Wikipedia talk:WikiProject Mathematics

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Integer factorization

The article integer factorization lists a lot of algorithms. I think it should say what algorithms are the most practical. Somewhere (I couldn't find it again) I read that Shanks's SQUFOF is clearly the best for $n<2^{64}$, or something a little bigger than that; that Quadratic Sieve was best for some range, and that something else was better for some other range.

Can someone add that to the article? Bubba73 ^{You talkin' to me?} 17:54, 30 October 2023 (UTC)

And General number field sieve says that it is best for integers over 100 digits. So what are the approximate range where each is the best? Bubba73 ^{You talkin' to me?} 05:06, 1 November 2023 (UTC)

Also, I think it would be good to have an article about how to factor an integer. I remember a journal article with that title decades ago. Also, Knuth, volume 2, § 4.5.4 goes through a process, but it is out of date. He first does trial factorization and then switches to Fermat's method, which isn't currently the best thing to do. Bubba73 ^{You talkin' to me?} 01:06, 5 November 2023 (UTC)

What are the approximate range where each is the best? This strongly depends on the used computer architecture and the used basic algorithms (integer multiplication, linear algebra, ...). So, I doubt that any reliable encyclopedic answer can be provided.

I think it would be good to have an article about how to factor an integer. See WP:NOTHOWTO. D.Lazard (talk) 11:35, 5 November 2023 (UTC)

More accessible explanations near the front, some historical discussion, possibly an example or two, and some explicit comparison of various methods wouldn't necessarily make integer factorization into a "howto". What we have now is mostly a list of wikilinks to various methods with no explanation or context. A novice reader of the integer factorization page (say a high school student) isn't going to get much out of it, even if integer factorization is relevant to one of their interests or projects, because the page is written in a pretty inaccessible way.

But Bubba73, what did you have in mind? You can always try adding the information you are looking for to the article. Or if you propose something concrete maybe others can help. –jacobolus (t) 14:46, 5 November 2023 (UTC)

I would want some reference on which to base such an article. Something like try trial division up to some point, and if that fails to completely factor it, do a primality test on what remains. If it is composite, then try to factor it with method X. etc. Bubba73 ^{You talkin' to me?} 22:15, 5 November 2023 (UTC)

The Joy of Factoring, by Wagstaff, page 247 gives several tips. It also says that the Quadratic Sieve is best for numbers with 50-100 digits and the Number Field Sieve is best for numbers with more than 100 digits. At least that should go in the integer factorization article. Bubba73 ^{You talkin' to me?} 22:30, 5 November 2023 (UTC)

Is that ten-year-old comparison still accurate, though? As of a different 2011 comparison the crossover was more like in the low 90s [1] and since then some NSF implementations have had significant speedups [2]. Later posts in the same thread put the crossover at around 100 again. —David Eppstein (talk) 23:18, 5 November 2023 (UTC)

Probably not. I'm not up to date on this. But it may give a rough idea. It could be in there with the date of the statement, and note that things change, and it depends on other factors. Bubba73 ^{You talkin' to me?} 00:48, 6 November 2023 (UTC)

I tried to make the first paragraph of integer factorization a bit more accessible. I think it would still be good to add another paragraph to the lead (and another section to the article) describing pre-computer pen-and-paper factorization algorithms and efforts. –jacobolus (t) 23:47, 5 November 2023 (UTC)

Regarding "how to", it shouldn't have the name "how to factor an integer", but something like "method to factor an integer". Also, greatest common divisor tells you "how to" find the GCD. Bubba73 ^{You talkin' to me?} 03:09, 21 November 2023 (UTC)

"factoring algorithms" —Tamfang (talk) 05:39, 25 November 2023 (UTC)

That's the right idea, but the more idiomatic phrase (maybe 2.5x more popular in Google Scholar) is "factorization algorithm" (singular because that's what Wikipedia tends to do). —David Eppstein (talk) 08:04, 25 November 2023 (UTC)

Something like that, but not just a list of factoring algorithms, but something to give guidelines about what should be used and when. I don't have a reference for this, but, unless you know that there are no small factors, I always start with some trial division. Sometimes trial division is enough. But after a certain number of small factors have been checked, do a primality test on what remains. (And you may want to do a primality test right after some factors have been found.) If it is composite, then what you do next depends on how large the remaining number is. Bubba73 ^{You talkin' to me?} 21:53, 25 November 2023 (UTC)

Summarizing Wagstaff's tips:

1. Do at least a probable prime test first

2. Look for small factors

3. Trial division first, then ECM with a small bound and increase the bound, if necessary

4. Parallelize if you can

5. Remember that Pollard Rho and ECM may not find factors in order

6. Choose the best algorithm for the size of the number. Quadratic Sieve is fastest for 50-100 digits and Number Field Sieve is fastest for more than 100 digits.

7. when a factor is discovered, check it for primality. Use the BPSW test. If you need to prove a number to be prime, use the Elliptic Curve Prime Proving method, or if you can factor p-1, use one of those methods.

