Wikipedia talk:WikiProject Mathematics

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Hard Unknots?

I noticed in the Requested Articles page there is one for the Millett Unknot. Would it be acceptable to create a page on "Hard Unknots" featuring specific examples, and redirect the Millett page to that one? This would be a distinct page from the unknotting problem. I can put this together but I could use some help explaining why it's notable. Another option would be to add a section to the unknotting problem page about specific hard unknots. ProfKnots (talk) 17:37, 21 October 2025 (UTC)

I think either of those would be fine. I'd recommend you pick whichever you think seems best and boldly go for it, and if anyone disagrees they can start a discussion about changing to a different approach. –jacobolus (t) 19:11, 21 October 2025 (UTC)

An article on hard unknots would require having some sourcing that covers hard unknots as a class rather than merely studying individual examples, to avoid WP:OR. If that sourcing can be found, though, then go for it. —David Eppstein (talk) 23:17, 21 October 2025 (UTC)

A quick search turns up multiple papers with "hard unknot" in the title, so I think that should be doable. One of these has: "We say that a diagram of the unknot is hard if, in any sequence of Reidemeister moves to the trivial diagram, the crossing number has to first increase before it decreases. They are of particular interest because they might provide counterexamples to potential unknot detection algorithms. Hard unknot diagrams are difficult to construct, ..." ProfKnots: After having done this quick search, I would recommennd hard unknot diagram as the title, and I think this topic seems pretty clearly notable. –jacobolus (t) 02:03, 22 October 2025 (UTC)

Question about this, are images from arXiv papers allowed on Wikipeda (given that they often have a CC-esque license) or do they count as copyrighted? (copywritten?) ProfKnots (talk) 07:36, 22 October 2025 (UTC)

If the paper is released under a compatible license, then pulling the image out is probably fair game; if it's not clear, I'd contact the author. Sesquilinear (talk) 04:01, 30 October 2025 (UTC)

Anywho, I went ahead and made the page, still need to make a few tweaks. Happy to have help! ProfKnots (talk) 10:30, 22 October 2025 (UTC)

After a quick search, it looks like hard unknot diagram is a more common name then hard unknot. That is, it's the diagram which is "hard"; the knot is always the same here. But I'll leave it to folks who know about the topic to make decisions about their preferred nomenclature. –jacobolus (t) 04:14, 30 October 2025 (UTC)

Mean value theorem

A small thing, but on the article Mean value theorem, under the statement heading, the theorem is stated as :$f'(c)={\frac {f(b)-f(a)}{b-a}}.$. On the adjacent image illustrating the theorem, however, it shows the theorem as using xi instead of c. Obviously it means the exact same thing, as the argument for the derivative is just a dummy variable in this case. It does seem a little confusing for people trying to get the concept, and I even I as someone with prior understanding was given pause over it, so I think it would be best if they were the same. My question is, which do we change, the prose or the image? Which notation is standard. C does seem more streamlined by I was taught it as xi. Which do we think would be best? WikiMacaroons^Cinnamon? 22:52, 25 October 2025 (UTC)

The best place to discuss issues specific to the Mean value theorem article is Talk:Mean value theorem. Regards, Mgnbar (talk) 03:03, 26 October 2025 (UTC)

Looks like there is on the talk page a section proposing changing ξ to c. Gramix13 (talk) 04:58, 29 October 2025 (UTC)

Organization of symmetry categories

The organization of symmetry group categories is a mess. For example, Icosahedral symmetry is in Category:Rotational symmetry, which is in Category:Lie groups even though the icosahedral group is finite. It isn't in Category:Finite reflection groups (although a redirect to it is), which is probably redundant because all finite reflection groups with at least two generators also contain a rotation. –LaundryPizza03 (dc̄) 22:51, 30 October 2025 (UTC)

I was tempted to respond to this by giving my general opinions of the entire categorization project on Wikipedia, but I figure that's not what you're looking for; is there a query or request for assistance here? I agree that having the category for rotational symmetry being inside the category for Lie groups doesn't seem super sensible (rather, some particular articles like 3D rotation group and Special orthogonal group definitely belong in both categories). --JBL (talk) 17:36, 31 October 2025 (UTC)

