Talk:Main Page

New $\LaTeX$ macros for your convenience and our internal consistency
There are a number of new $\LaTeX$ macros which have been developed recently as a result of a lot of discussion some time back which never ended up happening at the time.

They can be found on the page Symbols:LaTeX Commands/ProofWiki Specific, transcluded here:

Others may happen as and when I think of them. --prime mover (talk) 03:48, 25 August 2018 (EDT)


 * This is awesome! Thanks!! — Timwi (talk) 19:27, 27 August 2018 (EDT)

Public outreach
I have found on this site that a certain user was motivated to set up a blog about this site. I know how the story ended, and that setting up a blog was denied. I wonder, to what extent a no-go for the blog was given due to special circumstances? I understand that this additional task requires extra time, while a decentralized option would slow down the process due to opinion incompatibility. I myself would rather work on maths material rather than encourage discussions on a social platform. However, to what extent can one's contribution here be advertised elsewhere, barring appropriate disclaimers? --Julius (talk) 18:59, 9 May 2020 (EDT)


 * Which user was that? --prime mover (talk) 01:21, 10 May 2020 (EDT)


 * Thinking further on this, discussions on a social platform should not be a bad thing. If there's something wrong with the site, that's how we'd find out. We have a poor reputation everywhere (which is obviously understandable), and it would give would-be contributors the chance to rant about how bad we are.


 * Though I seem to recall you taking some issue with StackExchange, is frequently mentioned there and got quite a positive review when users were asked about its reliability. Not to mention the fact that it is usually quite high up in google searches. I think this website is much better rated than you realise - though very few of these viewers go on to regularly contribute. It's expected that the majority of viewers will just find what they're looking for, and leave, (as is the case with Wikipedia) but it'd be useful to find out if there's any specific reasons people pass on contributing. Caliburn (talk) 09:07, 10 May 2020 (EDT)


 * Ouch. Yes, I've been looking at, and it's seriously embarrassing. If only the people who have found all these mistakes had told us about them. This is appalling. --prime mover (talk) 09:28, 10 May 2020 (EDT)


 * Here's an example:


 * Wow I do not like the way those statements are worded at all. I also think it is a grave mistake to package the statement that ∅,S∈τ with empty unions and intersections. – Cameron Williams May 5 at 12:23
 * @CameronWilliams: I agree it's not pretty (in fact, the whole concept of ProofWiki is not very pretty at all). But logically it's correct. – Lee Mosher May 5 at 12:32
 * @LeeMosher 100% agreed for sure. I've had my issues with ProofWiki in the past but this one blows those out of the water for me. Pedagogically this drives me up the wall haha. – Cameron Williams May 5 at 12:40


 * which says it all, really. Basically, then, is rubbish. --prime mover (talk) 09:51, 10 May 2020 (EDT)


 * There are only a few hundred results down that link. Let's say that each of them involves 10 distinct people (which is very generous). Assuming all of them have problems with this site, this would sum up to a few thousand negative comments over several years. Now based on Alexa ranking we have roughly 6k-10k visitors per day. So, we are drawing conclusions from a severely misrepresented sample. Instead, we can turn this into a constructive thing. Once in a while scan the field, look for a misunderstanding, if possible, fix it, and report it there. This way the question is addressed, and people notice that feedback works. As for others who do not like our style - this is not a theorem beauty contest, and they are welcome to look elsewhere. 6k per day is a good achievement, especially for such dry material.--Julius (talk) 11:19, 10 May 2020 (EDT)


 * I tried to get involved with StackExchange once. You can't actually contribute to a thread unless you have earned so-much reputation. To do this you have to ask questions. So I tried that. I posed up a question, had no replies, no messages, nothing. When I went to take a look at it a week or two later, I found someone had deleted it. At that point I decided life was too short to waste time on it. If it were only a little more accessible, then maybe it would be worth doing, but I'm obviously not of a high enough mathematical level for it. So I can fix stuff, but it will be up to someone else to post up a reply to SpackExchange. --prime mover (talk) 11:40, 10 May 2020 (EDT)


 * Note we already have a facebook group, as well as a twitter account. Nobody really seems to use them much. --prime mover (talk) 04:19, 10 May 2020 (EDT)


 * If someone (admin or someone who could be made an admin) is more social media savvy than me, they're welcome to post on our social media accounts. --Joe (talk) 11:44, 11 May 2020 (EDT)


 * I had this user in mind.


 * He had a lot of "good ideas" including incorporating this site into another, affiliating us with a university, selling us to a company, completely changing our presentational style and philosophy, and so on. We eventually had to disallow his continued contribution, as (among other things) he would not stop taking on colossal restructuring tasks and not following through, leaving the site in an inconsistent state, requiring a large amount of work of other people to put it right.


 * As for the comment about the blog, as I say, I started writing a blog but was told I was not allowed to, so I stopped. Don't know what the problem is myself, but I like my internet access and I didn't want it to stop just because of the whim of some fascist. --prime mover (talk) 05:52, 10 May 2020 (EDT)


 * Who was this down to? Sounds like an extremely counterintuitive move. This is a public wiki, we're not exactly sworn to secrecy. Caliburn (talk) 09:16, 10 May 2020 (EDT)


 * Someone on Quora, I believe. Something about copyright and attribution and so on, and the fines that can be imposed upon violators. --prime mover (talk) 09:28, 10 May 2020 (EDT)


 * It sounds weird that someone had a problem with a blog and not with the site itself. I mean, this site generates small income to keep the server running. Maybe low traffic actually helps us to avoid big fish. Either way, what can one man do?--Julius (talk) 11:19, 10 May 2020 (EDT)


 * I don't follow the issue with the blog. Who's not allowing who to have a blog? From ProofWiki's prospective a blog is fine so long as our license is not violated. I'm also not against starting/hosing our own blog, but we'd need to have people actually want to contribute to it. --Joe (talk) 11:44, 11 May 2020 (EDT)


 * As I say, I did start a blog at one point as a subpage of my user page but got told to take it down, so I did. That would have been back in, I dunno, 2012 or 2013 or something. It might have been because it was substandard or something. I'll see if I can reinstate it if you like. Don't know how it would increase site visibility though. --prime mover (talk) 12:04, 11 May 2020 (EDT)