Now, this isn't what I do because I always check for some small factors before a primality test, etc. Bubba73 ^{You talkin' to me?} 22:22, 25 November 2023 (UTC)

Making mathematical articles more broadly accessible

During the Q&A of one of the lightning talks at WikiConference North America this last weekend, it came up that a lot of us, even those with degrees in mathematics, find the bulk of en-wiki's math articles among the least penetrable on the site, even when compared against other STEM topics. There were a few constructive suggestions (I'm paraphrasing, of course):

• Have in mind who your audience is. You aren't writing to show your professor that you understand the material. You are writing to teach the material to someone who is at the level in mathematics to be able to understand the relevant concept, theorem, etc., but to whom this particular material is new.
• Consider that the lede should be as broadly comprehensible as possible. In particular, it is OK if the lede oversimplifies a bit, as long as it is clear that the details can be found later in the article. Gödel's incompleteness theorems is a great example of getting this right. Note, in particular, that it does not even touch upon arithmetization of syntax, perhaps the most striking feature of Gödel's proof. Why? Because it's not an article about the proof, it's an article about the theorem, and someone with only a moderate background in mathematics is likely to be a lot more interested in what the theorem says than how, exactly, it is proved. Conversely, in the generally good lede of manifold: would it have killed anyone to explain the word "lemniscates" with "(e.g., figure-eights)" rather than make maybe 90% of readers click through if they want to understand what is being said?
• We might want to create a template which, in the session, we jokingly called {{Prerequisites}}. The idea would be to list, up front of either a section or of the article as whole, the concepts you need to understand the article, instead of having the reader discover them as they encounter links scattered through the article. E.g. Manifolds usefully says, at the end of the lede, "The study of manifolds requires working knowledge of calculus and topology." The is is the sort of thing {{Prerequisites}} could do; also, if the way we deploy such a templates were designed well, we might be able to make clear that most of the article can be understood with just topology, and only certain sections require any calculus.

Thoughts? - Jmabel | Talk 20:32, 14 November 2023 (UTC)

It's hard to argue with the general goal or the first two points. The third point, about prerequisites, has been proposed many times and shot down every time. In essence, the counter-argument seems to be: The prerequisites are already encoded into the links, and anything that does much more than the links is veering into Wikipedia:Wikipedia is not a textbook territory. I don't find that argument fully convincing — maybe there's a perfect balance to be discovered — but I've never seen it described in adequate detail. Regards, Mgnbar (talk) 21:33, 14 November 2023 (UTC)

As to the second point, I think that while Jmabel's example (the intro to Gödel's incompleteness theorems) is fine, the wording could be abused. There's a difference between leaving out detail, and saying things that aren't true. We should never oversimplify to the point that what we're saying isn't true, at least not without an explicit warning that that's what's happening.

That's the vexing thing about these discussions. On the one hand, it's definitely true that many math articles are written in a way that makes them much less useful than they could be. On the other hand, we see people throwing drive-by {{technical}} templates on articles that are written quite reasonably for any audience that has the background necessary to have a chance at the inherent subject matter. And there's a contingent that even thinks it's OK to tell lies to children if it makes readers feel like they understand. --Trovatore (talk) 21:53, 14 November 2023 (UTC)

I personally find throwing "maintenance templates" on tops of articles to be obnoxious and almost entirely useless. However, it would be nice if confused readers would take the time to start a talk page conversation with a concrete critique when they have an issue like this, so that someone can try to make improvements. If the subject gets discussed somewhere I see it, e.g. at this wikiproject talk page, I'm happy to give some effort to working on accessibility of subjects that I feel nominally competent to write about. (Or if that seems too off topic for here, I'd be happy to discuss such cases at some other venue.) As one example, someone recently mentioned on the talk page of WP:TECHNICAL that they found the lead section of our article Bijection to be hard to follow, so I tried to rearrange and expand its lead section a bit to hopefully be clearer to a general audience (people with more expertise than I have are welcome to rework that further). –jacobolus (t) 02:07, 15 November 2023 (UTC)

As a small aside, I'm curious about these potential readers who know topology but not calculus.... --Trovatore (talk) 22:14, 14 November 2023 (UTC)

Me at age 16. I took things in an odd order. - Jmabel | Talk 04:37, 17 November 2023 (UTC)

The biggest error that we make on Wikipedia in this area is summary style, whose author regretted it years later as I recall, since really it's a newspaper thing not an encyclopaedia thing. Introductions should introduce; not be a Britannica style Micropedia to the Macropedia that is the rest of the article. If you try to compress the rest of the article into the introduction, and then summarize the introduction into its first paragraph, the information density becomes ridiculous; and this is partly the cause of articles that end up as "subject is the namedrop jargon jargon of namedrop jargon when jargon namedrop jargons are encofstulated.^{[1][2][3][4][5][6][7]}". I don't think that prerequisites are the answer, but I do think that dropping the idea of compressing everything ever tighter again and again, and instead just introducing a subject, would avoid the "leaving out detail" problem. Uncle G (talk) 23:18, 14 November 2023 (UTC)

I am very strongly in favor of putting in the effort to make our articles as accessible as possible. But it is significant effort, and not the kind of effort that mathematics students are often well-trained in. That said, the original post has a serious inaccuracy in its framing. It writes:

Have in mind who your audience is. You aren't writing to show your professor that you understand the material. You are writing to teach the material to someone who is at the level in mathematics to be able to understand the relevant concept, theorem, etc., but to whom this particular material is new.