I do think that removing "symmetry group" from the first sentence of this thread would leave a correct sentence. But I agree that the distinctions between finite/infinite or discrete/continuous are more fundamental than rotational/non-rotational. —David Eppstein (talk) 20:38, 31 October 2025 (UTC)

Restriction conjecture

I think Restriction conjecture and Fourier extension operator#Restriction conjecture explain the same topic, so I think Fourier extension operator#Restriction conjecture may be merged into Restriction conjecture.--SilverMatsu (talk) 02:31, 3 November 2025 (UTC)

DFT

Is Wikipedia really helpful for the audience about mathematics, particularly in Discrete Fourier transform? Have found this YouTube link this morning. [1] Dedhert.Jr (talk) 02:45, 4 November 2025 (UTC)

What are you asking? Are you asking whether our Discrete Fourier transform article is better than that YouTube video (which I haven't watched)? Are you suggesting that Wikipedia should no longer cover the DFT? Or are you complaining about the current state of the DFT article? I don't get it. Mgnbar (talk) 02:49, 4 November 2025 (UTC)

Should have said this explicitly. I was trying to say that the Wikipedia article is terrible at explaining math, according to YouTube. Caption of images, technical terms, etc. Dedhert.Jr (talk) 02:58, 4 November 2025 (UTC)

There was a discussion about this today on the discord, where we discussed how we could begin to restructure our articles on mathematics, so they cater to people of all education levels - right now it's not consistent, but most of them require you to have pretty advanced background knowledge, but the average person doesn't even have first-year calculus. We are not Wikiversity, so there's no need for us to take a teacherly tone, and go on overlong explanations of advanced concepts in order to make them accessible to the layperson, but I think it would be worth having each article build up from fundamentals to get to the concept - perhaps (though this is perhaps a little radical, no pun intended) even have this content be collapsible so mathematically inclined readers needn't wade through the stuff that's obvious to them to get to the concept the article is about. Vigilantcosmicpenguin likened it to the "Background" section in an article about a historical event, in that its content, while not strictly relevant to the article's subject, is relevant insofar as an understanding of it will greatly aid one's understanding of the concept the article is about. We also may need to consider recruiting non-mathematically inclined people to read the new restructured articles, to see if they find them more digestible. We may all disagree on exactly how much hand-holding we should be doing, but I think we can all agree that the maths articles are in a dire state due to their inaccessibility (in case you're not convinced, Dedhert.Jr linked a great video about it above ^) WikiMacaroons^Cinnamon? 22:57, 4 November 2025 (UTC)

@WikiMacaroons "I think we can all agree that the maths articles are in a dire state due to their inaccessibility" → Well, I disagree.

Obviously, many (if not most) of the math articles could be improved. But for many (if not most) of these, I don't think accessibility is the main issue.

Mathematics is a cumulative discipline: something that's been proved will always be true. As a result, much of the progress in mathematics consists in defining new objects and asking new questions — sure, it also includes simplifying proofs and unifying concepts, which is why the average college student of today is able to get to concepts that were once at the forefront of research. But overall we are adding things on top of one another.

As a result, (1) things can quickly get technical and abstract as one goes up the stack — more so than someone who hasn't really tried doing so will usually assume; and (2) to truly understand something, one needs to understand what's at its base.

But, of course, sometimes — in fact most of the time — we do not need to "truly understand" things; some vague idea will suffice. If the growth of the stack was motivated by some natural phenomenon or by some very concrete problem, it will usually be possible for the motivated layman to get such a "vague idea" about a topic. In that case, an effort should definitely be made to make part of the corresponding article — at the very least the lead — accessible. However, if the growth of the stack was motivated by the pure joy of making it grow (which I guess is the main motivation for many mathematicians)... then honestly none of it will make sense unless you start from the base and go up the levels one by one.

