (talk) 12:44, 5 April 2024 (UTC)

This is an area that I also have some familiarity with, but I am not a great expert. If I have time, I will try to clean up some of these articles and keep you posted. Tito Omburo (talk) 19:11, 6 April 2024 (UTC)

Copy-pasting proofs in articles

I just deleted some technical proofs in the Coppersmith's attack article, because they were pretty much copy-pasted from some of the sources with a few words changed. Given the article is not about most of the proofs, they're probably better deleted anyways, but I was wondering what the policy was on proofs in articles and how similar they're allowed to be, because obviously Wikipedia can't have its own proof of every subject, but I think it's a copyright violation if you copy-paste (even with a few words changed) the exact wording. Does anyone know if that's correct? I looked for any policy pages but could not find them. Mrfoogles (talk) 00:56, 11 April 2024 (UTC)

Sounds about right. Review Wikipedia:COPYVIO. Perhaps if you spent a lot of time and determined that the sources were CC-by-SA, and so they could be copied, the question still remains if mathematical proofs in articles adds any value. Usually, they don't. See Category:Article proofs and Category:Articles containing proofs and Wikipedia:WikiProject Mathematics/Proofs. My knee-jerk reaction is you're dealing with a form of spam from a novice editor with good intentions but lacking experience. On closer inspection, that article was created whole, including the proofs, in 2011, by an editor who created two WP articles in two days, and never-ever edited WP ever again. The intricacy of detail suggests that the author is copying directly from their own thesis, i.e. they are probably copying their own work. Probably. But since they're anon, can't quite tell, and since they're not active, can't ask them.

One more comment about proofs. If they are added by a clear subject-matter expert (as is the case here) then they're OK (because likelihood of correctness is high, and the maintenance burden is low.) When they are added by a college sophomore studying for a mid-term exam, then they must be exterminated with prejudice. If they clutter the article, they can be wrapped with one of the auto-expander click-thru box templates, so that they don't take up space when not expanded. 67.198.37.16 (talk) 02:46, 11 April 2024 (UTC)

Please see Wikipedia:Expert editors:

• "Wikipedia does not grant additional powers or respect to subject-matter experts."

Who added content or under what circumstances is not relevant. Johnjbarton (talk) 16:22, 11 April 2024 (UTC)

True. And it is also the case that a contribution from a subject-matter expert is more likely to be correct than one from a "college sophomore studying for a midterm exam". :-) PatrickR2 (talk) 23:19, 11 April 2024 (UTC)

Hodge conjecture

Hello, I think there was some vandalism on the Hodge conjecture. I tried to revert to what looked to be last good version of the article. However, I am not sure given the maths in the article. If someone can take a look at the diffs and content of the article that would be much appreciated! Classicwiki (talk) If you reply here, please ping me. 22:03, 3 April 2024 (UTC)

Current version looks OK. It's identical to [1] Tito Omburo (talk) 22:12, 3 April 2024 (UTC)

@Tito Omburo, thanks for reviewing! Classicwiki (talk) If you reply here, please ping me. 22:13, 3 April 2024 (UTC)

@Classicwiki and Tito Omburo: I notice that the article was edited heavily by Darcourse in 2023 and 2022; this editor is singularly incompetent in my opinion (though not a vandal), so if you're checking what the last good revision is, you might inspect their edits, too. (The I looked a little bit and wasn't convinced one way or the other.) --JBL (talk) 18:20, 4 April 2024 (UTC)

I'm not really qualified to judge most of the article for mathematical accuracy, but I would say that on a cursory reading it seems ok. Tito Omburo (talk) 19:01, 4 April 2024 (UTC)

@JayBeeEll - unfortunately I am in the same boat and can not judge the mathematical accuracy, so it is tough for me to determine if the edits are appropriate. Classicwiki (talk) If you reply here, please ping me. 22:09, 4 April 2024 (UTC)

Ok, no worries -- thanks both. --JBL (talk) 17:45, 12 April 2024 (UTC)

Featured article review for 0.999...