This is written as if there can and should be only one audience for each mathematics article, someone ready for the material who does not yet understand it. This is untrue. That is one kind of audience, but not the only one. We need the leads of our article to be readable by someone who is not ready to learn the material in any depth but is interested in finding out what it might be about (WP:ONEDOWN). And we need the later parts of our article to be usable by people who do understand the material already but want to refer to it anyway as reference material, as a quick reminder of what they already know, or as a collection of pointers to more in-depth literature to use as references for other works. Writing articles that can be used for all these purposes is even more effort, and is something typically not appreciated by readers at a different level who either think the material is too technical or too oversimplified. —David Eppstein (talk) 23:30, 14 November 2023 (UTC)

I would like to call out another aspect of Jmabel's (generally reasonable) post. I disagree that we are writing to "teach the material". This is a reference work, not a textbook. We want to facilitate self-teaching, but we do not ourselves want to teach.

This is more a question of organization than level of difficulty. I've never come up with the perfect way of expressing it, but maybe -- we want to be random-access rather than serial? We're not presenting an order in which you're supposed to learn the material. We want to make the material available to you, so you can look up an individual fact quickly if you need it, or find a path through it to learn it if that's your goal. --Trovatore (talk) 01:36, 15 November 2023 (UTC)

I can appreciate the motivation for a {{Prerequisites}} template, but in the end it sounds like a gimmick: one more thing to have to maintain, and one more way that complicated relationships get flattened for the sake of having a box in the sidebar. It's rather like the "influenced" and "influenced by" items in {{Infobox philosopher}} and {{Infobox scientist}}, which were recently removed. Our time would be better spent improving the text than futzing about with new sidebars.

I'll echo the sentiment above that an article can in general have more than one audience, which makes for hard writing. Moreover, this is one way that we differ from textbooks: a typical textbook is made with an audience in mind and a sense of where in the curriculum it's going to be used. Another difference alluded to above is that textbooks are generally sequential, rather than "random-access". The standard approach in teaching a course is to begin at the beginning of the book and work through the chapters pretty much in the order they're printed, maybe not getting all the way to the end. The challenges of writing well in that style are going to be different than the challenges of writing well on this platform, just by the nature of the platform itself. My sense of the overall situation is that the top priority for some articles, like short ones on highly specialized upper-level topics, should be to expand, organize, and reference them. A lot of improvement in those corners of the encyclopedia would involve writing at a similar level to what the existing text presumes, but just being less half-ass about it. In other places, the top priority ought to be providing a solid opening. XOR'easter (talk) 08:24, 15 November 2023 (UTC)

It has always been thus, as I know from the day I joined the project in 2007. Wikipedia is not a textbook, and this is particularly stark in advanced mathematics articles... but we should – and often do – strive to make mathematics articles as accessible as possible to the dedicated reader, by making our explanations as clear as possible, and by linking to sources and resources that will help readers learn more and then appreciate the value and beauty of maths. Geometry guy 00:43, 16 November 2023 (UTC)

There is a conflict between your first and second points here IMO. If it is true that we "are writing to teach the material to someone who is at the level in mathematics to be able to understand the relevant concept, theorem, etc., but to whom this particular material is new" then it just can't be that the lede should be "broadly comprehensible" by appealing to intuition and lacking rigor. No one ever learned a mathematical concept without seeing its actual definition, or something very close to it. This is exemplified by many analysis articles on WP nowadays. For instance, Distribution has a long lede that attempts many times to give the reader an intuitive idea of what distributions are, without ever actually saying explicitly what they are. It is not informative at all. One has to dig down quite considerably to learn anything on this page. The problem is hardly unique to that article.

I think Gödel's incompleteness theorem(s) is a bad example. This is a topic which has great significance in popular mathematics and about which much philosophy, etc. has been written. It is accurate to the RS to present it as that article does. The same is not true of most math topics. ByVarying | talk 22:35, 16 November 2023 (UTC)

Just a comment on Distribution (mathematics): It is wrong that the definition of distributions is not given in the lead. It is clearly explained that distributions are linear forms on the infinitely differential functions with compact support, and that a function can be considered as a distribution by integrating the product of the function with infinitely differential functions with compact support. This is a complete and accurate definition.

If some readers do not recognize this as a definition, it is probably because passing from functions to distributions is a fundamental change of paradigm that requires some work to be well understood. Probably, the lead could be improved on this point, but this is not an easy task. D.Lazard (talk) 10:29, 17 November 2023 (UTC)

To be clear, the reason I didn't understand that the definition of distribution wasn't in the lede was that I didn't scroll all the way down to the last paragraph to find it. It's a little embarrassing that my statement turned out not to be true, but I think the example is still relevant in that the four intervening paragraphs, which I assume attempt to prepare one for the definition, do not do so. ByVarying | talk 11:07, 17 November 2023 (UTC)

OK: let me be more concrete. Does anyone think my example above about adding a parenthetical phrase to explain "lemniscates" is not the sort of thing that makes an article more useful? Can it really be better to make someone click through to get a concept we can explain pretty well in a couple of words?