 * I found your old blog and tried to pick though past discussions. I think this was around the time we were trying to figure out our format. Also maybe not all your blog posts were purely mathematical in nature :). Also, to be honest, that was a long time ago. Feel free to blog about whatever (within reason obviously) you want on your blog subpage. With respect to the above mentioned user. I suspect the primary issue (though I don't remember) was that we often get new people who don't stay around who want to totally rearrange everything. At this point in time I'd be happy to start of official blog assuming: primve mover and Lord_Farin agree, and we have someone(s) willing to write. --Joe (talk) 12:44, 11 May 2020 (EDT)


 * If anyone wants to start a blog, let them feel free, if they have something clever and interesting to write on a regular basis. I've looked back at my own effort and I can't see anything there particularly worth enshrining or even preserving, but if someone feels like rattling off a regular essay I'm not stopping them. --prime mover (talk) 17:59, 11 May 2020 (EDT)


 * I'm a bit confused too - ProofWiki's license is incredibly flexible (basically, do whatever you want as long as you attribute it and release it under the same license) so I can't see how a problem with licensing or the like would arise. I'd be happy to contribute. Caliburn (talk) 11:47, 11 May 2020 (EDT)


 * I believe that at the moment people simply cannot distinguish us from the noise. Those at higher echelon, like PhD students simply cannot find their stuff here, while undergrads often are not aware of our existence, or they consider us to be yet another personal wiki hobby. I am a PhD student in theoretical physics, and none of my colleagues have heard of this site. I even had to stress that we are not a branch of Wikipedia.


 * Considering how linear maths is (everything you learn builds upon what you've learnt before), to start adding in substantial amounts of research material you'd need a good amount of graduate material, for which you'd need a good amount of undergraduate material. Due to our lack of contributors, sometimes these foundations just don't exist and indeed users with the knowledge to add such content are scarce. You mention theoretical physics, but even our coverages of basic classical mechanics and electromagnetism are microscopic, so it'd take a lot of time and many dedicated contributors to build it up enough to have PhD-level content on there. Most of our problems boil down to a lack of contributors, for which I agree outreach can only be a good thing. Caliburn (talk) 09:07, 10 May 2020 (EDT)


 * I've started on both E-M and QM but the work is slow because that's what I'm like. I get to a concept that I think: that's too much like hard work I'm going to do something easier, and I go and do something else. Takes a while to get back to it. --prime mover (talk) 09:10, 10 May 2020 (EDT)


 * Yeah that's fair enough, these things take time and with a lot to do you can easily find yourself skipping between things. It's not really anything we're doing wrong (that I know of) - it's just a lack of active contributors. Caliburn (talk) 09:16, 10 May 2020 (EDT)


 * Then I also had a short conversation on Reddit. There were few responses, most of them telling "good job", and one with more basic comments about the website (e.g. what does $\leadsto$ mean, and why we don't have a link to a dictionary of such symbols on the main page). I think that regularly throwing out something on FB or TW, should it be more serious or just a shared meme, or referring to us in YouTube comment section in a reasonable way would show the world that we are alive (20000 theorems don't matter if nobody knows, that there are 20000 theorems).


 * I can't comprehend the mindset that claims (in the context it is seen) that they don't know what $\leadsto$ means. I mean, we make a pretty good job of declaring our definitions as we use them (this being one of the main bones of contention), but it becomes a bit much when our usefulness is seen to be limited because there are people who not only don't know, but can't even work out, what this most basic bit of mathematical punctuation means.


 * The reason why we don't have a link to a dictionary of such symbols on the main page is because we honestly never dreamed that mathematicians would need it. Although we say we include links to definitions of all terms used on every page, we never went to the detail of defining every single mathematical symbol used, on the grounds that some stuff we think everybody knows. But if it is indeed the case that mathematicians do need such an aid to understanding, then I expect it will fall to someone to have to take on this reponsibility. --prime mover (talk) 06:01, 10 May 2020 (EDT)


 * Also, there are more statistics-based tools to gauge this, like alexa ranking. Even on https://www.alexa.com/siteinfo/proofwiki.org I can see which websites we are giving up to, which keywords are missing here, and what traffic comes from various countries. Working on these numbers would improve popularity in an indirect way. --Julius (talk) 05:18, 10 May 2020 (EDT)


 * I'll leave that up to someone else, because, seriously, I don't care in the tiniest slightest. It is acknowledged that this site is of limited worth, and the only thing that will improve it is a complete redesign from the ground up (or even, as many commentators say, deleting the thing and completely removing it from the web), but for those of us with limited opportunity to interact with the outside world, it gives us something to do to fill our twilight years. --prime mover (talk) 06:01, 10 May 2020 (EDT)

Sorry I didn't respond earlier to this discussion, I completely missed it. If someone has ideas with what we should doing better for community outreach and wants to help, please let me know. --Joe (talk) 11:46, 11 May 2020 (EDT)


 * I made a Reddit community. If any of you use it, I can make you a moderator. It will take some time before it starts looking solid. In the nearest future I am planning to design it according to some simplified version of more established math communities. As for Twitter, I saw that Wikipedia simply puts fun facts on theirs. Not sure how to make this work for math. Since messages there are supposed to be brief, I guess a more humorous approach would work. Otherwise, I believe that we simply have to visit various forums and YouTube comment sections and refer to this site whenever there is a genuine relevance. I believe that since we would start this from scratch, we should simply look for channels used by more authoritative sites.--Julius (talk) 18:21, 11 May 2020 (EDT)


 * Good idea! Will be lurking on there. (supposed I should reduce my footprint so I'll be commenting under /u/PWCaliburn) Caliburn (talk) 18:34, 11 May 2020 (EDT)


 * I lurk on Reddit as /u/joejoebob —Joe (talk) 20:14, 12 May 2020 (EDT)

A bit late to the party, but please know I've caught up with all the above and I'm totally fine with any blog/Reddit/YT/FB presence as long as it is made in good faith. &mdash; Lord_Farin (talk) 13:47, 24 May 2020 (EDT)

Stackexchange
I've now been through the list of issues raised at Stackexchange. (Or at least, most of them, there was a lot.) I've fixed the obvious mistakes, but I'm not sure how to handle all of them. But issues fixed or not, there's no denying that we are very much considered substandard, to such an extent that it is believed we should not exist.

