User talk:Prime.mover

New approach to natural number section
I'm sure you've noticed my new approach to getting the natural numbers section to a higher level. I hope you like the direction I'm going in :). As always, feel free to inform me about any remarks or other concerns that you might have.

If you want to keep an eye on the progress, just take a look at the items on User:Lord_Farin/Sandbox/NN Refactoring that have been marked done (you can also add items you think relevant to the list). &mdash; Lord_Farin (talk) 20:37, 22 October 2014 (UTC)


 * I'm keeping an eye, no worries ...


 * I'm going to let you get on with it, I'm happy with this approach. It's a project I found was beyond my powers of extended concentration. Meanwhile, I'm going to continue trying to make head or tail of Euclid's books on number theory. --prime mover (talk) 20:44, 22 October 2014 (UTC)


 * You've probably noticed that I haven't been doing much; I have been occupied by other things. Fortunately, my explicit planning list helps to keep track of where I was and what needs to be done, so that we can still progress in the end. &mdash; Lord_Farin (talk) 16:40, 7 February 2015 (UTC)


 * The Warner work was among the first work I did for and hence has the least corroboration from other sources. I have it in mind to completely revisit the entire thread in due course. Flag up any possible changes to the flow by means of source review templates and I will make sure I get to them when I re-approach this work.


 * At the moment I have my head down completing this first pass through Euclid, which I am going to concentrate on while I am still motivated. After a while I will need to take a break, but while I'm still making progress I'm sticking with that. After that I have no firm plans. It may depend on what "the people" want. --prime mover (talk) 17:36, 7 February 2015 (UTC)


 * I consider the big Warner revisiting due, at least regarding the NOS. Just so you know. I'm fine with postponing until you get around to it (I haven't been able to find a digital copy, so I can't do it myself). &mdash; Lord_Farin (talk) 21:13, 25 February 2015 (UTC)


 * As I say, leave me be to finish off Euclid (tantalisingly close now, nearly the end of book XI, only two more and some bits and pieces left to do) and then I'll revisit Warner.


 * A medical condition (and utter boredom with parallelepipeds) leaves me unable to concentrate for long periods at the current moment, so work is progressing more slowly than it ought. --prime mover (talk) 21:27, 25 February 2015 (UTC)

Feedback for Proofs
I noticed that the "Maximum Modulus Principle" has no proof, and I am working on one in my sandbox right now. But I am not comfortable posting it yet, and have a few questions I'd like to ask first (primarily looking to see if certain results are already proved on the site, and I just can't find them). Where is the best place to get feedback on works in progress before posting them? The main talk page? The questions page?

By the way, my two issues are:


 * I'm trying to find if what I've called the "Mean Value Property" (in my sandbox proof) is already proved on the site.
 * I know that my reference to "Absolute Value of Complex Integral" doesn't quite apply in this situation, but a very similar principle does. I'm wondering if the more general measure-theoretic version is proved somewhere on the site.

-- Ovenhouse (talk)


 * Feedback on specific works in progress can either be got by posting up on the talk page of the actual page in question, or if you believe there is a wider application for the question, then on the main talk page. Note that all edits can be seen on the "Recent changes" page, so everything that is posted up anywhere is visible by anyone who regularly trawls through this.


 * As far as I know, the "Mean Value Property" has not been done yet. I agree, it merits its own page. Note that every statement used needs to be backed up by a link to a demonstration of the truth of that statement; in particular there are references to integration by substitution and change of variables, neither of which have been covered in the complex domain on . Then of course we have the question of what further statements do these rest on? It's tortoises all the way down.


 * The more general result / similar principle to the "Absolute Value of Complex Integral" page do not as far as I know exist. It would be a worthwhile exercise to put them in place.