I have nominated 0.999... for a featured article review here. Please join the discussion on whether this article meets the featured article criteria. Articles are typically reviewed for two weeks. If substantial concerns are not addressed during the review period, the article will be moved to the Featured Article Removal Candidates list for a further period, where editors may declare "Keep" or "Delist" in regards to the article's featured status. The instructions for the review process are here. voorts (talk/contributions) 20:58, 11 April 2024 (UTC)

On the subject of FAR, would anyone like to try fixing the mild under-citation at Emmy Noether? XOR'easter (talk) 14:14, 13 April 2024 (UTC)

Possible duplicate logic articles

Does it really make sense for "Logical connective", "Boolean function", and "Truth function" to all be separate articles? If I were more sure, I wouldn't be asking. I get how "Logic gate" is a separate article, but the other ones seem to cover the same territory, although I'm not sure which one(s) should be merged into which. Thiagovscoelho (talk) 12:21, 17 April 2024 (UTC)

Hi @Thiagovscoelho: These are distinct topics. Logical connectives are used to connect (logical) expressions in first-order logic, second-order logic and set theory in general, to express axioms, theorems, inference rules, etc. By contrast, Boolean functions are functions (things having input and output) that operate on elements of a boolean algebra. These may be finite, countable or uncountable sets. See Stone representation theorem for details. Note that the elements of a boolean algebra are NOT logical expressions! The truth function article deals with the most limited, narrow case, where "truth" is taken to be a single value T or F. A single letter, a single bit. This is distinct from elements of a boolean algebra, which can be large complex things, and it is also distinct from logical connectives, which apply to the text strings of a term algebra or a model theory. So, very distinct concepts which magically happen to have the same notation. Which, yes, can lead to confusion. Perhaps the ledes of these articles should be amended to clarify this, state this up front. 67.198.37.16 (talk) 20:46, 19 April 2024 (UTC)

(p.s. The word "magic" is fun. Formally, it is called the semantic/syntactic distinction, and there are a collection of theorems from the 1930's that clarify this relationship. Turing's incompleteness theorem is perhaps the most famous; there are others. e.g. Skolem-Lowenheim upward/downward, the completeness theorem, and the assorted variants of it from Godel, Post, Gentzen, Bernays, Kleene.) — Preceding unsigned comment added by 67.198.37.16 (talk) 21:03, 19 April 2024 (UTC)

Hi @Thiagovscoelho: I noticed that you just went through a major, massive rewrite of the very long article on propositional calculus. Are you sure that this is a good idea, given the confusion you expressed above? The old version of the article seemed to get to the point, right after the first two introductory paragraphs; the new version seems to take some tortuous detour, before starting to explain what it is half-way into the article. I cannot help reviewing this, but perhaps more wisdom and fewer facts would help. 67.198.37.16 (talk) 22:27, 19 April 2024 (UTC)

I've replaced disorganized sections that did not cite sources with organized sections that do cite sources, and I'm doing this by reading all the sources. The sources are naturally textbooks on logic, and they mostly only mention connectives, which, semantically, are only defined by means of their associated truth functions, whereas syntactically they are of course not properly "defined" at all, but may have their behavior described by inference rules prescribing their introduction or elimination. As it stands, the Logical connective article has no coverage of introduction/elimination inference rules, so its coverage overlaps a lot with what Truth function ought to cover. As to Propositional calculus, the old version "got to the point" by failing to define terms that are defined in all the sources, introducing notation without explaining what it means, failing to keep syntax and semantics clearly distinct, failing to distinguish between a formal language and the proof system used with it, and, most of all, not citing any sources at all and therefore not describing any of the variation between authors on the topics. You are welcome to edit it and improve it, but there is no Wikipedia standard by which the old article was better. Thiagovscoelho (talk) 23:05, 19 April 2024 (UTC)

Yes, the current version of Boolean function actually specifies "truth function" as an alternative name for it. If you are sure that there is such a sharp distinction, it would be good for you to edit the article and cite the Reliable Sources that you are familiar with for this statement. I have not read any of the literature that specifically refers to "Boolean functions", but the idea of such a sharp division surprises me, since George Boole was a logician, after all, and I mean, just look at the article, it features all the normal connectives from logic. Thiagovscoelho (talk) 23:27, 19 April 2024 (UTC)

My experience is that at least in computer science "Boolean function" is usually, but not invariably, used to mean functions that take values in the two-element Boolean Algebra. This is how the textbooks I currently have access to use the term. I don't currently have access to Rudeanu's classic on the subject, but I could check his terminology in the library next week if needed. At least Steinbach and Posthoff do also explicitly mention truth function as a common synonym. Felix QW (talk) 08:15, 20 April 2024 (UTC)

References

Accessibility of Newton's method

Can someone here take a look at the recent changes at Newton's method and discussion at talk:Newton's method, and maybe help resolve the edit war there? user:Fangong00 insists on a substantial rewrite, especially of the first few sections, which I think makes the article significantly worse, most importantly rendering it, in my opinion, almost completely illegible to most of the intended audience. They don't seem too interested in having a discussion about the trade-offs involved in of various possible choices of scope/focus for the article or its early sections, but I don't really want to spend all day revert warring. Maybe someone else can phrase concerns about this in a way that gets through? –jacobolus (t) 02:23, 21 April 2024 (UTC)

The previous page for Newton's method is outdated. That version only presents Newton-Raphson method. In today's numerical analysis, Newton's method most often means Simpson's extension and also include the Gauss-Newton iteration and the newly discovered the rank-r Newton's iteration. Furthermore, the crucial convergence theorems such as Kantorovich Theorem and alpha theory were not included.