Is this the point where we decide to wrap it up and admit there's not much point trying to keep this site going? --prime mover (talk) 01:11, 11 May 2020 (EDT)


 * No, it isn't. For each successful fix the complaint is resolved and the old negative conclusion becomes irrelevant. I also find any constructive comment (or the link thereof) from elsewhere copied on the respective discussion page useful. Then everybody can see what is there to be done. --Julius (talk) 02:26, 11 May 2020 (EDT)


 * I think we can address issues that make sense to us. I'm sure we'll never be able to make everyone happy. --Joe (talk) 11:47, 11 May 2020 (EDT)


 * SE is also a place where everyone can disagree with whatever they want because their own style is supposedly superior. But without the responsibility of actually making something worthwhile that could be argued about. In that sense high-level criticisms are often not much more than a matter of taste. &mdash; Lord_Farin (talk) 13:47, 24 May 2020 (EDT)

Terminology: Real Number Line or Real Number Space?
For reasons of clarity of understanding, I am trying to emphasise the difference in concept between:
 * the real number line, which is the set of real numbers under the usual ordering

and:
 * Real Number Line is Metric Space, where $\R$ has the Euclidean metric imposed on it (which I never got round to defining as a definition page)

and:
 * the real number space, which is the topological space induced by the Euclidean metric $\map {d_2} {x, y} := \sqrt {x^2 + y^2}$ on the real number line.

I'm at a loss as to what to call these things. I have found that at one point I started to use real number space without any authoritative backup, and it can of course be confused with for example $\struct {\R^n, \d_2}$.

There's a good reason for separating them out: because for example:
 * you can define a metric space with the real number line with different metrics on than just the Euclidean one

and
 * you can define a topological space with the real number line without reference to a metric at all (for example, the real number line under the discrete metric, as is done for example in Double Pointed Discrete Real Number Space is not Lindelöf‎.

I want to refactor this whole area so as to be consistent, and merge a number of pages which are the same, for example Definition:Real Number Space and Definition:Euclidean Space/Euclidean Topology/Real Number Line but I need advice as to what to name these things for maximum clarity and minimum unwieldiness.

Advice? --prime mover (talk) 06:05, 31 May 2020 (EDT)


 * Further thoughts: Definition:Real Number Line, Definition:Real Number Line (Metric Space), Definition:Real Number Line (Topological Space) and similar for the plane and $3$-space? --prime mover (talk) 07:25, 31 May 2020 (EDT)


 * Actually I'm not sure about Definition:Real Number Line (Topological Space) because that doesn't leave it open for different topologies to be applied.


 * Could also go with Definition:Real Number Line (Euclidean Space) but that doesn't distinguish between that and the RNL as a Metric Space.


 * Yes I know they are the same thing, but as I say, I want to leave it open for different metrics and different topologies, and lumping the real number line, the RNL as a metric space under the Euclidean metric and the RNL under the topology induced by the Euclidean metric leaves us in danger of not being able to establish the specific chain of proofs which demonstrates rigorously that they *are* all the same thing. We also want to make sure we can provide the framework to equally rigorously prove that an "open set" in the context of a topological space is an "open set" in the context of a metric space is a union of "open intervals" in the real number line. Yes I know we've already got that chain of proofs in place, the point is the framework into which it is included is unstructured and duplicated. Similar applies to compactness. --prime mover (talk) 08:08, 31 May 2020 (EDT)


 * My suggestions would be Definition:Real Number Line, Definition:Real Number Line (Euclidean Metric) or Definition:Real Number Line (Euclidean Metric Space), Definition:Real Number Line (Euclidean Topology) or Definition:Real Number Line (Euclidean Topological Space) --Leigh.Samphier (talk) 10:04, 31 May 2020 (EDT)


 * Another thought: Definition:Real Number Line, Definition:Real Euclidean Line, Definition:Real Euclidean Topological Line --Leigh.Samphier (talk) 10:17, 31 May 2020 (EDT)


 * Or Definition:Real Number Line under Euclidean Metric, Definition:Real Number Line under Euclidean Topology -- but I wonder what the correct preposition or conjunction ought to be: "under", "with", "endowed with", "bearing", something along those lines, maybe. The "main" page perhaps ought to be Definition:Euclidean Space/Euclidean Topology/Real Number Line with a redirect, and similarly for the metric space. Then the 2d and 3d versions can be implemented similarly. --prime mover (talk) 15:28, 31 May 2020 (EDT)


 * In differential geometry, one usually endows or equips a manifold with a metric. In colloquialisms "with" is used very often, i.e. "A manifold together with a Riemannian metric is called a Riemannian manifold". I cannot remember anyone use "under" or "bearing".--Julius (talk) 17:42, 31 May 2020 (EDT)


 * "with" works for me then. When I'm less tired I'll go with that.

--prime mover (talk) 18:22, 31 May 2020 (EDT)


 * I think "with" is standard. Its typically "real numbers with usual topology". --Leigh.Samphier (talk) 20:07, 31 May 2020 (EDT)


 * Another concern I have is with the term "Euclidean Space" which currently refers to the "Metric Space". The term "Euclidean space" refers to the fact that it satisfies the axioms of "Euclidean Geometry" which to me suggests that is should at least refer to the "Inner Product space". Subtly different things. We should think about how to distinguish the "metric space", the "normed vector space", the "inner product space" and the "Hilbert space" --Leigh.Samphier (talk) 20:07, 31 May 2020 (EDT)


 * Okay scratch "Euclidean Space", replace it whenever used with "Euclidean Metric" and "Euclidean Topology" or "Euclidean Norm" etc. unless used to encompass all concepts. OTOH there's already a refactor request for Definition:Euclidean Space, which should be either a (better crafted) top-level parent page for a complicated system of transclusion (hard work) or a disambiguation page (less optimal). --prime mover (talk) 01:25, 1 June 2020 (EDT)

So far so good. We now have Category:Real Number Line with Euclidean Metric and Category:Real Number Line with Euclidean Topology, and all references to "Real Number Space" in the context of the real number line have been removed.

I suppose the next exercise would be to do the same for "real number plane", although I can see that Definition:Rational Number Space is also crying out for the same treatment.