Or let's take a couple of sentences in the (generally decently written) article Topological space.

• "Here we also allow $X$ to be empty." Why not just "$X$ may also be an empty set" (we might even omit "also")?
• "In other words, each point of the set $X$ belongs to every one of its neighbourhoods with respect to ${\mathcal {N}}$." I believe it would not be instantly obvious to someone who has not already studied topology that "its" refers back to "each point", not to $X$. I had to read it twice myself, and it states a fact that I knew perfectly well. It could be clarified by writing (for example) "In other words, each point $x$ of the set $X$ belongs to every one of its ($x$'s) neighbourhoods with respect to ${\mathcal {N}}$."

These are trivial examples, but I'm picking them because I think everyone reading this can presumably understand them. This sort of thing drives me up a wall when I try to read Wikipedia articles about math topics where I certainly have a background that should suffice to read a well-written article on the topic.

Non-rhetorical question: don't most of you find yourself with the same problem when you try to read a Wikipedia math article outside of your own specific field? Because last week in Toronto, when we were discussing this in a room where probably half the people had STEM degrees, most of us at least a Masters, it appeared to be a pretty universally shared experience. - Jmabel | Talk 05:18, 17 November 2023 (UTC)

Adding parenthetical phrases (short pieces of text that distract from the main sentence (a complete thought, usually formatted with a capital starting letter and ending with a period) by separating pieces of it by long distances from other pieces) can make text harder to read. Using unfamiliar jargon also makes text hard to read but we should avoid piling on one problem to fix another. In some cases a single level of short glosses may be ok. In others, a more jargon-free choice of wording may be a better choice. In the case in question, explaining that a lemniscate is an algebraic curve studied by Bernoulli would not help. All we really need to say is that curves that cross themselves, like a figure 8, are not manifolds. —David Eppstein (talk) 08:46, 17 November 2023 (UTC)

Yes, there is plenty of agreement that we should try to work on this. But it's a huge amount of difficult work researching, synthesizing, writing, drawing or soliciting diagrams, and making decisions and compromises; it takes both mathematical subject knowledge and writing skill/taste; and ideally it involves collaborating with other Wikipedians to settle on something that is precise enough for a specialist audience while also being as broadly legible as practical to a wider audience. There are hard choices to make about inter- and intra-article organization, tone, emphasis, level of detail, etc. If you want to pick an article or cluster of articles and get to work, "be bold" about making changes, with the understanding that your changes might be disputed or reverted, so you are likely to need to engage in sometimes frustrating discussions and do more meta-work than typical when writing for yourself. If you get stuck or need help or feedback, you can recruit volunteers here. –jacobolus (t) 23:09, 17 November 2023 (UTC)

If anyone wants ideas for what to improve first, I took the "Vital articles" for mathematics and found their 30-day page view counts. These topics are likely to have broader audiences than many of the more esoteric articles, and only a few of them have any GA/FA stamp of approval.

Extended content

General (5 articles) Mathematics (Level 1) - 149,119 Algorithm - 70,127 Mathematical proof - 14,046 Set - 34,892 Function - 47,338 Counting and numbers (12 articles) Combinatorics - 18,384 Number (Level 2) - 70,474 Real number - 41,579 e - 142,557 π - 173,469 Fraction - 20,799 Integer - 46,072 0 - 241,262 (compare with 1, not on this list, which has 248,145) Natural number - 51,345 Prime number - 123,187 Complex number - 53,705 Number theory - 26,054 Algebra (5 articles) Algebra - 59,303 Equation - 12,364 Variable - 8,242 Linear algebra - 31,252 Abstract algebra - 13,047 Analysis (5 articles) Mathematical analysis - 16,889 Calculus - 77,679 Infinity - 45,475 Limit - 25,049 Series - 15,213 Arithmetic (4 articles) Arithmetic (Level 2) - 21,069 Exponentiation - 31,434 Logarithm - 67,018 nth root - 9,462 Geometry and topology (12 articles) Geometry (Level 2) - 42,034 Angle - 18,884 Trigonometry - 36,550 Area - 15,529 Conic section - 20,720 Circle - 40,105 Line - 9,302 Polygon - 22,187 Triangle - 36,061 Three-dimensional space - 20,725 Volume - 19,626 Topology - 35,479 Probability and statistics (2 articles) Probability - 31,176 Statistics (Level 2) - 51,094

Also in this range are Quadratic formula (70,039), Sine and cosine (52,339), and probably others that aren't springing to mind right now. XOR'easter (talk) 22:56, 17 November 2023 (UTC)

XOR'easter, how much trouble would it be to get a similar view count for all of the math wikiproject articles of top/high(/mid) priority? –jacobolus (t) 23:14, 17 November 2023 (UTC)