The reason why I am writing all this because, as a researcher in mathematics, I'm pretty used to the fact that there are some concepts/objects I simply cannot understand, even on a superficial level, because I don't have the prerequisites. The time I would need to invest to understand them depends on how abstract/technical and how far away from my field those topics are; and for many of them I would need to start by taking semester-long courses at the Master's level. That seems completely normal to me. So it is sometimes frustrating when people who (quote) "[don't] even have first-year calculus" complain about not understanding something: yeah, neither do I, and I don't think I should.

The bottom line is that while I think many math articles are terrible, I don't think there is a generic problem with accessibility; the reason why many people perceive such a problem might be that — like author of the video linked by Dedhert.Jr — they are (quote) "trying to learn math from Wikipedia". It's an encyclopedia, not a textbook.

Also: "I think it would be worth having each article build up from fundamentals to get to the concept" → with all due respect, I think this suggestion is too unspecific to even make sense. What should those "fundamentals" be? They can't be absolute (otherwise good luck starting from basic arithmetic and reaching master's-level topic); and if they are relative to the topic, then how is that different from what many articles already try to do?

PS: "in case you're not convinced, Dedhert.Jr linked a great video" → I watched (part of) that video and didn't find it great. It does a good job at pointing out a few absurdities in some articles (e.g, the figure in Average); but to me a lot of the complaints seemed a bit unjustified (perhaps even arrogant). For instance, in the initial rant about discrete Fourier transforms alone: (1) "you know the article is going to be good when the word appears in its own definition", well, it doesn't because "discrete Fourier transform" $\neq$ "discrete-time Fourier transform" — there is even a warning about that literally one line above that sentence (+ a link on "discrete-time Fourier transform"); (2) "the FFT algorithm is mentioned, even though we aren't told what FFT stands for until later in the article", yeah, but the name doesn't matter at this point, what matters is what it does, and the sentence kind of explains that (and if you really want to know what FFT stands for, you can just click on the link); and (3) "despite mentioning sinusoids several times [...]" well, someone unable to recognize a complex exponential as containing sinusoids may not have the required background to understand this topic — at least not from an encyclopedia article... Pretty much all the video was like this for me, so I stopped watching ~7 minutes of it.

Malparti (talk) 04:21, 5 November 2025 (UTC)

Well said. There is no royal road to mathematics! PatrickR2 (talk) 05:22, 5 November 2025 (UTC)

I don't think it's possible to learn mathematics just from reading an encyclopedia. I doubt it's possible to learn mathematics just from passively reading a textbook, either. The student has to get actively engaged and put pen/pencil to paper. A textbook gives the illusion of making it easier by putting the whole development in one place, whereas on Wikipedia, one has to click around and read at least a few different articles. So, yes, there's a lot we could be doing better, but some of the complaints about the current situation seem misguided. Stepwise Continuous Dysfunction (talk) 17:48, 5 November 2025 (UTC)

In the ninth grade I found Thomas's "Calculus and Analytic Geometry" easier to understand than popular expositions of Calculus. As a teenager, I found a number of texts to be quite accessible including Halmos's "Finite Dimensional Vector Spaces" and Kelley's "General Topology". Keep in mind that the space constrains in a text book are a lot more liberal than the space constraints on a wiki page. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:50, 5 November 2025 (UTC)

@WikiMacaroons One flawed premise in your proposal, I think, is to believe that non-mathematically inclined people will have an interest in trying to learn about mathematics. Most such people don't and that's perfectly fine. (The same could be said about most other scientific fields, and again it's perfectly fine.) Trying to make the articles "more accessible" to them by adding a lot of dumbed down cruft seems a futile quest that will just make the rest of us annoyed and make the articles harder to read for those who are really interested. PatrickR2 (talk) 05:34, 5 November 2025 (UTC)

To editor WikiMacaroons: I do not like people who discuss from outside Wikipedia how Wikipedia editors should work. They would better choose some badly written Wikipedia articles (there are too many) and try to improve them. By doing so, they will learn that it is not so easy to write a mathematical article that is together accessible, mathematically correct, and not misleading. By doing so, they would learn that, if there are so many bad articles, this is because we lack of editors having the competence for writing good articles. D.Lazard (talk) 09:17, 5 November 2025 (UTC)

@WikiMacaroons: "Things should be made as simple as possible, but not any simpler."