"Real number plane" has the added complication that there is more than one way to structure such a plane, standard Cartesian coordinates, oblique Cartesian coordinates, polar coordinates ... so I will also have to take into account Definition:Cartesian Plane and all subsequent entities. Not today, I need a break. --prime mover (talk) 09:08, 1 June 2020 (EDT)

Linking books to pages
Is it allowed to endow chapters in the book section with a link to a page, say, corresponding to the first entry from the given chapter? This would make the navigation easier.--Julius (talk) 15:45, 25 June 2020 (EDT)
 * The trouble with giving a page number is that a different printing may result in the citation referring to the wrong page. It should not matter. A reader who is following a book by means of the previous and next links will already know what page he is on. And if the user is not able to follow the book, then it will not matter anyway what page the result is on. In most texts, the chapter, section and item number (if given) should be enough to pinpoint the page anyway, to a pretty good extent. A book which does not have section numbers, and no means of identifying the location of the citation than a chapter number, would probably be quite rare.
 * In the interests of consistency, if we started giving page numbers for one work, we would need to do it for them all in order to regain consistency, and I just don't have the steam. --prime mover (talk) 15:57, 25 June 2020 (EDT)


 * I should have put this differently. Here is an example (compare with this):

...

Preface

1. Normed and Banach spaces


 * 1.1 Vector spaces


 * 1.2 Normed spaces


 * 1.3 Topology of normed spaces


 * 1.4 Sequences in a normed space; Banach spaces

...


 * Here every section is linked to the first item I decided to extract from the given section. No page numbers are involved. This would be a one-way trip (book $\to$ page on ProofWiki) unless we develop a template to go back and forth between different printings. Table of contents format visually would not be affected very much, but it would be more interactive. --Julius (talk) 06:26, 26 June 2020 (UTC)


 * Oh yes I see what you mean. Yes, we've done that for some books. It's just tedious to do, and you can only really do it after you have processed the book. --prime mover (talk) 08:04, 26 June 2020 (UTC)

Problem with Chessboard
I've noticed that the file File:Chessboard480.svg is not being rendered properly at the moment. It comes onto the page as a black square:


 * [[File:Chessboard480.svg]]

But if you click on its image in the actual image itself (e.g. in the "Original file" link) it appears as it should:

https://proofwiki.org/w/images/d/d7/Chessboard480.svg

You can see an example of a black chessboard on page Definition:Chess/Chessboard.

Has something been changed in the rendering software?

--prime mover (talk) 13:08, 3 July 2020 (UTC)


 * I've fixed this for now by replacing File:Chessboard480.svg with File:Chessboard-480.png which seems to work as well, except for in the doc file for Module:Chessboard, which seems to have rendering problems. --prime mover (talk) 21:32, 3 July 2020 (UTC)


 * There are still a couple of chess pieces whose svgs are not being rendered properly: [[File:Chess qdt45.svg]] and [[File:Chess rdt45.svg]]


 * ... and those dodgy chess pieces have now been fixed. The svg source code was faulty. For some reason they worked well enough before this latest recent change. The rendering s/w must be more stringent. No worries. --prime mover (talk) 22:00, 3 July 2020 (UTC)


 * Probably related to a recent server upgrade I did. I'll look into it later this week. --Joe (talk) 00:50, 4 July 2020 (UTC)
 * This should be resolved now. You'll need to be sure to clear your browser cache. --Joe (talk) 16:12, 5 July 2020 (UTC)


 * And it is done. Thank you. --prime mover (talk) 19:19, 5 July 2020 (UTC)

Formulæ displication sans “client-side scripting?”
Presently, it seems to me that displication of formulæ from requires “client-side scripting.“ A user might prefer (possibly æsthetically worse) formulæ displication sans “client-side scripting.” Indirigible (talk) 00:50, 29 July 2020 (UTC)


 * That would be nice to have, though I haven't looked into options for this yet. --Joe (talk) 15:17, 3 August 2020 (UTC)

Cartesian Coordinate Systems
I'm in the process of refactoring the Definition:Cartesian Coordinate System and related work. The original way it was done was by lots of transcluding, which made a lot of large unwieldy pages. I'm simplifying it down so as not to do so much transclusion, and instead using Also see to bring in the various cases (in particular, Cartesian Plane and Cartesian 3-Space).

This was made necessary by the fact that I am embarking on the work to bring 3-dimensional analytical geometry into play, as well as some of the basic physics of ordinary space. All of this needs a fairly solid underpinning of 3-d coordinate geometry. When we start to address curvilinear coordinates there will be another layer of complexity, so we need our ducks in a row here. --prime mover (talk) 22:36, 11 October 2020 (UTC)


 * It was all so clear in my mind this afternoon. Now I've lost it again. I'm going to abandon my mission to rationalise the work on vector spaces, there are too many short-sighted and oversimplified interpretations of what a vector is and I can't think my way through them all. It will have to wait till I've slept. --prime mover (talk) 16:20, 21 October 2020 (UTC)

Refactoring around Limits and Continuity on real number line
I have simplified the structure around Definition:Continuous Real Function at Point and Definition:Limit of Real Function. The 2 definitions for continuity that we had have been moved into 2 different definitions for limit points, and the definition of continuity is now defined solely in terms of limit points.

Hence the sources which did not start with the concept of limit point, but merely defined continuity directly in the language of the epsilon-delta of the limit definition, may now not be accurately reflected in the structure of the pages now here. Not sure how much this matters. Probably best to invoke the Limit Point definition 1 in the source citation flow somewhere, despite the fact that the actual concept of a limit may not actually be included in those source works. --prime mover (talk) 21:37, 15 November 2020 (UTC)

Refactoring Inverse Hyperbolic Functions
I have restructured and renamed as necessary the pages concerning inverse hyperbolic functions as follows:

a) I have been sternly schooled as to when it may or may not be appropriate to use, for example, $\cosh^{-1}$ and $\arcosh$. Current thinking is that $\cosh^{-1}$ is "not nice" because it does not adequately allow for the fact that (in the real case) it has two branches, and the same with $\sech^{-1}$. Hence for the usual case where the single branch is required, $\arcosh$ is proper.

b) There is no such thing as "hyperbolic arc-functions". They are properly "area hyperbolic functions". Many sources denote them as, for example $\operatorname {arccosh}$ and so on, but on we are not going to do that.

I have written a number of explanatory "also known as" pages which lay all this out and present it on the appropriate pages.

It remains to amend the pages which reference these functions, which I will proceed to address in due course.

This was all brought on by my attempts to resolve the last few remaining incomplete proofs of primitives in this area. I confess I had to resort to asking questions on Math StackExchange over the last couple of weeks. --prime mover (talk) 13:28, 3 January 2021 (UTC)

Complete Frustration
I cannot figure out specifically where to enter a question.

I contributed an entry to Proofwiki which required an explanation. That entry did not appear on the edit page.