 * There has been embarrassingly little work done on complex analysis on this site. I have several times made a start but (such is my nature) I have each time become bogged down in the minutiae of the fundamentals to the extent that I have not yet got as far as complex calculus. Apart from me, the only other person to have done any work on the area was Anghel, who did a considerable amount of excellent work on Jordan curves.  However, the basics of complex calculus are still basically untouched.


 * A new approach from someone new would be welcome. If there are existing results, then (apart from the basic stuff and the aforementioned Jordan curve work) it is incomplete and does not form a coherent body of knowledge. If you are of a mind to flesh it out, then your efforts will be greatly appreciated. If results become duplicated, and as a result some redundancy creeps in, then no matter, we can merge existing pages into any new work that is being done.


 * Please note that there is a stringently-applied house style (there are reasons for this which have been discussed in some detail -- and at some heat -- elsewhere), so please do not be put off by a rash of "tidy" and "refactor" and "MissingLinks" templates appearing all over everything you do :-) -- these are more reminders for us janitors to work on than for you to be constantly going back and reworking stuff. --prime mover (talk) 10:57, 30 December 2014 (UTC)


 * If you are in doubt or cannot find a specific result, do not hesitate to use the MissingLinks template, describing what link is missing. Someone else might find it, or write it up. In this way, at least the reference is recorded. &mdash; Lord_Farin (talk) 12:46, 30 December 2014 (UTC)

Minimal Infinite Successor Set vs. Natural Numbers
Yesterday, I started on the next task on the list, Definition:Minimal Infinite Successor Set. I posted up Definition:Natural Numbers as Elements of Minimal Infinite Successor Set. Regarding this double nature of the MISS, it would make sense to separate the MISS approached from the context of set theory, and its interpretation as $\N$.

In particular, it would be better for clarity in naming pages to have "Natural Number" only refer to the abstract properties of $\N$ as described by Peano's Axioms (and, to a lesser extent, the NOS). Currently, however, the term also refers to what is more rightly called "Element of MISS". True, this obfuscation is ubiquitous in set theory and related areas, but I still think there's a case to be made for keeping the two separate on. Are you fine with keeping the name "Element of MISS" or would you prefer something else? We can also opt for the minimum of results necessary to prove the equivalence with "Finite Ordinal" and use that name for all other results. There are probably sources defining $\N$ as the collection of finite ordinals, which would be easier to cover if we choose the latter option. I'm curious to hear your thoughts. &mdash; Lord_Farin (talk) 18:59, 5 March 2015 (UTC)


 * The optimum approach from my point of view is also the most work! All of the following are done:


 * a) We have an object called "the set of natural numbers", which is your go-to page for any invocation of "natural numbers" which are more-or-less intuitively understood by people, along with a set of conditions which they fulfil (which can be defined as their "axioms"): they have commutative, associative and well-defined addition and multiplication; the latter distributes over the former; they are countably infinite (yes I know, this is how countable infinity is defined, etc.) And whatever other axioms they have.


 * b) We have an object called a "naturally ordered semigroup", which fulfils certain conditions, and can be shown to fulfil all the axioms of the natural numbers, and that the natural numbers are a NOS, and the natural numbers and a NOS are unique and can be identified as the same thing (the basic approach of Warner, and as it has been implemented on.


 * c) We define the MISS and do the same for it as b) (and we demonstrate that the MISS is derived from the ZF axioms).


 * d) We show that Peano's Axioms also give rise to the MISS.


 * e) We demonstrate that Peano's Axioms, the MISS and NOS all are interderivable.


 * Consequently I see the abstract properties of $\N$ being proven just once, but under whatever structure. The fact that they hold for the other structures is a consequence of the other structures being isomorphic. If the fact that these properties hold is a prerequisite for the proof of those isomorphisms, then yes, they can be proved for those structures.


 * But ultimately it depends on whatever we can find / derive as proofs. If we find (published somewhere) these properties proven using the one structure, then we can show that proof.