Good question. Long enough that I'm not going to try by hand; there are 911 "high-priority" mathematics articles and 244 "top-priority" ones. XOR'easter (talk) 23:21, 17 November 2023 (UTC)

Oh, I thought you had a script. Maybe someone can figure out how to automate this. (I could probably, but I'm not feeling super motivated to try right now.) –jacobolus (t) 23:23, 17 November 2023 (UTC)

Funny story about that: I remembered doing something along these lines for physics a while ago, and I thought there was a script for it. Then I finally found my notes and discovered that I had done it by hand. XOR'easter (talk) 23:31, 17 November 2023 (UTC)

There's apparently an existing tool. So that's convenient. For example, high-priority math article views in the past year. –jacobolus (t) 23:38, 17 November 2023 (UTC)

Thanks. "Massviews"... not the most obvious name! XOR'easter (talk) 23:44, 17 November 2023 (UTC)

Highest-view articles that are concrete math concepts per se are apparently Roman numerals, Poisson distribution, Golden ratio, Fibonacci sequence, Taylor series, Exponential distribution, Gamma distribution, Gamma function, ... Seems like people interested in introductory statistics topics do a lot of Wikipedia lookups. –jacobolus (t) 23:48, 17 November 2023 (UTC)

One of those high-viewcount and high-priority articles, arithmetic, has been undergoing major changes this month from a couple of editors whose names I don't recognize. Might be for the better, I'm not sure, but could at least benefit from some checking. —David Eppstein (talk) 07:32, 18 November 2023 (UTC)

The previous state of arithmetic wasn't very inspiring, and definitely needed some help. I haven't looked super closely but at least it doesn't seem to be getting worse. It wouldn't be bad to have some more group effort go into organizing and fleshing out the article though, if anyone else wants to dive in there. –jacobolus (t) 07:45, 18 November 2023 (UTC)

I still see issues:

1. Arithmetic#Definition, etymology, and related fields has an extremely dated reference to Algebra; Abstract Algebra deals with more than numbers.
2. No mention of transcendental numbers.
3. Arithmetic#Number theory is too narrow, e.g., no mention of Gaussian integers.

These are areas that deserve links to existing articles but not lengthy expositions here. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:23, 20 November 2023 (UTC)

One of the two (Phlsph7) is responsible for bringing Logic to FA status recently. --JBL (talk) 17:21, 18 November 2023 (UTC)

Okay, here's a sorted list of all the top/high priority articles with more than 700 daily views:

More than 5k views per day: Albert Einstein, Isaac Newton, Srinivasa Ramanujan, 0, Pi, Normal distribution, 1, Aristotle,

More than 3k views per day: Standard deviation, Bayes' theorem, Galileo Galilei, Mathematics, Quantum mechanics, E (mathematical constant), Roman numerals, Prime number, Archimedes, Poisson distribution, John Forbes Nash Jr., Golden ratio, Time, Fibonacci sequence, Newton's laws of motion, Taylor series, John von Neumann, Fourier transform, 2

More than 2k views per day: Entropy, Physics, Pythagorean theorem, 4, Pythagoras, Bertrand Russell, Theory of relativity, 5, 8, Information, René Descartes, Variance, Aryabhata, 6, Exponential distribution, Calculus, Gamma distribution, Algorithm, Gamma function, Dot product, Principal component analysis, Game theory, Tesseract, Matrix (mathematics), Number, Maxwell's equations, Eigenvalues and eigenvectors

More than 1.5k views per day: Butterfly effect, Chaos theory, Quadratic formula, Chi-squared test, Monte Carlo method, 10, Logarithm, Logic, Conway's Game of Life, General relativity, Euclid, Linear regression, Statistics, Central limit theorem, Riemann hypothesis, Singular value decomposition, Euler's formula, Entropy (information theory), Derivative, Turing machine, Abacus, Complex number, Algebra, Leonhard Euler, Navier–Stokes equations, Sigmoid function, Natural number, Laplace transform, Dirac delta function, Exponential function, Spacetime, Spherical coordinate system, Probability density function

More than 1k views per day: Regression analysis, Fast Fourier transform, Matrix multiplication, Determinant, Sine and cosine, Carl Friedrich Gauss, Data science, Integral, Markov chain, Integer, Quadratic equation, Expected value, Confidence interval, Collatz conjecture, P versus NP problem, Fourier series, Chi-squared distribution, Blaise Pascal, Newton's method, Covariance, Trigonometric functions, Median, Student's t-test, Prisoner's dilemma, Cumulative distribution function, Infinity, Function (mathematics), Norm (mathematics), Natural logarithm, List of unsolved problems in mathematics, Gottfried Wilhelm Leibniz, Cryptography, Real number, Gradient, Cartesian coordinate system, Square root, Student's t-distribution, 9, Tensor, Cauchy–Schwarz inequality, Geometry, Euclidean distance, Geometric series, Kurt Gödel, Fermat's Last Theorem, Momentum, Special relativity, Triangle, Kinetic energy, Roger Penrose, Topology, Correlation, Kepler's laws of planetary motion, Circle, Factorial, Nash equilibrium, Data analysis, Trigonometry, Rational number, Accuracy and precision, Logistic function, Fields Medal, Hilbert space, Gödel's incompleteness theorems, Discrete Fourier transform, Polynomial, Maximum likelihood estimation, Big data, Euler's identity, Ellipse, Graph theory, Vector space, Probability distribution, L'Hôpital's rule, Discrete mathematics, Differential equation, Lagrange multiplier, Law of large numbers, Paul Dirac, Boolean algebra, Modular arithmetic, Set (mathematics), Manifold, Chain rule, Set theory