Some topics simply require a lot of background to understand. How would you make, e.g., Connections on a vector bundle, intelligible to a layman without massively exceeding size limits?

Also, shouldn't discussions on improving wiki articles be hosted on wiki, where they are accessible to everybody, rather than on obscure fora like WP:DISCORD? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:54, 5 November 2025 (UTC)

Wow, woke up to a lot of notifications this morning. Thanks so much to everyone who took the time to read it, and I really appreciate all your thoughtful responses. I will try to get everybody.

• @Malparti: "things can quickly get technical and abstract as one goes up the stack" - A really good point. It's always hard to know exactly how much background knowledge to assume when writing technical articles. In the interests of Wikipedia:Make technical articles understandable, if we're not explaining the background of things directly, at least pointing people towards a lower, more fundamental part of a stack in an obvious way in an article would help. I'm not saying we should necessarily give a background to concepts on articles, but we should at least make them less impenetrable by pointing people towards the things lower down on the stack towards a given concept.
• "there are some concepts/objects I simply cannot understand, even on a superficial level, because I don't have the prerequisites." + what you said about getting a "vague idea" - I agree with this. Wikipedia is a collection of summaries of secondary sources, (and sibilance) and there's only so much we can do before we just have to say, "listen, you need to take a master's degree course to understand this topic". But some higher-level concepts that I believe I could explain to people with GCSE maths under their belt if really given the chance are written in an esoteric way when they simply needn't, (again WP:MTAU).
• "trying to learn math from Wikipedia" - Who are maths articles on Wikipedia for? Not for people who already understand a topic, and not for people who don't understand a topic, then who? Everyone is on some spectrum from no understanding to full understanding before they read a wikipedia article, and there are some where we needn't assume that they, say, know what a derivative is when we can easily explain it.
• "I think this suggestion is too unspecific to even make sense." - yes, sorry that was vague. I plan this weekend to pick an article that I feel has serious issues with clarity to people with less than a master's degree and make a sandbox version of it more digestible for people with a passing interest in maths and above. I'll say right now that I don't think we should be starting from 1+1=2, I think depending on how much education a topic requires to explain, we should use that as a jumping-off point by looking at an article and thinking, "what's the lowest level of maths education you could have to still understand this concept, if it was explained in the right way?", because I think many articles could be explained in much more fundamental ways than they are now. I hope that makes sense.
• @PatrickR2: By not mathematically inclined, I meant people who have an interest in mathematics but not much formal education in it. "Not much" is doing some heavy lifting here, but as I said in the point right above this ^, I think how much education we should assume will vary with how advanced a concept is from article to article, and while Malparti is right that there's no generic accesibility problem, I think plenty of articles are a bit too esoteric.
• As a tangent to @D.Lazard: and @Chatul: - I totally agree with you about offwiki discussion, which is why there’s a strong consciousness on the discord among its members, enforced by moderators, that canvassing is strictly forbidden, and any discussion that takes place there is not a substitute for on wiki discussion. We had an idle, five minute “wouldn’t it be interesting if,” conversation before I felt it was appropriate to move it on wiki for the real discussion, not just vague idle chatter on a discord server.

Just to sum up my thoughts, as I wrote my original message rather hastily, not expecting this much traction, and I really appreciate all your thoughts:

• Many maths articles on Wikipedia are only penetrable to people of very high maths education levels, and while some of them are unavoidably so, as they are extremely advanced concepts, some of them are unecessarily difficult for most people to understand, which goes against WP:MTAU.
• Chatul put it best: "Things should be made as simple as possible, but not any simpler." I believe for each article, we should consider the least possible maths education someone would need to be able to understand the concept as a starting point. We should consider the amount of space we have, and think, could we explain this to an A-level maths student in this space? What about a degree level student? And try to go with the lowest one.
• Over the weekend, I plan to demonstrate what I mean with an example of what this “background” section would look like. I don’t think we should be afraid of putting information that is not directly relevant in an article, because as I said, history articles have information not directly relevant as a background section, but the information is indirectly relevant, as understanding it gives you context for the article’s subject matter. We just have to strike a balance between assuming all knowledge, and inventing the universe from scratch.