Below is the challenged statement:

"The Collatz sequence consists of instances of a single odd term interspersed by one or more even terms. The sequence grows if and only if there is only a single even term between two successive odd terms.

Consider the following sequences: 31→94→47	2051→6154→3077		The ratio between the last and first terms is 1.5 41→124→62→31	233→700→350→175		Here the ratio is 0.75 2429→7288→3644→1822→911		Now it is 0.3785 I For a sufficiently large set of 3 term sequences, approximately 50% will have one even term and 50% will have two or more even terms.

Question.png This article, or a section of it, needs explaining, namely: What is the reasoning behind that last statement? You can help Pr∞fWiki by explaining it. To discuss this point in more detail, feel free to use the talk page.

If you are able to explain it, then when you have done so you can remove this instance of from the code.

I explain it thus. In a CS you divide the "3x+1 term by 2 until it becomes an odd number. 50% of the even terms are divisible only by 2. The other 50% are divisible by 4,8 16, ... . Everybody should know that? Senojesse (talk) 00:12, 23 January 2021 (UTC)


 * Nobody else seems to have problems navigating. What browser are you using? You should see a link at the top of every page you navigate to saying "Discussion". You will find that link at the top of the Collatz Conjecture page.


 * Half of all even numbers are divisible only by 2. The other 50% are divisible by 4,8 16. Yes, no arguments there.


 * But what you have not proved is that half of the even entries in a Collatz sequence are divisible only by 2, and that half of the even entries of a Collatz sequence are divisible by 4.


 * It is not true of all sequences. Take for example the sequence: $\sequence {s_n}_{n \mathop \in \N} := 2^n$. It goes: $1, 2, 4, 8, 16, \ldots$ and it is trivially noted that only the second term is divisible only by 2. The subsequent terms are all divisible by $4$.


 * So if it not true of all sequences, then it is not necessarily true of a Collatz sequence. It is not true, for example, of the Collatz sequence which starts from, for example, $2048$. Investigate that sequence and work out exactly how many are divisible by $4$. --prime mover (talk) 08:04, 23 January 2021 (UTC)

next direction
I took another hack at completing some of the topology work left undone from way back when. Been working through the questions in Mendelson, but they are so repetitive and laborious I have completely lost interest, and can't bring myself to continue to finish off chapter 2.

I'm at a loss as to what to do next. Anyone any ideas as to what they think I ought to do so as to progress this site? --prime mover (talk) 22:42, 28 January 2021 (UTC)


 * I would suggest something from differential geometry or smooth manifolds. Not only less abstract, but also quite important in studies of physical systems, ranging from the motion of constrained particles to general relativity and black holes. --Julius (talk) 23:14, 28 January 2021 (UTC)


 * To get there I have to go through Topology. Okay, I'll see what I can do, I have that Auslander and MacKenzie book, but I don't know what I'm doing yet. --prime mover (talk) 07:04, 29 January 2021 (UTC)


 * Two books that I found accessible were the books:
 * Book:John M. Lee/Introduction to Topological Manifolds
 * Book:John M. Lee/Introduction to Smooth Manifolds
 * --Leigh.Samphier (talk) 10:52, 29 January 2021 (UTC)


 * This is puzzling: it seems that I transcribed the contents list for this myself over the course of the last 6 years or so, which means I must have a copy of this on my shelf -- but I can't find it anywhere, and don't actually remember ever seeing it. As it is Springer, I'm expecting it to be a big yellow thing. Can't find it anywhere though. --prime mover (talk) 12:25, 29 January 2021 (UTC)


 * A couple of things that I've thought might be of interest to someone visiting the site and might provide a break from the formality of most of the site:
 * Some expository pages on writing mathematical proofs and how mathematical proofs are written on.
 * Some pages on false proofs. These are always entertaining and illuminating. Although keeping false proofs separate from real proofs may prove to be a challenge.


 * Given other comments you've made, I suspect you aren't interested in doing another compendium or two, but I thought I would mention:
 * Proofs from THE BOOK by Martin Aigner and Gunter M.Ziegler: This is 268 page book that covers a broad range of topics in mathematics. I'm sure most of the theorems will already exist on, but there may be some new proofs.
 * The Princeton Companion To Mathematics edited by Timothy Gowers: This is a huge book and is encyclopedic rather than a typical mathematics texts.


 * I'm no expert on what people want to see on this site, and I'm certainly not interested enough to write these pages, so I understand if these ideas end up in the bit bucket.
 * --Leigh.Samphier (talk) 10:52, 29 January 2021 (UTC)


 * Generally speaking, my bookshelf is populated with findings from my local thrift shop (it's a good one, it specialises in books, and it has been used by many mathematics professionals to offload their own shelves when they have retired, or whatever -- there are some quite interesting names in the flyleaves of some of them). The result is that the collection is eclectic and varied, but completely untargeted. It has been accumulated haphazardly. The few books I *have* acquired intentionally, by ordering them online or asking for them for xmas, have often been nowhere near as good as the ones I've got "by accident" (that appallingly stodgy Mendelson comes to mind). An exception is when I went to Montreal for a job a few years ago and found a university bookshop in town with two shelves crammed with Dover reprints, which I practically cleared over the course of 2 months).


 * I won't have you dising Mendelson. It was the first book that I read on General Topology :-) --Leigh.Samphier (talk) 13:36, 29 January 2021 (UTC)


 * Next to W.A. Sutherland's "Introduction to Metric and Topological Spaces" (which is where I started) it does not compare very well. Both tread the same sort of ground, but Sutherland is far better explained; he provides ample motivation and historical notes, and far more engaging and stimulating exercises. He doesn't go into such technical detail as Mendelson does as regards indexed families, however, and does not go into the subject of neighborhood spaces at all (for which I am somewhat grateful). --prime mover (talk) 15:51, 29 January 2021 (UTC)


 * Hence a lot of the classics which I am "supposed" to be using as a source work, by people who have better knowledge of what is right for a mathematician to learn from, are just not there on my shelf. I look out for them when I am out and about (which I have not been able to do for the last year, wonder why that is?) and when that happens I will be able to cover them, or at least, to start work on them.


 * Is a list of Books Wanted maintained somewhere? --Leigh.Samphier (talk) 13:36, 29 January 2021 (UTC)


 * I will make the effort to get the works you suggest, although as they need to be ordered online I will need to get clearance from the boss first, and that may take some time. I do have "Riddles in Mathematics" by Northrop containing some of the more interesting paradoxes, most of which are couched as "false proofs", I may make a start on them.