Why does jacobolus insists on keeping the outdated version? When someone tries to look up Newton's method, he or she is entitled to see the what Newton's iteration is today. Fangong00 (talk) 02:35, 21 April 2024 (UTC)

@Fangong00 the basic real-function version of Newton's method is not "outdated", but is used ubiquitously, and anyone working with computer software involving numerical calculations is likely to come across it sooner or later. It is taught to early undergraduate students and frequently encountered by people with relatively limited pure math background. It is essential that a Wikipedia article about such a basic and widely used tool start out in its first few sections with explanation which is legible and accessible to the broadest possible audience. If you want to include detailed technical discussions of advanced niche generalizations, then that is fine, but it must be done much further down the page and clearly contextualized so that readers can figure out what is being discussed and why.

As a simple example, a computer game programmer with a high school level math background might plausibly read fast inverse square root and come across a wikilink there to Newton's method; if they click through they must not be confronted with a wall of jargon expecting several years of preparation they don't have. –jacobolus (t) 02:48, 21 April 2024 (UTC)

Raphson's method is not out dated, it is now a special case of what we call "Newton's method" in numerical analysis. The most widely used Newton's iteration is not Raphson's but Simpson's and Gauss-Newton. To make the page a useful reference for the broadest possible audience.

Why do you not want a vistor to see the most widely used Newton's method? Fangong00 (talk) 12:56, 21 April 2024 (UTC)

Fangong00, you are incorrect. The page in its previous state does have a section for systems of equations and for Banach spaces, where the Gauss-Newton iteration and (what you call in a nonstandard way) "Simpson's extension" belong. The rank-r Newton iteration seems to be from a 2023 paper and it is not at all clear that it is notable enough for mention. The Kantorovich theorem is also already mentioned there.

Smale's theorem is certainly appropriate for inclusion, and could go for example in the Analysis section. Gumshoe2 (talk) 02:54, 21 April 2024 (UTC)

I know the page mentioned Simpson's version serveral pages later. Users visiting the page is unlikely to see it when they thought only Raphson's method is Newton's iteration. Fangong00 (talk) 12:58, 21 April 2024 (UTC)

The Gauss-Newton method and the Banach space Newton method are also already mentioned further down the page in the previous version. I don't think there would be any objection to expanding the text in that context.

I also don't think there would be any objection to drawing attention to this by mentioning, perhaps in a couple sentences, in the lead section that there are important multidimensional extensions of Newton's method. The lead section, after all, is meant to be a brief summary of the article. (See Wikipedia:Manual of Style/Lead section; I think neither version of the article has a satisfactory lead section.) Gumshoe2 (talk) 13:17, 21 April 2024 (UTC)

I think this content should be restored. I have commented at Talk:Newton's method. Tito Omburo (talk) 12:18, 21 April 2024 (UTC)

Adjoint functor theorem

I think that people searching for "Adjoint functor theorem" are looking for explanations about the Freyd's adjoint functor theorem, so I suggest changing the redirect target to the Formal criteria for adjoint functors. SilverMatsu (talk) 15:35, 24 April 2024 (UTC)

Ths is a possibility. However, there is an anchor "Freyd's adjoint functor theorem" in Adjoint functors. I have changed the redirect for pointing to this anchor instead of to the lead. Note that Formal criteria for adjoint functors is linked to just above this anchor. I have no clear opinion about your proposed change of target, but, in any case, Freyd's adjoint functor theorem and Adjoint functor theorem must have the same target. D.Lazard (talk) 16:24, 24 April 2024 (UTC)

Thank you for clarifying the redirect target. By the way, there are two versions of Freyd's adjoint functor theorem, which are sometimes called General adjoint functor theorem and Special adjoint functor theorem. --SilverMatsu (talk) 00:19, 25 April 2024 (UTC)

Requested move at Talk:Basic Math (video game)#Requested move 24 April 2024

There is a requested move discussion at Talk:Basic Math (video game)#Requested move 24 April 2024 that may be of interest to members of this WikiProject. RodRabelo7 (talk) 05:33, 28 April 2024 (UTC)

Wigner probability distribution

It might be good to have some people watch Wigner semicircle distribution, with someone having just added back some extensive material I deleted a couple months ago. I think it's pretty incoherent, and not good material for the page regardless. Gumshoe2 (talk) 16:22, 28 April 2024 (UTC)

"Distinct" definition

The use of the word "distinct" , should be reviewed , so that its usage becomes clear, here are the pages I have noticed them in: Constructible polygon, ,Carl Friedrich Gauss ,Exact trigonometric values ,Constructible number

The constructible polygon page says : A regular n-gon can be constructed with compass and straightedge if and only if n is a power of 2 or the product of a power of 2 and any number of distinct Fermat primes.