 * Hope this hasn't confused too badly. I'm distracted at the moment by a business trip tomorrow for which I am ill-prepared. :-( The good news is that it is in the delightful Luxembourg. :-) The bad news is that I fly in and out on the same day :-( --prime mover (talk) 19:58, 5 March 2015 (UTC)


 * Well, this is more or less what I see as well. However, unless you mean Peano's Axioms by (a), there's not a good way of doing that (we cannot prove things about an intuitive concept with desired rigour). Rather, we can only postulate axiomatisations and constructions which seem reasonable and prove them equivalent (the painstaking process of which I have been intermittently working on for months now).


 * I have carefully proved that NOS and MISS form Peano Structures, and the uniqueness of PA and NOS (well-definedness of MISS upcoming). A proof that PA gives rise to NOS is also due, but still missing at this point (to finish the equivalence of PA and NOS proper).


 * There is room for replacing (or rather, augmenting) the very formal treatment of the operations on $\N$ with intuitive descriptions, and including excerpts of those in the intuitive section of Definition:Natural Numbers.


 * I have given it some more thought and will proceed to split the set-theoretic meaning of "natural number" to "finite ordinal", prove "element of MISS $\iff$ finite ordinal" and reduce the occurrences of "element of MISS" to the minimum.


 * That all said and agreed, I wish you best of luck on your trip. Too bad you won't stay on mainland Europe for too long, it's great :). &mdash; Lord_Farin (talk) 20:54, 5 March 2015 (UTC)


 * I'll leave it completely up to you to accomplish what you believe needs to be done to sort out the Peano Structure approach. My personal view is: as it is so important historically, and as it provides such a usefully accessible entry point for the beginner in axiomatics and the foundations of mathematics (and because I like it so much), it would be good to place it prominently in the dissertation.


 * When all is complete I will go through the various works I have access to (Deskins, Warner, Halmos etc.) and review their contributions in light of what we have -- and the cycle will repeat. :-)


 * I took a deliberate career decision mid last year to specifically not have to work abroad so much as I had been doing -- I'm okay with travelling, but my wife's health is poor and I need to be here for her. Last year I spent considerable time in Luxembourg, and would move there given half a chance. But if I want to keep certain aspects of my life together, that won't be immediately possible ... I love Europe, I always have. --prime mover (talk) 21:04, 5 March 2015 (UTC)

Theorems about multi-defined concepts
I ran into Natural Numbers are Transitive Sets in the ongoing rewriting operation. We already have Equivalence of Definitions of Minimal Infinite Successor Set, so we already know that the $\N$ in Halmos' eyes is equal to the set of finite ordinals. Since ordinals are by definition transitive, there is not really a theorem needing proof.

I'm bringing this up generically: What should we do in these cases? Post up the proof as if the equivalence hadn't been demonstrated? Include a trivial a fortiori reasoning as proof 1? &mdash; Lord_Farin (talk) 10:40, 12 April 2015 (UTC)


 * This result is part of Halmos's presentation of the natural numbers. He starts by setting up the numbers in the usual way: $n^+ = n \cup \{n\}$, introduces the Peano axioms, and then uses this result as part of his proof that $\N$ is a Peano set. That is:


 * "The proof of (V) is not trivial; it depends upon a couple of auxiliary propositions. ... (i) no natural number is a subset of any of its elements, and (ii) every element of a natural number is a subset of it. Sometimes a set with the property that it includes ($\subset$) everything that it contains ($\in$) is called a transitive set. More precisely, to say that $E$ is transitive means that if $x \in y$ and $y \in E$, then $x \in E$. ... In this language, (ii) says that every natural number is transitive."


 * In my mind, this result precedes Equivalence of Definitions of Minimal Infinite Successor Set, coming from the direction that the construction of $\N$ via Peano and Zermelo/Fraenkel. In fact, at this stage of the development of numbers, Halmos has not even defined "ordinal" -- he does not get that far till chapter 18. I believe I stopped at 17, having been unable to get past differences of opinion as to how this material was presented (as usual I am far too easily distracted and derailed by trouble with detail) and I started work on something else instead.