 * Another thought along similar lines, having just finished Matt Parker's Humble Pi, is a compilation of mathematical mistakes due to the incorrect use of a theorem/proof. Of course the mistakes has to have a comic or tragic result. --Leigh.Samphier (talk) 13:36, 29 January 2021 (UTC)


 * The thought of writing an exposition on "how to write a proof" has never occurred to me to do. Mainly this is because I wouldn't know where to start.


 * A quick search on Mathematical Proof will turn up books about how to write mathematical proofs along with other resources that might provide some inspiration if you are so inclined. --Leigh.Samphier (talk) 13:36, 29 January 2021 (UTC)


 * The Auslander and MacKenzie "Introduction to Differential Manifolds" was one of the books I got on my Montreal trip, and I haven't really opened it until just now, because it's a lot more advanced than the simple stuff which I can only just get my head round as it is.


 * ... which I'm going to have to abandon at this stage because I'm not 100% sure what some of the objects are that it describes on page 1. looks like I need to dig into a different book. --prime mover (talk) 21:25, 30 January 2021 (UTC)


 * What about differential geometry? It should be easier. I've noticed that you tried Book:T.J. Willmore/An Introduction to Differential Geometry and Book:C.E. Weatherburn/Differential Geometry of Three Dimensions/Volume I but did not progress much. Same issues?--Julius (talk) 11:28, 31 January 2021 (UTC)


 * Yes, I've started again with Weatherburn, as suggested. I think I must have stopped at some point because I was uncomfortable with the coverage of vector spaces, and wanted to rationalise it, then never got round to continuing with it again. Reconciling the different approaches to vectors (i.e. as "arrows in space" with "representation as coordinates in a cartesian frame" with the abstract concept approaching it from module theory) proved a bit challenging over the summer and I left it to do something else. Never got back to it. Suppose now is the time. --prime mover (talk) 14:26, 31 January 2021 (UTC)

Different approach
Some time ago it was decided (don't know who it was) that the proper process of documenting a source work was to do it completely. That means, all examples and all exercises. I have done my best to do a meticulous job on this, going back through previous works and re-processing them, filling in the gaps, but some books have exercises and examples which are just so tedious and make-work that I wonder what the point is. It has got me to the stage where I am getting fed up with it, to be frank. I asked the question on MathStackExchange and got the answer that the examples and exercises are there for a reason, and there is never an excuse to skimp the work because they are there for a reason, the book was written by a professor, and so by seniority he has the superior position, and my attitude sucks and it is doubtful that I would be allowed to remain at whatever school I was at. Unfortunately this has resulted in me not doing such a good job of documenting these examples and exercises that they deserve, and this site is starting to suffer badly as a result. So: do I have the authorisation to continue to just casually flap my way through stuff, picking the bits I like and not giving a damn about completeness, or is it on me to pull myself together and get on with the job?

What say we relax that rule? On what grounds are we justified in skimping the exercises? Or are we justified in so doing at all? I wanna go out and play on my bike. --prime mover (talk) 07:04, 29 January 2021 (UTC)


 * I think we can relax it. In my opinion, the most important stuff are definitions and theorems, followed up by some examples stated in the main part of the text. Exercises sometimes are too niche, and coming up with a nice title for them is a headache. We are here not to take a test. We can have our reasons.--Julius (talk) 07:32, 29 January 2021 (UTC)


 * Is what I've done so far okay, or have I got to improve, do you reckon? For a start, I've been told my attitude towards my elders and betters needs to improve, what about things like my attention to detail and accuracy of presentation and stuff? --prime mover (talk) 08:48, 29 January 2021 (UTC)


 * I also think we can relax the rule you speak of. Skimping on the exercises won't/shouldn't create holes in the main body of the work, so if skimping means you retain interest in doing what you are doing then so be it. And who is going to know? And if they care enough they can add the exercises. To me, the purpose of the books is to provide a source for the main theorems being proved. Every book has diversions that were of interest to the author at the time, but will end up as dead ends at some point in the future. I don't think it is necessarily important to include the entire book. Also, I doubt that any course given that uses a particular book will cover the entire book. And the author of a book has made a decision about what to include about a topic and what not to include. This is done because there is a lack of resources to include it all. And this is the reason you may want to skimp on exercises. --Leigh.Samphier (talk) 10:52, 29 January 2021 (UTC)

Collatz Conjecture/Supposed Proof again
Can somebody else take the time to take a look at this and add their thoughts? I'm afraid my patience is so limited that I can no longer do so without resorting to a style of discourse which would be termed "immoderate".--prime mover (talk) 08:33, 30 January 2021 (UTC)


 * Conjecture: The CS chooses its elements from the set of all positive integers.

Senojesse (talk) 19:10, 2 February 2021 (UTC)


 * I don't think there is much to say. Is it an interesting observation? Probably so. Is it a proof? Definitely no. I can pull out few dozens, if not hundreds, of attempts to prove Riemann hypothesis. Even though none of them are accepted, they have a logical structure where there is a sequence of seemingly connected "lemmas" and "theorems". This statistical approach proves nothing in the same way as all zeros of zeta function that lie on the critical line that we know of so far. But surely it can inspire to think of some kind of statement involving limits like it has been done for the prime counting function. Until such a rigorous treatment is presented, this stuff should not have its own page. I don't have an issue if anyone wants to try something as long as they keep it in their own sandbox. If this sounds too harsh, the author is more than welcome to post the results in vixra, where everyone can post their discoveries without the dangers of peer review. --Julius (talk) 10:35, 30 January 2021 (UTC)


 * I agree with Julius and would add that is not the place for original work. I have reviewed the page and added comments. --Leigh.Samphier (talk) 11:33, 30 January 2021 (UTC)


 * I'd qualify that -- original work is fine, as long as it is correct and rigorous. Many of the proofs of things have had to be carved by the contributors themselves, because it has been difficult to find their proofs online.
 * I would also point out that Westwood's Puzzle is entirely original, and (as far as anyone on knows) appeared here first. --prime mover (talk) 11:48, 30 January 2021 (UTC)


 * Point taken. But original work (i.e. unsourced work) needs to pass a higher level of scrutiny --Leigh.Samphier (talk) 12:10, 30 January 2021 (UTC)


 * Most of us have the practice of preparing pages in our own sandbox until they are good enough to publish. Would Senojesse be prepared to move the page to their sandbox until it has achieved the rigour and presentation style expected. --Leigh.Samphier (talk) 12:41, 30 January 2021 (UTC)


 * He's having difficulty getting to grips with how to use a wiki, so it may not be possible for him to set up a sandbox. --prime mover (talk) 13:59, 30 January 2021 (UTC)

General overview of Vector Analysis work
I have started working through the vector analysis stuff, doing a rigorous infodump of the remarkably intense little book of, delightfully easy to understand, and covers some important fundamentals that we either hadn't covered or had taken for granted. During the course of doing this, I am making an attempt to rationalise the existing material on here, which has the side-effect of allowing the focus to be mainly on the conventional $3$-dimensional Cartesian real space.