More than 700 views per day: Probability, Riemann zeta function, Directed acyclic graph, Grigori Perelman, Laplace operator, Quantum field theory, 3, Histogram, Russell's paradox, Linear programming, Eratosthenes, Fibonacci, Sphere, Trapezoid, Parabola, Permutation, Integration by parts, Fundamental theorem of calculus, Dimension, Bayesian inference, Number theory, Information theory, Halting problem, Tuple, Inverse trigonometric functions, Field (mathematics), Exponentiation, Velocity, Gaussian elimination, Summation, Irrational number, Stochastic process, Heat equation, Mean value theorem, Information security, Liberal arts education, Combination, Polar coordinate system, Likelihood function, Mean, Taylor's theorem, First-order logic, Principia Mathematica, 1000 (number), Dirac equation, Time complexity, Greatest common divisor, Omar Khayyam, Divergence, Partial differential equation, Brahmagupta, Orthogonal matrix, Polygon, Bell's theorem, Euclidean algorithm, Law of cosines, Power of two, Axiom, Platonic solid, Euclidean geometry, Divergence theorem, Radian, Wave equation, Euler's totient function, Tensor product, Perfect number, Euclidean space, Arithmetic, Random variable

–jacobolus (t) 18:45, 19 November 2023 (UTC)

Thanks for organizing the list this way.

The article Circle seems particularly lackluster: many bulleted lists instead of prose, few references. XOR'easter (talk) 03:18, 20 November 2023 (UTC)

Circle is definitely on my list of pages to eventually dramatically expand. It's a bit tricky to figure out how to organize because there is so much to say and so many relevant sources. For inspiration take a look at A Treatise on the Circle and the Sphere. –jacobolus (t) 04:23, 20 November 2023 (UTC)

I seriously thought about starting by writing a section that summarized Book 3 of Euclid.... XOR'easter (talk) 05:50, 20 November 2023 (UTC)

We should start with a dedicated article about Book 3 of Euclid. Ideally we'd also have dedicated articles about Book I (parallelograms / affine geometry), Book II ("quadrature"), Book V (proportions), Book VI (similarity), Books VII–IX (number theory), Book X (incommensurable lines), Books XI–XIII (solid geometry). (Aside: I've been reading a translation/commentary and other sources about Theodosius' Spherics of which the first half is more or less "let's make every analogy of every proposition we can from Elements III on the sphere". I'm still not quite ready to expand the Spherics article though.) –jacobolus (t) 06:14, 20 November 2023 (UTC)

Regarding the lead sentence of Yoneda lemma, it says the Yoneda lemma is arguably the most important result in category theory.. This seems to be due to the addition of Template:Technical to the article, but I wonder if it helps with accessibility. By the way, nominating some articles to WP:DYK may be more useful than adding maintenance templates to the top of the articles. --SilverMatsu (talk) 16:42, 18 November 2023 (UTC)

WP:DYK basically doesn't want technical articles. They just sit in a queue before eventually someone decides they aren't of general enough interest and get dropped. Not worth the trouble to nominate things there. –jacobolus (t) 17:05, 18 November 2023 (UTC)

I got affine symmetric group there -- I think it was helpful that along the way I was repeatedly pushed to keep writing the introduction in a more and more gentle way, well beyond what I thought possible. --JBL (talk) 17:18, 18 November 2023 (UTC)

New editor with a draft

• Draft:Mal'cev's criterion
• JesseStraat (talk · contribs)

This is going to be a difficult AFC review for whoever picks it up. I've given the article creator a few tips and pointers, but some help from others would not go amiss. Uncle G (talk) 22:58, 14 November 2023 (UTC)

I'd be happy to review it. Felix QW (talk) 09:42, 17 November 2023 (UTC)

Disambiguation of Two-dimensional space

Could you help to disambiguate links to Two-dimensional space? There are a lot of articles (shown in this list which link to the dab page. Not all would fall under this wikiproject but many are either related to Surface (mathematics) or Plane (mathematics) and I don't have the expertise to know which.— Rod ^talk 21:45, 26 November 2023 (UTC)

At a glance, this looks like a merge opportunity? I don't see why surface (mathematics) and surface (topology) shouldn't be a single article, particular when the "mathematics" one is glossed in the disambig page as "a topological space of dimension two". Planes are more specific than 2D spaces and probably should not be in the mix here at all. --Trovatore (talk) 22:42, 26 November 2023 (UTC)