 * I believe there is a point to establishing this basic result, because in the context it is made, it needs to be made clear on the road towards proving that $\N$ is a set of ordinals. --prime mover (talk) 11:36, 12 April 2015 (UTC)


 * Fair enough, I'll keep it in. I got an idea, you will get the e-mails about my actions :). &mdash; Lord_Farin (talk) 11:40, 12 April 2015 (UTC)


 * Cf. Element of Minimal Infinite Successor Set is Transitive Set. I hope you like it. &mdash; Lord_Farin (talk) 11:46, 12 April 2015 (UTC)


 * I think so -- I'm going to revisit Halmos once I've done Warner and make sure it all hangs together. The crucial line is the one in Chapter 11: "A natural number is, by definition, an element of the minimal successor set $\omega$. This definition is the rigorous counterpart of the intuitive description according to which they consist of $0, 1, 2, 3$ "and so on." Incidentally, the symbol we are using for the set of natural numbes ($\omega$) has a plurality of the votes of the writers on the subject, but nothing like a clear majority. In this book that symbol will be used systematically and exclusively in the sense defined above."


 * Thus if we replace all references to "natural numbers" in the results and definitions on this thread by "minimal infinite successor set" as is in progress, we will then know that we are specifically building up the required knowledge about MISS and only after that has been established do we then "equate" them with the Natural Numbers by playing the "equivalence of definitions" card. --prime mover (talk) 11:54, 12 April 2015 (UTC)

Ok, I will add an entry to your worklist (towards the bottom of the NN refactoring page) for going through Halmos. You can add anything coming out of that as a sublist under that item.

Relatedly, have you finished source-reviewing Warner? Probably not. I will move that one to your task list as well. &mdash; Lord_Farin (talk) 11:57, 12 April 2015 (UTC)


 * No I haven't, not yet. I'm still bogged down on rationalising the material around index laws. --prime mover (talk) 11:58, 12 April 2015 (UTC)

Closure notation
Presumably missed in the plethora of other changes: Definition talk:Upper Closure. &mdash; Lord_Farin (talk) 15:47, 13 April 2015 (UTC)

Category:Closure Operators
You have set out to change the category of pages pertaining to upper/lower closure to this category or its definitions counterpart. I disagree. Closure operators are specifically self-maps, whereas upper/lower closure are mainly a general proof tool for ordered sets. &mdash; Lord_Farin (talk) 08:48, 18 April 2015 (UTC)

Nuance: I see it for closure on sets, but on elements it's inappropriate. I have made the necessary edits. &mdash; Lord_Farin (talk) 08:55, 18 April 2015 (UTC)


 * Okay, I'll go along with that. But we need to do something to break down the huge category that is Category:Definitions/Order Theory, though. What do you suggest? --prime mover (talk) 08:58, 18 April 2015 (UTC)


 * Split off the various ordered structures, as well as the number sets. A big win would be a way to deal with the multitude of subpages in a good way. &mdash; Lord_Farin (talk) 09:12, 18 April 2015 (UTC)


 * Okay, I'll start with a tried and tested technique for sorting out subpages. --prime mover (talk) 09:21, 18 April 2015 (UTC)

Principle of Recursive Definition
Another huge chunk of the refactoring has been finished. I hope you like it. Slowly, we are getting there. &mdash; Lord_Farin (talk) 12:14, 5 May 2015 (UTC)


 * Yes, that all hangs together well. Good job.
 * I hope you excuse my change of direction back to topology. --prime mover (talk) 12:34, 5 May 2015 (UTC)

in terms of vs. in Terms of
Searching "in terms of" returns both methods of expression. Which one should be used? --kc_kennylau (talk) 13:39, 20 May 2015 (UTC)