Clearly this is a special case of the general multi-dimensional case in the general vector space. However, I believe that it would not at this stage be helpful to try to make the exposition as general as can be. This is the context in which most of applied maths and mechanics is concerned with, and in order to cover classical physics (including electromagnetism), this is where we need to start.

I would appreciate other sets of eyes on this work as I go through it stage by stage over the next however-long until I get stuck again (or bored), and greatly appreciate someone else filling in some of the points which have been glossed over (which is mainly because Hague has gone into no further detail himself -- it's a tiny book, 120 pages of A6) and perhaps commenting wherever his approach is old-fashioned and outdated.

This is all stuff which I did some decades ago in my electrical engineering degree, and later in my MMath. The results themselves are memorable enough -- what I am having difficulty making truly sure of is the rigorous proof of equivalence of the general "vector space" approach, and the "vector quantity" approach (magnitude and direction). --prime mover (talk) 23:35, 15 February 2021 (UTC)


 * As far as I am concerned, everything looks fine, and I agree that we can postpone the more general approach. This way we will understand what we are about to generalize. As for equivalence, I understand that we are mostly working with real scalar- or vector-valued functions, and they exist in 3d Euclidean space over real numbers with Euclidean norm. At least to me, this sounds sufficient to start state all of this as a special case of a vector space over a division ring equipped with a norm or a metric. To have a magnitude, we need a way to measure lengths, so "vector quantity" approach should be compared to either "normed vector space" or "metric space". Then any vector can be written down as a magnitude (supplied by the norm) times some unit vector in the same direction. Is anything else missing?--Julius (talk) 18:48, 16 February 2021 (UTC)


 * For a magnitude and a direction you need an inner product. So it would seem to require that the "vector quantity" be compared to an inner product space over a subfield of $\C$, possibly finite dimensional, and possibly (metrically) complete --Leigh.Samphier (talk) 08:59, 17 February 2021 (UTC)


 * I am completely out of my depth. I may have to acknowledge that I'm not really qualified to continue on this site. Every time I think I have a handle on things, it slips out of my hands. It's just too daunting. --prime mover (talk) 09:47, 17 February 2021 (UTC)


 * Maybe I'm a little bit pessimistic here, but I'm definitely afraid I won't be able to cover that gap in our vector analysis coverage that Leigh talks about. I'll keep on posting stuff up but I will have to rely on those competent and accomplished to ensure it is correct and complete. --prime mover (talk) 17:20, 17 February 2021 (UTC)


 * I don't think that the situation is that bad. We already have that ordinary space is modeled by Euclidean space. This should be rigorous enough for classical mechanics and vector analysis. If needed, we can always replace it by a complex analogue. This should also be enough to bridge "vector quantity" and "normed vector space" (or "metric space", or "vector space with inner product").--Julius (talk) 17:59, 17 February 2021 (UTC)

Update: I've worked my way nearly to the end of the chapter on the del operator and its applications. A number of examples from physics are inevitable at this stage, but these (purposefully) lack detail and will remain so until we get our teeth into works detailing such matters. (I have already got a few tomes on electromagnetism and electronics and wave mechanics and fluid mechanics lined up which have been waiting for someone to fill in the background work on vector analysis. But it's been hard and tedious work to make sure all the detail of the basics have been covered.

At some stage it will be necessary to put together school-level physics (moving blocks, friction, swinging things, heat, fluid flow, and so on) and I suppose that needs to happen. So far we've got no further than SUVAT. --prime mover (talk) 14:57, 21 February 2021 (UTC)

Banach-Alaoglu Theorem
I have tried to consolidate our coverage of the Banach-Alaoglu Theorem. The exposition as given here is compact and difficult to follow, and uses some terms which are as yet undefined on.

I also found Alaoglu's Theorem which, in all the literature I've seen (not much, I confess, while I may have some books covering this field, I haven't got round to studying them yet), is "supposed" to be the same as Banach-Alaoglu Theorem. Yet the original exposition, presented in 2013 by User:Giallo, seems to prove a subtly different result, similar to Banach-Alaoglu Theorem but deliberately not specifying that the normed vector space be separable.

I have bundled the two up together into the page Banach-Alaoglu Theorem, separating the lemmata and proofs out into their separate pages with the usual transclusion technique, with a view to leaving it open to someone who understands what they are doing who may be able to complete them (filling in missing links, explaining notation and terminology, and so on).

But now I wonder whether I should have bundled them together, and whether my source (the Borwein and Borowski dictionary) is actually correct.

The status of these proofs is one of the reasons why (in my opinion) it is better to build a proof on an established layer of stuff which has already been posted up, and also our insistence on on linking to established results and definitions.

However, we may now be in a position to have built sufficient material up in the field of normed vector spaces that we may be able to fill this all in.

Is anybody up to this task? It's beyond my immediate reach, I'm afraid. I still have a long way to go before I am able to give this field more than cursory study. --prime mover (talk) 09:06, 28 February 2021 (UTC)


 * Although I am not yet in the position to actually work on this, I noticed that later parts of the chapter I am working on contain terminology present in these theorems. I also consulted a few other books and found two versions, which to my untrained eye appear to be slightly different. I will follow the order in which all these topics are presented in Sasane, and if by the time I reach those latter chapters the question is still up for grabs, I will gladly fill in the gaps.--Julius (talk) 14:00, 28 February 2021 (UTC)

Khan Academy
I have gone through and replaced all remaining references to the old Khanacademy template with the new version. (Khan Academy completely changed the structure of its website some years back, breaking our template, so I developed a new one, but never got round to changing the references.)

Some of those old references to Khan Academy on some of those pages had to be abandoned, as I could not find the corresponding pages on the new site, and so ended up deleting the citation.