Once again, I’m so glad to share this community with people as thoughtful as you all, and you’ve given me a lot of food for thought. :) WikiMacaroons^Cinnamon? 22:36, 5 November 2025 (UTC)

Please go ahead and pick any article that you think is written in an over-complicated way (or is missing information, or is biased, or out of date, or has whatever other issue) and rewrite it in a better way. Make sure that everything claimed is accurate, that the rewrite still supports the needs of well-prepared readers, and that all claims are supported by reliable sources. This effort doesn't need to be any special ceremony, and if you ask for help someone will probably be happy to help / discuss, but do notice that this only scales linearly with the amount of available effort: many articles have already seen such a treatment, but there are thousands and thousands of such articles remaining to work on, and each one takes weeks to months of dedicated effort to bring up to a high standard. –jacobolus (t) 07:37, 6 November 2025 (UTC)

+1. As far as I can tell no one in the math wikiproject objects to other people making articles better, and we're a healthy community as far as resolving "what if you and I disagree about what better means?" through robust discussion. --JBL (talk) 18:33, 6 November 2025 (UTC)

For what it's worth, I think some of the video's criticism of the DFT article is fair, and we could make this article (and many others) more accessible – in particular the lead section is quite problematic – but also the bulk of the video author's suggestions and analysis seem misguided and substantially unhelpful (e.g. about an NIST paper and the author's own website tutorial), and beyond identifying some parts that need attention, I don't think the video gives particularly valuable advice about how to make the article good for a variety of audiences.

Fundamentally the problem is that making clear explanations of technical topics is hard, especially if trying to accommodate a variety of audiences with variable background knowledge, and writing each bit of one takes significant effort and finesse (and sometimes arguing back and forth about the trade-offs).

If anyone knows a way to recruit like 10x more mathematics subject experts who are also excellent at writing accessibly, drawing professional diagrams, etc., and patient enough to engage with a variable pseudonymous community, please do try. –jacobolus (t) 07:56, 5 November 2025 (UTC)

Another approach is a companion introductory article, see examples. fgnievinski (talk) 21:07, 7 November 2025 (UTC)

Concretely, the DFT article should probably lead by saying that the DFT converts back and forth between two different representations of a trigonometric polynomial: (1) a representation in terms of the function values at equispaced sample points, and (2) a representation in terms of the polynomial coefficients. The DFT (especially in its fast form) is useful when we have any sufficiently smooth periodic function, because by taking high enough degree (i.e. enough sample points) a trigonometric polynomial will approximate it to any desired degree of accuracy, and trigonometric polynomials represented in terms of their coefficients are very convenient (computationally inexpensive and easy to reason about) objects to work with. Ideally we'd have a picture showing a real periodic function decomposed in this manner, showing the several basis sinusoids in one part of the picture (scaled by coefficient) and showing the function values in another part of the picture, perhaps as equispaced vertical lines from the x-axis up to the function value at each sample point. –jacobolus (t) 08:00, 5 November 2025 (UTC)

Requested move at Talk:Universal set#Rename to Universe of sets?

There is a requested move discussion at Talk:Universal set#Rename to Universe of sets? that may be of interest to members of this WikiProject. CNC (talk) 21:54, 5 November 2025 (UTC)

Due diligence

In 2006, someone who has long since wandered away from Wikipedia uploaded file:Gaps.jpg (update: now File:Wolf conjecture about prime gaps.jpg) to illustrate... some stuff about prime numbers. From what I can tell, this was reverted as Original Research, he was chided for citing his own papers, etc.