 * Please use the former, "in terms of". (PM is having a break, so I have taken the liberty to respond in his stead.) &mdash; Lord_Farin (talk) 16:05, 20 May 2015 (UTC)

Fun with Source Reviews
I made the call yesterday to push my prepared work on the Principle of Mathematical Induction to the main namespace. This naturally has generated some issues with various source sections and flows. It would be appreciated if you could look into this. &mdash; Lord_Farin (talk) 11:37, 31 May 2015 (UTC)


 * Aha, I have my edit privs back, I didn't think I had. I will do some tidy up work. --prime mover (talk) 19:37, 2 June 2015 (UTC)

Regarding Primitive of Reciprocal of x cubed by a x + b cubed
With all due respect sir, wolfram alpha tells me that the expression below and the expression above are different, and it is not just a difference of constants. --kc_kennylau (talk) 12:54, 9 June 2015 (UTC)


 * If you can work out what's wrong with what I did, feel free to correct it. --prime mover (talk) 18:45, 9 June 2015 (UTC)


 * I don't really think that it's your problem. I suspect the above expression is wrong. (Sorry for my informal language) --kc_kennylau (talk) 23:19, 9 June 2015 (UTC)


 * As I say, if you think you may be able to resolve the issue, please feel free. --prime mover (talk) 05:00, 10 June 2015 (UTC)

Trigonometry headline
Should we add the line "where $\sin$ denotes the sine function" in Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 14/Integrals Involving Sine of a x? --kc_kennylau (talk) 09:33, 11 June 2015 (UTC)


 * No, I don't think so. The individual results should have all the appropriate definitions in them -- the curious reader will find out at that level. Spiegel doesn't include them here, and the idea is to reproduce his presentation to a reasonable level of accuracy. --prime mover (talk) 09:37, 11 June 2015 (UTC)

A useful tool
Just a note, that I use regex to make the additions faster. --kc_kennylau (talk) 09:39, 11 June 2015 (UTC)

"square of a x squared plus b x plus c"
I notice that Primitive of Square of Arcsine of x over a uses "Square" instead of "square". --kc_kennylau (talk) 09:41, 11 June 2015 (UTC)


 * Yes I noticed that -- I'm not too concerned at this stage, maybe we can go through and tidy up later. Flag them if you want. There are a few like that. --prime mover (talk) 09:44, 11 June 2015 (UTC)

Category:Definition Equivalences
Am I permitted to create such a category? --kc_kennylau (talk) 10:06, 12 June 2015 (UTC)


 * Good idea. I have created Category:Definition Equivalences which I put as a direct child of Category:Proofs. --prime mover (talk) 11:03, 12 June 2015 (UTC)


 * Can you help me to populate it as well? --kc_kennylau (talk) 11:07, 12 June 2015 (UTC)


 * You get started, it was your idea. :-) --prime mover (talk) 11:08, 12 June 2015 (UTC)

Laplace transform
The author of this book (Spiegel) deliberatly (or not) swapped $f$ and $F$... Should I follow her (or his) notation or the conventional notation? --kc_kennylau (talk) 16:24, 13 June 2015 (UTC)


 * Just lay off for the moment and do something else, please. --prime mover (talk) 17:41, 13 June 2015 (UTC)


 * Okay. I'm sorry. --kc_kennylau (talk) 17:44, 13 June 2015 (UTC)

How does this need the stabilizer to be a subgroup?
Stabilizer is Subgroup/Corollary 2 --Mathmensch (talk) 07:46, 20 September 2015 (UTC)

On $f^\gets(S)$
I notice this flashy new notation $f^\gets(S)$ and $f^\to(S)$ has been replacing $f^{-1}[S]$ and $f[S]$ in many places. Do you really think it's better than what it was? Square brackets were unambiguous in my book, while $f^\gets$ and $f^\to$ are rarely ever used. I'm worried this might harm the accessibility and therefore usefulness of the site. &mdash; Lord_Farin (talk) 21:22, 7 October 2015 (UTC)


 * Of all the works cited on the page Definition:Image/Mapping/Subset, two of them (Kasriel and Devlin) use $f \left[{A}\right]$ and $f^{-1} \left[{A}\right]$, and one (Blyth) uses $f^\to \left({A}\right)$ and $f^\gets \left({A}\right)$. Not one single other work that I have access to differentiates between them -- they use $f(x)$ and $f(A)$.