I agree that merging surface (mathematics) and surface (topology) is probably a good idea, but I don't think that planes should be excluded from the disambiguation of 2D spaces. If someone is searching for 2D spaces, they'd probably expect planes to show up as a prominent and canonical example. --MtPenguinMonster (talk) 08:45, 27 November 2023 (UTC)

I don't think these should necessarily be merged. There's plenty to say separately about topological vs. other kinds of features. If we want a merge target for a top-level article about surfaces, it should probably be a merger to surface, which is currently very mediocre. Someone who knows/cares about surfaces in a reasonably wide range of contexts should try to think carefully about how various topics related to surfaces should be be organized into multiple articles, and then split those topics between a few articles with clearly defined scope, which can all be linked from surface as an overview article. –jacobolus (t) 10:47, 27 November 2023 (UTC)

I think this should redirect to Plane (mathematics), which (perhaps along with Euclidean plane or Coordinate plane) is the intended target for the vast majority of inbound links, and which could also discuss general surfaces in a section. @Fgnievinski recently switched from that to a disambiguation page in special:diff/1183701575 without any discussion or consensus, and I think it was a significant mistake. –jacobolus (t) 10:38, 27 November 2023 (UTC)

I reverted to the version which is a redirect. If there is going to be a disambiguation page, it should be under a title like two-dimensional space (disambiguation), with two-dimensional space redirecting to plane (mathematics) which is the commonly intended target. –jacobolus (t) 10:54, 27 November 2023 (UTC)

Surely somebody looking for information on, e.g., ellipsoids, would find a redirect to plane to be perverse. Certainly spheres is a commonly intended target. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:55, 27 November 2023 (UTC)

Why would someone wanting to direct readers to ellipsoids put a wiki-link to two-dimensional space? In my opinion that would be ridiculous, and it seems vanishingly unlikely. –jacobolus (t) 12:59, 27 November 2023 (UTC)

The same question applies to planes; why would someone wanting to direct readers to planes put a wiki-link to two-dimensional space? That seems just as ridiculous and unlikely. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:22, 27 November 2023 (UTC)

In my mind the words "line", "plane", "space" are shorter synonyms for "1-space", "2-space", "3-space" (which originally arose in a context where "space" always meant 3-space).

What do you take 2-space to mean? –jacobolus (t) 18:13, 27 November 2023 (UTC)

I think "plane" by default means Euclidean plane, whereas 2D space definitely does not imply Euclidean, basically ever. I not only don't think "plane" is a reasonable redirect target, I don't even think it should be linked from this target. I think surface (mathematics) and surface (topology) should be merged, and two-dimensional space should redirect to the merged article. Or possibly the merged article should just be named "two-dimensional space". --Trovatore (talk) 18:28, 27 November 2023 (UTC)

I agree with jacobolus. Here are the first few links to two-dimensional space grabbed from https://en.wikipedia.org/wiki/Special:WhatLinksHere/Two-dimensional_space:

• Cartesian coordinate system is about planes
• Real coordinate space is about planes
• Gas electron multiplier maybe shouldn't be linked to any of these articles but if it should then it should be about planes
• many of them (like Piranha 3D, XLfit, Photo psychology, and List of Aqua Teen Hunger Force characters) shouldn't be linked to any mathematical article, but certainly should not be linked to articles about topological surfaces
• Plane of rotation is about planes

Maybe somewhere in Wikipedia someone has linked two-dimensional space when they should have linked something about surfaces, but mostly people are trying to link to something specifically about planes (possibly not in a truly mathematical sense). --JBL (talk) 18:44, 27 November 2023 (UTC)

"Plane" means a wide variety of different things depending on the context, and in my impression is a precise and almost completely interchangeable synonym for "two-dimensional space". For example "hyperbolic plane" = "two-dimensional Riemannian space of constant negative curvature" and "projective plane" = "two-dimensional projective space". Without qualification or context both of these most often mean a Euclidean plane (flat 2-space with a concept of distance), but sometimes mean an affine plane or some other kind of plane. (One slight difference in common use I have observed is that often "two-dimensional space" is used to mean the whole context for the geometry, while "plane" is often used for, more specifically, a "two-dimensional subspace".) –jacobolus (t) 18:57, 27 November 2023 (UTC)

Comment Without any comment on the substance of the matter, I would just like to point out that the intended targets of the links to the page will be heavily biased towards planes simply because that is where the term lead until recently. So anyone who would have wanted to link anywhere else may well have considered linking to Two-dimensional space and then seen that the target is not where they intended to point. Felix QW (talk) 19:49, 27 November 2023 (UTC)

Yes, I think most of these links to "two-dimensional space" just need to be fixed at the link.

Jacobulus points out that "plane" doesn't always mean "flat plane", but I think it does by default, whereas "two-dimensional space" does not mean flat by default.

Maybe stickier, what about $S^{2}$? That's a two-dimensional space, but I think is very unlikely ever to be called a "plane".

