Wikipedia talk:WikiProject Mathematics/Archive/2024/Jun

Degenerate bilinear form is unreferenced

Request for an esteemed colleague from WikiProject Mathematics to please review and find a source for Degenerate bilinear form, which has been tagged as "Unreferenced" since August 2008. Cielquiparle (talk) 09:58, 25 May 2024 (UTC)

I see this has been fixed; surely though the right title for this topic is Nondegenerate bilinear form? They're the important ones .... 64.26.99.248 (talk) 18:24, 30 May 2024 (UTC)

I'm guessing there is a stupid Wikipedia reason for this bizarre state of affairs. Tito Omburo (talk) 21:35, 30 May 2024 (UTC)

The reason is probably history rather than policy. IAC, rather than renaming the article it might be better to merge it into Bilinear form with {{R to section}} in the redirects. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 11:39, 31 May 2024 (UTC)

That sounds reasonable. XOR'easter (talk) 00:26, 6 June 2024 (UTC)

Uncited statements at 0#Computer science

A few statements at 0#Computer science need support from manuals, textbooks, and/or histories. I know math people aren't necessarily computer people, but it seemed a good idea to raise the signal here too. XOR'easter (talk) 02:42, 6 June 2024 (UTC)

Doi will be added to the Theory and Applications of Categories

See this blog post. SilverMatsu (talk) 15:31, 6 June 2024 (UTC)

SVG rendering bug is fixed

I'm happy to announce that MediaWiki has finally updated their SVG rendering library to a less obsolete version, and as a result plenty of bugs were fixed, including the one that sparked a discussion here back in March. Tercer (talk) 20:23, 6 June 2024 (UTC)

Thanks for the good news! —David Eppstein (talk) 20:31, 6 June 2024 (UTC)

History of the definition of the real numbers

I am confused by the Wikipedia description of the history of the definition/construction of real numbers:

• In Real number § History: The first rigorous definition was published by Cantor in 1871. No indication on the method (infinite decimals?)
• In Construction of real numbers: Nothing
• In Foundations of mathematics § Real analysis: In 1858, Dedekind proposed a definition of the real numbers as cuts of rational numbers
• In Dedekind cut, the note 3 refers to Dedekind, Richard (1872). Continuity and Irrational Numbers. Apparently, 1972 is the date of the English translation, not of the original in German. This seems confirm that Cantor's definition was not the first one.
• In fr:Charles Méray (translated): In 1869 he is the first to give a rigorous construction of the real numbers. This construction is based on equivalence classes of Cauchy sequences of rational numbers.

Similarly, it depends on the Wikipedia article whether the first (ε, δ)-definition of limit must be attributed to Bolzano, Cauchy or Weierstrass.

Could someone provide a clarification? D.Lazard (talk) 18:28, 12 June 2024 (UTC)

I'd hazard that the 1858 date is the erroneous one for Dedekind. Stetigkeit und irrationale Zahlen was published in 1872 [1]. However I think the question of priority is the wrong frame for the construction of the real numbers. One first needed integers (Peano), rationals (maybe Dedekind), infinite sets (Cantor), by which point of course "the real numbers" were already in some sense defined! Tito Omburo (talk) 22:20, 12 June 2024 (UTC)

According to Kline, Dedekind had given lectures in 1858 where he realized real numbers hadn't been properly formalized, but these ideas weren't published until 1872. It also looks like Meray (1869), Heine (1872), Cantor (1871) and Dedekind (1872) all published some constructions of the irrationals in around the same time frame, but its difficult to locate the primary sources. Weierstrass claimed to have presented a rigorous construction in 1859 that was never published. Tito Omburo (talk) 22:39, 12 June 2024 (UTC)

Clarification should be in the form of a reference to a history. Johnjbarton (talk) 22:25, 12 June 2024 (UTC)

There's an issue of publication vs. discovery. See the following (bolding for emphasis):

Dedekind worked out his theory of Dedekind cuts in 1858 but it remained unpublished until 1872.

Weierstrass gave his own theory of real numbers in his Berlin lectures beginning in 1865 but this work was not published.

The first published contribution regarding this new approach came in 1867 from Hankel who was a student of Weierstrass. Hankel, for the first time, suggests a total change in out point of view regarding the concept of a real number [...]

Two years after the publication of Hankel's monograph, Méray published Remarques sur la nature des quantités in which he considered Cauchy sequences of rational numbers [...]

Three years later Heine published a similar notion in his book Elemente der Functionenlehre although it was done independently of Méray. [...] Essentially Heine looks at Cauchy sequences of rational numbers. [...]

Cantor also published his version of the real numbers in 1872 which followed a similar method to that of Heine. His numbers were Cauchy sequences of rational numbers and he used the term "determinate limit". [...]