 * (The now-removed unwieldy notation $f_g \left({A}\right)$ for the induced mapping on the powerset of $g$ I can't find anywhere, and I now believe I must have made it up when I was writing my book on the subject some 10 years ago.)


 * Two gets one: so by democratic vote squares should beat arrows. However, Blyth is unusually thorough and detailed on a lot of this detail which many gloss over (including Halmos, which is disappointing), so I borrowed his notation as it emphasises that it is a different mapping (that is, it's a mapping from the powerset to the powerset), while all the other sources don't say much on it.


 * The work that influenced me most here was in fact McCarty (one of the first books I bought for independent study after having just started my MMath), who is the only one apart from Blyth and Halmos to actually state the nature of what is being talked about: "A function $f: A \to B$ defines (or induces) in a natural way a new function, usually denoted by the same symbol $f$, from $\mathcal P(A) \to \mathcal P(B)$ ..." and then goes on to define similar for inverses.


 * The "induced mapping" which appears as Definition:Mapping Induced on Powerset by Mapping and so on is explored in some detail in Blyth, McCarty (who dabbles his toes in category theory and relation theory but finds the water a bit cold) and briefly Halmos; I did briefly go a little further in exploring the difference between the inverse of an induced mapping and the mapping induced by an inverse mapping, but found no source works able to back me up. I did at one point derive a couple of results of my own which I conceitedly named "Westwood's Theorem for Surjections" and "Westwood's Theorem for Injections" (don't worry, such arrogance has now been deleted from the universe -- you're not allowed to name things after yourself) but it didn't lead anywhere and in hindsight I believe that Blyth's great attention to detail is possibly misguided.


 * In short, I'm happy to delete all references to "induced mappings" and, like all the other works, merely gloss over the detail of how the notation "$f[X]$" conceals under its brevity the concept of a mapping from a powerset to a powerset.


 * I can start ripping all of that out tomorrow -- as of now I have some sleep to be getting on with, or I'll fall asleep in the meeting tomorrow like I did last week. --prime mover (talk) 22:00, 7 October 2015 (UTC)


 * I believe that the issue at hand is that while there is added value in the realisation that these are induced mappings, it is not until one has a firm understanding of the "naive" $f[S]$ that this value really shines. Before that point, such finesse might cause more confusion than clarification. Much like category theory only becomes a good idea after an appropriate smorgasbord of subjects are explained in a set-theoretic fashion. &mdash; Lord_Farin (talk) 16:19, 8 October 2015 (UTC)


 * I think I put it all back the way it was, sort of, but I may have mucked up. --prime mover (talk) 05:35, 9 October 2015 (UTC)

More notational quabble
What's with $\lim\limits_{x \mathop \to \infty}$ instead of $\lim\limits_{x \to \infty}$? I thought the arrow was quite unambiguous and needed no further style adjustment. I won't revert any more of these edits, but I won't adhere to the stylistic choice either... Let's make this one optional, ok? :) &mdash; Lord_Farin (talk) 17:37, 21 October 2015 (UTC)


 * Thought I could get away with that bit of petty vandalism. I like a gap. --prime mover (talk) 18:02, 21 October 2015 (UTC)

Tensor product
Hello, I often lurk and ocassionally assist but I have a question, why are there little to no topics on Tensor Product? Have I missed them? Or is it simply that none has ever gotten around to them?