However, it also seems as if he's a legitimate figure in the field... although I'm not really fit to assess that.

So, first: is Gaps.jpg useful enough that it'd be worth exporting to Commons? And, second: if yes, what would be a better filename? DS (talk) 15:45, 6 November 2025 (UTC)

This file is not self-explanatory, and the associated explanation says that it is about a function ⁠$h_{N}(g)$⁠ without saying what ⁠$h,N,g$⁠ represent. Because the name of the file, I guess that it is about gaps between prime numbers, but this is not sufficient to understand what is supposed to be represented. IMO you may delete the file. D.Lazard (talk) 18:00, 6 November 2025 (UTC)

On the other hand, commons is pretty indiscriminate about what they take. There is no reason to host it locally, so it should either be moved to commons or deleted, but if it could be explained better it would probably be a valid contribution to commons. —David Eppstein (talk) 18:37, 6 November 2025 (UTC)

If you check his contributions, is that enough to understand what h, N, G are? (I can't, but that's why I'm asking here.) DS (talk) 19:24, 6 November 2025 (UTC)

The function $h_{N}(g)$ is the number of pairs of consecutive primes $p,p'<N$ such that $p'=p+g$. For instance, $h_{N}(2)$ is the number of pairs of twin primes small than $N$. Malparti (talk) 19:25, 6 November 2025 (UTC)

I'll also add that while this person seems to know the mathematical literature on prime numbers, they are not a mathematician and their paper is not a math paper: it only contains conjectures based on simulations. However, as far as I can tell (I don't know anything about prime numbers), their numerical study seems rigorous and has been cited in math papers. So I think these conjectures are at least somewhat notable. Malparti (talk) 19:40, 6 November 2025 (UTC)

Here was the original context:

Marek Wolf has conjectured that the number of the number of pairs of consecutive primes p_n, p_n+1

with difference p_n+1 - p_n= g:

$h_{N}(g)={\rm {number~of~pairs~~}}p_{n},p_{n+1}<N~{\rm {with}}~g=p_{n+1}-p_{n}$

is given by the formula:

$h_{N}(g)=2c_{2}{\pi ^{2}(N) \over N}\prod _{p\mid g,p>2}{p-1 \over p-2}e^{-{g\pi (N)/N}}+error~term(N,g)~~{\rm {for}}~g\geq 6~~~~~~(*)$

Here π(N) is the number of primes below N. Below is the plot showing the dependence of the histogram h_N(g) on g at N=2²⁰, 2²², ..., 2⁴⁴. There is a logarithmical scale on the y-axis , while on the x-axis there is a linear scale. The values of h_N(g) obtained from the computer search are represented by small circles. The points oscillate with main period 6. Let us mention local spikes at g=30 ( = 2 • 3 • 5), 60 ... . Especially well profound are spikes at g=210=2 • 3 • 5 •7, and for N=2⁴⁰, 2⁴², 2⁴⁴ also at its multiplicity g=420 (second harmonic)

Assuming that the largest gap G(N) between primes < N occurs only once from (*) it follows that

$G(N)\sim {N \over \pi (N)}(2\log(\pi (N))-\log(N)+c_{0}),$

where $c_{0}=\log(2c_{2})\approx 0.278$ and $c_{2}$ is a Twin_prime_constant. Taking the usual approximation π(N)≈N/log(N) the Cramer conjecture follows:

G(N) ≈ log²(N).

Here is the comparison of this formula with compter data:

File:Gmax.jpg

If this image were copied to Commons, those paragraphs could be added to the image description page as a summary. I'd recommend a filename like "Wolf conjecture about prime gaps" or the like. –jacobolus (t) 20:50, 6 November 2025 (UTC)

Thank you; I've added that as a description, and used that filename. DS (talk) 22:01, 6 November 2025 (UTC)

Background: Wikipedia_talk:WikiProject_Mathematics/Archive/2014/Mar#Cramér's_conjecture. --JBL (talk) 19:30, 7 November 2025 (UTC)