Whereas , the Constructible number page says:

• powers of two
• Fermat primes, prime numbers that are one plus a power of two
• products of powers of two and any number of distinct Fermat primes.

Notice here the second bullet point is separate to the third ; is that to say that "any number of distinct Fermat primes" does not include one Fermat prime appearing on its own. And would zero Fermat primes be considered a distinct number of Fermat primes?. This should be specified. EuclidIncarnated (talk) 13:48, 28 April 2024 (UTC)

The formatting of the post above is difficult to read. As far as I can tell, the issue is more about "any number" than about "distinct". I think that it is best treated by editing those specific pages to address that specific issue. Mgnbar (talk) 14:03, 28 April 2024 (UTC)

Sorry about my bad formatting , I am relatively new to Wikipedia writing and thank you for bringing to my attention , "any number", which should be defined more clearly. I would say that so does "distinct". For example , consider one number is it distinct? or is there required a second number for it to be said to be distinct?. Such things should be made more clear. EuclidIncarnated (talk) 14:44, 28 April 2024 (UTC)

As an example, I have edited Constructible polygon#Conditions for constructibility. I did not clarify what "distinct" means, but I did clarify (some might say too explicitly) what "any number" means. What do you think of this solution? Does "distinct" still require clarification? Mgnbar (talk) 15:03, 28 April 2024 (UTC)

I changed the ·bullets to asterisks to make a proper list. —Tamfang (talk) 19:38, 28 April 2024 (UTC)

The last bullet point includes the first two bullet points as special cases. –jacobolus (t) 14:22, 28 April 2024 (UTC)

I don't see how the first bullet point is a special case of the last bullet point , could you explain what you mean? EuclidIncarnated (talk) 15:03, 28 April 2024 (UTC)

In the third bullet point, let 2^j be the power of 2 involved, and let k be the number of distinct Fermat primes involved. The first bullet point is the special case where k = 0. The second bullet point is the special case where k = 1 and j = 0. Mgnbar (talk) 15:44, 28 April 2024 (UTC)

Yes this is going off of the definition that the product of a number is itself and thus a power of 2's product is itself. This is what Product (mathematics) says is the definition of products : "Originally, a product was and is still the result of the multiplication of two or more numbers." Therefore your definition of product is not this. EuclidIncarnated (talk) 17:47, 28 April 2024 (UTC)

Sorry; I don't quite understand your post. No one here has defined the word "product", have they? The Wikipedia article Product (mathematics) is not a Wikipedia:Reliable source. Anyway, products and powers can take on slightly different meanings in different contexts. When stating a theorem, it is a good idea to make the intended meaning explicit and clear.

Have you seen my recent edit to Constructible polygon#Conditions for constructibility, which I mentioned above? Is it not clear? Regards, Mgnbar (talk) 18:01, 28 April 2024 (UTC)

It seems fine to me , your edit. EuclidIncarnated (talk) 18:11, 28 April 2024 (UTC)

@EuclidIncarnated Mathematicians define the "product" of any (possibly empty) collection of elements all belonging to some structure where multiplication is well-defined. An empty product is equal to the multiplicative identity, which is 1 in the case the quantities being multiplied are numbers. The "product" of a single quantity is just the quantity itself. –jacobolus (t) 00:09, 29 April 2024 (UTC)

All true, but this is a point we should be careful of when writing articles for non-mathematicians who may become confused by 0-element and 1-element products. —David Eppstein (talk) 00:29, 29 April 2024 (UTC)

Is there anywhere in Wikipedia that has such a definition. EuclidIncarnated (talk) 07:12, 29 April 2024 (UTC)

@EuclidIncarnated This is described in Product (mathematics) § Product of a sequence. While a sequence per se is an ordered list of numbers (or other quantities), if multiplication is commutative (true in many but not all contexts) the order doesn't matter and you could just as well take the product of an unordered collection like a multiset. –jacobolus (t) 07:15, 29 April 2024 (UTC)

And Product (mathematics)#Empty product is all about the case where 0 numbers are being multiplied. Mgnbar (talk) 11:59, 29 April 2024 (UTC)