EmperorZelos (talk) 12:51, 24 October 2015 (UTC)


 * Nobody's done it yet. I have to see any exposition on the subject that has even been able to explain worth a damn what a tensor is. Until that happens I'm not touching it. --prime mover (talk) 13:24, 24 October 2015 (UTC)


 * Does "Big Modda fukkah module" suffice as description? :P Jokes aside, would you mind if I gave it a shot? EmperorZelos (talk) 13:27, 24 October 2015 (UTC)


 * Give it a go. I recommend you back it up with a published source of some kind, otherwise you may find your work challenged by other contributors who may believe they have a better idea. :-) --prime mover (talk) 13:28, 24 October 2015 (UTC)


 * I have 2 books if I recall correctly on the issue and I know of a fantastic online document for it that I'll link to, i'll get working on it. EmperorZelos (talk) 13:30, 24 October 2015 (UTC)


 * How do I link to online sources written? EmperorZelos (talk) 14:05, 24 October 2015 (UTC)


 * You don't. You reference printed materials. If the online source is an online version of a printed resource, then implement it like all the other books. If the material only exists online, then we would prefer not to reference it because we have discovered the hard way that such work is often tediously ephemeral. --prime mover (talk) 15:44, 24 October 2015 (UTC)


 * While that is true and I agree, is it possible to still use it if it's better? This one This one is exceptionally well written for tensor products in both aspects of it and I figured anyone who wishes to read it's a fantastic read. EmperorZelos (talk) 16:04, 24 October 2015 (UTC)

I'm afraid I don't find it particularly readable. I can see there are definitions in there, somewhere, but they are not easily digested because they are surrounded by loads of extra material, and it is difficult to extract what is important, and part of the definition, and what is merely explanatory and therefore not actually part of the definition. It is also not laid out too well -- it is very densely packed on the page and no attempt has been made to make it screen-readable. But if you like it, then fair play. --prime mover (talk) 16:09, 24 October 2015 (UTC)


 * My two cents: Please get started, using an incremental approach is better than doing nothing. The material will likely need refactoring some time, but this needn't hinder us. But, as I suspect there will be some basic material missing, please create a work-page somewhere in your own workspace (subpages of your user page) where you keep track of what pages and areas are still to be covered. This makes it easier not to get lost. &mdash; Lord_Farin (talk) 13:52, 25 October 2015 (UTC)


 * I'll take those two cents and raise it with 1 penny! I am working on the basic definition as we speak on subpage, once I feel it is sufficient I'll start around it. EmperorZelos (talk) 13:59, 25 October 2015 (UTC)

On the definitions for Multiplication
We now have:


 * Definition:Multiplication/Natural Numbers
 * Definition:Multiplication/Naturally Ordered Semigroup
 * Definition:Multiplication/Natural Numbers/Naturally Ordered Semigroup
 * Definition:Multiplication/Natural Numbers/Minimal Infinite Successor Set
 * Definition:Multiplication/Natural Numbers/Peano Structure (but I only created this yesterday)

which basically all define multiplication in terms of addition. Therefore my proposal is to merge all these definitions, keeping a transclude of only Definition:Multiplication on 1-Based Natural Numbers. What say you? &mdash; Lord_Farin (talk) 18:47, 26 October 2015 (UTC)


 * I'm trying to cook up a reason why not, and can't find one. Please feel free to go ahead ... --prime mover (talk) 19:16, 26 October 2015 (UTC)


 * Do you subsequently agree that Multiplication in Naturally Ordered Semigroup is Commutative etc. can go? Because that decision would imply considerable consolidation of the multiplication pages. Basically, it'd mean that we can drop the exact axiomatisation of $\N$ beyond the realm of addition, which would be a great achievement in my opinion. &mdash; Lord_Farin (talk) 20:55, 26 October 2015 (UTC)


 * On balance: yes, probably -- feel free to move the citations to whatever other pages hold the results that they are replaced by. If not, then set a delete flag and a source review flag and I'll attend to them when the mood strikes me. --prime mover (talk) 21:03, 26 October 2015 (UTC)