I don't know if anyone currently contributing is interested in following up Khan Academy and making sure all links are correct and complete. I'm not that person, as I find Khan Academy slow and tedious, and I lack the patience. --prime mover (talk) 12:42, 5 March 2021 (UTC)

Viewing Mathematicians in Alphabetical Order?
Is there a way to view the ProofWiki list of mathematicians in alphabetical order?

Wanted to see if John von Neumann and Alain Connes were in the ProofWiki list. Tried using the search function, but that didn't work for me. The search on von Neumann gave me too many results, and none of them were on the Mathematicians page. The search on Connes gave me nothing at all. --DeaconJohn (talk) 16:12, 21 March 2021 (UTC)


 * You can try:
 * a) When searching for "Von Neumann" and getting loads of results, look on the upper RHS and see "Check: All | None". Click on "All" and press "Search" again. You will see Mathematician:John von Neumann at the top.
 * b) Press "All Categories" on the menu bar down at the left, then go "Content Categories" --> "Historical Content" --> "Mathematicians" which gives you the raw category in which the mathematicians are listed. These of course will be in alphabetical order of their first names, not their surnames, for obvious reasons.

Thank you, Prime Mover, both of your suggestions worked. Your first suggestion got the ProofWiki von Neumann page for me, and I found Alain Connes in the ProofWiki list of mathematicians by following your second suggestion.

However, the ProofWiki search function did not find Alain Connes in the ProofWiki list of mathematicians, or anywhere else on ProofWiki either. Is that because there is a bug in the ProofWiki search engine? --DeaconJohn (talk) 21:04, 21 March 2021 (UTC)
 * No it's because I only just added him --prime mover (talk) 21:34, 21 March 2021 (UTC)


 * Thank you, Prime Mover--DeaconJohn (talk) 21:48, 21 March 2021 (UTC)


 * I suppose we could add yet another suite of pages where the mathematicians are all held in alphabetical order, but that would create another maintenance burden and take some considerable time to do, for an insignificant payoff. I for one would rather not get sucked into that. --prime mover (talk) 16:29, 21 March 2021 (UTC)


 * I would want to get sucked into that either. --DeaconJohn (talk) 21:04, 21 March 2021 (UTC)

Revenue
I realize that the revenue from the ads on this site help offset the costs of keeping it up and running, but have any other revenue streams been explored? Perhaps posters, tee shirts and/or coffee mugs with equations and/or proofs? Just curious. --Robkahn131 (talk) 18:15, 21 March 2021 (UTC)

Suggested default for macro \norm
I was just wondering if the default behaviour of the macro \norm, namely \norm {} could be made to be $\norm{\, \cdot \,}$, the same as \norm{\, \cdot \,}

I don't know of any reason why anyone would want \norm {} to produce $\norm{}$, but this is what you would lose.

If this seems like a good idea and someone could make the change (or tell me how to make the change), I'll clean up all existing occurrences of \norm{\, \cdot \,}

Similar defaults might be useful for macros \size and \card. --Leigh.Samphier (talk) 01:56, 2 April 2021 (UTC)


 * Do you know how this would be done? I like the idea (it's irritated me in the past as well) but I haven't a clue how to do it. --prime mover (talk) 05:05, 2 April 2021 (UTC)


 * I'll investigate and see what if something is possible --Leigh.Samphier (talk) 05:27, 2 April 2021 (UTC)

Self-attributed trivial results
I want a ruling on this, because I'd like to establish right now what our approach here is going to be.

We appear to be getting a whole series of results which are themselves trivial, each given a name attributing it to the supposed inventor of these trivialities.

I appreciate that children discovering such results are justifiably proud of themselves for having done so, but I'm not sure that is the place to document them.

When I was at school, I gleefully discovered that $\dfrac 1 2 - \dfrac 1 4 + \dfrac 1 8 - \dfrac 1 {16} \cdots = \dfrac 1 3$. I have never seen that result in a maths book attributed to someone, so perhaps I ought to grab that one for myself and post it up with my name against it, and get it published in an IEEE journal, and send it to all the maths websites in the world for instant acclaim. Or maybe not.

Then again, I have done exactly the same thing myself: posting up (and naming for me) a simple and amusing little result on in the field of geometry a decade or more ago. This result is apparently now being used as an interesting teaching exercise in some schools.

So what's the ruling here?

Thoughts? --prime mover (talk) 16:26, 19 April 2021 (UTC)


 * I think that when it comes to academic knowledge, the correct way would be to wait for someone else to call the result by your name. I have published some results in peer-reviewed academic journals, but I would not dare to give them my name. I would rather avoid any names unless one of the "giants" is involved. When it comes to puzzles, I am not so sure. You can either sell a book or post on a forum and wait for the word to spread, and if later you find the name being used in greater circles, I guess you could use it here as well. However, for trivial results, I can only see them as variants at most, and no names should be attached. There are lots of websites like vixra, art of problem solving etc. where one is free to post whatever is preferred. I expect the bar to be higher here.--Julius (talk) 17:35, 19 April 2021 (UTC)


 * I agree with Julius - no names. --Robkahn131 (talk) 17:44, 19 April 2021 (UTC)


 * So to be consistent, we really should delete Westwood's Puzzle, as it's always been controversial. --prime mover (talk) 19:41, 19 April 2021 (UTC)


 * No need to delete it. This is a good example of how to post an interesting result. The problem arises when people use a double angle formula and then slap their names on top. A puzzle can have its place here provided is not a one-step problem. In your case, the title does not bug me. We can keep it as a legacy. I think that creators of this site deserve at least that.--Julius (talk) 20:27, 19 April 2021 (UTC)

Well-Founded and Strictly Well-Founded Relations
Any aficionados of set theory may wish to glance at the tidying up that I have recently done on the subject of Definition:Well-Founded Relation and Definition:Strictly Well-Founded Relation, based on a close reading of, which is under my microscope at the moment.

I have taken care to remove all context of whether it is defined in set theory or class theory (it's pretty basic in both).

Hence the page Definition:Foundational Relation has been superseded by Definition:Strictly Well-Founded Relation (despite the fact that the term is more unwieldy), and Definition:Strongly Well-Founded Relation (coined by a previous contributor) subsumed into Definition:Well-Founded Relation. Equivalence proofs will possibly be done over the weekend, if I feel mentally fit enough.

There is still plenty of work to do to tidy up this area, but I hope that putting these definitions on a solid and unambiguous footing will be a start. --prime mover (talk) 21:18, 30 April 2021 (UTC)