As we mentioned above, Dedekind had worked out his idea of Dedekind cuts in 1858. When he realised that others like Heine and Cantor were about to publish their versions of a rigorous definition of the real numbers he decided that he too should publish his ideas. This resulted in yet another 1872 publication giving a definition of the real numbers.
— O'Connor, John J.; Robertson, Edmund F. (October 2005), "The real numbers: Stevin to Hilbert", MacTutor History of Mathematics Archive, University of St Andrews

I think this is also covered in some of MacTutor's cited references. So Dedekind is often credited with the first construction in 1858, the first publication is credited to Hankel in 1867, the first publication with a "rigorous construction" is credited to Méray in 1869 or Cantor in 1872 or Dedekind in 1872. — MarkH₂₁^talk 22:44, 12 June 2024 (UTC)

Many thanks (I have fixed the parameters in your reference to Mac Tutor). D.Lazard (talk) 08:29, 13 June 2024 (UTC)

Can someone explain what Riemannian circle is supposed to be?

My guess is that the article Riemannian circle has an incorrect definition; as it is described there it seems like an obfuscated synonym for great circle, which should just be redirected there. But it wouldn't make any sense to call a great circle a "Riemannian circle" instead, so I imagine the term is probably supposed to mean something different instead. However, I don't really have the background or patience to sift through old sources trying to figure out precisely what. Can someone who knows about Riemannian geometry figure out what is going on there? –jacobolus (t) 02:40, 13 June 2024 (UTC)

Just WP:BOLDly redirect. The article has offers references and even if it did the content would be better in great circle. Johnjbarton (talk) 03:01, 13 June 2024 (UTC)

I don't want to do that because my expectation is that Riemannian circle means something different; if so, it would be better to delete the page instead of redirect. However, it would be better still if someone can replace this with a more accurate definition. (Doesn't have to be anything fancy; it's fine if the page remains a stub.) –jacobolus (t) 03:12, 13 June 2024 (UTC)

To me, as defined there, it appears to be an obfuscated definition for the metric space of arc length around a circle. Embedding it as a great circle on a sphere and then using geodesic distance on the sphere doesn't change anything. Also the part about Gromov is described better at filling area conjecture.

Searching Google Scholar for this phrase finds varying definitions:

• This definition, the arc length metric on a closed curve of length $2\pi$
• Arc length metric on any closed curve
• Arc length metric on a closed curve embedded as a rectifiable curve in a Euclidean space
• "A curve in a Riemannian space whose development in a tangent space is a circle"

The first three are not different except for scale, and seem like the majority of uses.

We probably should have an article on the arc-length metric on simple closed curves, and this title seems like a plausible place to put it if it doesn't already exist elsewhere with better content. So my tendency would be to attempt a rewrite along those lines, removing the definition about geodesics on a sphere between points of a great circle except more briefly as the conjectured answer to the filling area conjecture. —David Eppstein (talk) 04:34, 13 June 2024 (UTC)

Rewrite done and moved to metric circle, somewhat more common and less ambiguous. —David Eppstein (talk) 07:50, 13 June 2024 (UTC)

Thanks! –jacobolus (t) 08:52, 13 June 2024 (UTC)

While we're here, is there any place where this topic can be put into context and related to nearby topics? I feel like our collection of circle-related topics are somewhat atomized and not fit together into any particularly coherent narrative, many are incomplete, they don't do all that much interlinking, etc., and we're lacking much high-level overview. We have Circle, Circle group, Angle (but no separate "Angle measure"), Turn (angle), Radian, Arc length § Arcs of circles, Directional statistics, Circular distribution, Circular mean, Periodic function, One-dimensional symmetry group, Trigonometric functions, Fourier series, Root of unity, Cyclic group, Modular arithmetic, .... Some kind of summary should be in a section Circle but that article also has to discuss the way circles fit into other spaces making it a poor fit for substantial expansion in this direction. I'm not sure if the name Metric circle is used widely enough or if that article quite fits as a central place for discussing the use of the circle as a geometric space though.

As a separate aside, should we have an article Periodic interval or the like? We currently don't, but it seems worthwhile (though it overlaps with many of the topics I listed above). –jacobolus (t) 21:11, 17 June 2024 (UTC)

Let's not forget Jordan curve, pseudocircle, and quasicircle as topological forms of circles.

Anyway, in my rewrite I wanted to focus on specifically the one-dimensional compact Riemannian manifolds (a phrase that unfortunately does not turn up much good sourcing). One can find circle-like objects as Euclidean shapes, objects in topological spaces, rings, etc., but I think trying to write a single article about all of them would be too incoherent. —David Eppstein (talk) 21:45, 17 June 2024 (UTC)