I think basically any link to "two-dimensional space" where the reader would be surprised to arrive at an article that includes $S^{2}$ — is just an incorrect link and should be fixed in the text of that article. --Trovatore (talk) 23:57, 27 November 2023 (UTC)

Both of these imply flat when left unqualified. Cases where someone wants a curved space are either called "surface" or e.g. "curved two-dimensional space". But anyway, the 2-sphere is definitely a type of "plane" just like the hyperbolic plane or elliptic plane (historically the sphere was occasionally called the "Riemannian plane" (example 1, example 2), especially when someone wanted to emphasize an intrinsic/axiomatic definition, but that's somewhat ambiguous and now I think usually means something else). It's typically called the "sphere" rather than any kind of "plane" primarily because the concept of a sphere is ~2000 years older than the concept of a "curved plane", but perhaps also because "elliptic plane" is used to mean the sphere with antipodal points identified. Confusingly, the ancient Greek term "line" means any curve and the flat version was explicitly called a "straight line", which is sort of the opposite situation from what we deal with in the 2-dimensional case; in French they picked a better abbreviation and call straight lines "straights" for short. All of these various inconsistencies and ambiguities in nomenclature came about because these names were established long long before they were generalized, and then the generalization was done piecemeal and incoherently by a variety of different authors and translators. –jacobolus (t) 02:47, 28 November 2023 (UTC)

I'm not sure what you mean by "both of these". The phrase "two-dimensional space" absolutely does not imply flat when left unqualified. --Trovatore (talk) 07:16, 28 November 2023 (UTC)

Yesterday I spent 20 minutes skimming through academic papers matching the search result "two-dimensional space", and when unqualified this is usually referring to the Euclidean plane. Examples include children learning about maps, computer algorithms for segmenting images, wallaper groups, modeling car traffic, surfaces projected onto their tangent space, fractals, graph algorithms for graphs embedded in the plane, imaging of thin slices of some three-dimensional material, etc. There are also lots of papers talking about "curved two-dimensional space", "two-dimensional space-time", "non-Euclidean two-dimensional space", etc., but even these are not too often used for just any old surface from what I can see. Examples where there is an arbitrary surface include e.g. robot kinematics where there's some "two-dimensional space" of available motion for some part, and computer graphics mapping an image parametrically onto a triangle mesh; these are examples of real coordinate space. –jacobolus (t) 15:13, 28 November 2023 (UTC)

I'm really having trouble seeing why you would ever not want to just say "plane" if you mean a flat 2D space. "Plane" very clearly says it's flat (a plane is a tool used to make things flat). "Two-dimensional", on the other hand, says nothing whatsoever about flatness, just about the number of independent ways you can move in the space. --Trovatore (talk) 17:15, 28 November 2023 (UTC)

What this demonstrates is that you've absorbed a particular expert perspective so far as to lose sight of how non-experts use words and language, rather than anything about the encyclopedia. The usage of "two-dimensional" to mean "planar" is common, widespread, and rarely causes confusion. (By contrast, the usage of "two-dimesional" to describe a curved surface embedded in three-space causes lots of confusion.) --JBL (talk) 17:46, 28 November 2023 (UTC)

I'm not trying to psychoanalyze anyone; I'm just telling you that in practice "plane" and "two-dimensional space" are typically treated as synonyms. But as I suggested below, I think we could make an article at two-dimensional space. The tricky part then is figuring out how to organize (and ideally expand) material between there, plane (mathematics), Euclidean plane, inversive plane, hyperbolic plane, sphere, affine plane, projective plane, coordinate plane, surface, surface (mathematics), surface (topology), etc., most of which are currently mediocre. The part I think is a bad idea is encouraging meat-bots to go "fix" every wikilink pointed at two-dimensional space so they don't land on a disambiguation page, which is apparently banned. –jacobolus (t) 17:47, 28 November 2023 (UTC)

Maybe it's useful to consider a different approach to the problem: under what circumstances would someone put a link to two-dimensional space? Are there scenarios where someone would be more likely to link to two-dimensional space than to an article like plane or surface? --MtPenguinMonster (talk) 09:08, 28 November 2023 (UTC)

One other possibility would be to make a short halfway-to-disambiguation page modeled on one-dimensional space. –jacobolus (t) 19:11, 27 November 2023 (UTC)

Thanks for all the edits, reverts and discussion on this, with lots of useful ways forward. If anyone had any good ideas about links to the dab page Parametrization (shown at Disambig fix list for Parametrization that would be great as well.— Rod ^talk 15:31, 28 November 2023 (UTC)

And links to Tensor category (shown here) are also giving me grief trying to disambiguate them.— Rod ^talk 16:19, 28 November 2023 (UTC)

Isbell conjugacy, Isbell duality

I think Isbell conjugacy may be an another name or subsection of the (article of) Isbell duality (See also Codensity monad#Relation to Isbell duality). So, I think it might be possible to merge those two articles. Also, if articles are merged, which is better, Isbell conjugacy or Isbell duality ? SilverMatsu (talk) 16:33, 30 November 2023 (UTC)

What is the second article? Isbell duality is a redlink. - CRGreathouse (t | c) 17:21, 30 November 2023 (UTC)

Oops, I overlooked that it was a red link. --SilverMatsu (talk) 17:38, 3 December 2023 (UTC)