Things are going well, I'm steadily working towards a cleaner repository. As part of that endeavour, consider:


 * Uniqueness of Recursive Mapping from Naturally Ordered Semigroup to Algebraic Structure with Identity
 * Uniqueness of Recursive Mapping from Naturally Ordered Semigroup to Algebraic Structure
 * Restriction of Recursive Mapping from Naturally Ordered Semigroup to Algebraic Structure

which, under the surface of convoluted language, only deal with Definition:Power of Element. Do you think they can go? Should they be rephrased for $\N$? If so, how? &mdash; Lord_Farin (talk) 21:15, 8 November 2015 (UTC)


 * For some reason I felt it was necessary to distinguish between stuff that concerns semigroups and stuff that concerns monoids. The power laws for a semigroup, for example, are different when there's an identity (because then you have to think about zeroth powers), and for a semigroup of which at least some elements have inverses is separate again (because then you have to think about negative powers), and as multiplication is just an implementation of the power law used on addition, a separate case was made for each. Perhaps I was overly influenced by Warner, who is thorough to the point of Cooperesque OCD, but if you have a clear vision through this morass without losing sight of what may later prove to be essential details, please feel free to make like Alexander with the Gordian Knot. --prime mover (talk) 21:24, 8 November 2015 (UTC)


 * Essential is basically only the page about restriction, which could be named something like Power of Element of Monoid is Compatible with Power of Element of Semigroup or Power of Element of Monoid is Well-Defined. But even then, it could be phrased for $\N$ and refer to Definition:Recursively Defined Mapping directly. But the proof is massively boring, so I'm not inclined to write it right now. Might put this on my to do list. &mdash; Lord_Farin (talk) 21:32, 8 November 2015 (UTC)

The pages on multiplication for $\N$ have been restructured to a point where they're more or less coherent. Not perfect, but good enough to move on. &mdash; Lord_Farin (talk) 20:14, 15 November 2015 (UTC)


 * Okay, once I've finished what I'm doing here I will return to this and check the source flow. --prime mover (talk) 20:15, 15 November 2015 (UTC)


 * I did my best to keep it in sync using my digital copy of Warner. A second pair of eyes is probably a good idea anyway. &mdash; Lord_Farin (talk) 20:20, 15 November 2015 (UTC)


 * Second pairs of eyes, and second sources of sources, are always better than one. --prime mover (talk) 20:20, 15 November 2015 (UTC)

Initial Definition, Tensor
How is this for an initial definition?

EmperorZelos (talk) 06:34, 24 November 2015 (UTC)


 * All you need to do now is to put it into the house style and give it a category. --prime mover (talk) 06:59, 24 November 2015 (UTC)

House style? What does that mean if I may ask here? As for category, do you reckon it's better getting a new category for tensor products along with a few obvious ones? EmperorZelos (talk) 07:00, 24 November 2015 (UTC)


 * When you initially registered on this site, on your talk page was posted a welcome message. In that message was included a link to the house style guide which you were encouraged to look at. Within that style guide, and elsewhere in the Help pages (also accessible via the menu that can be seen down the left hand side of the page), you should be able to get an idea of how the site is structured (although we do confess that we appreciate that these help pages could be structured considerably better).


 * You may of course be able to determine where the page you have written differs from house style by studying existing pages and comparing the presentational style of your own work with that of existing pages (hint: don't pick one which has the "tidy" template invoked on it), but unfortunately I have in the past discerned that contributors appear to have had traditionally had considerable difficulty with this technique, so maybe you will find this approach is not for you.


 * As for category, it would be named whatever the name of the category is that tensors form a part of. Tensor algebra? Tensor calculus? Is it a subcategory of linear algebra? These are all questions to which I do not know the answer, as I have never made the effort to study the subject. --prime mover (talk) 07:18, 24 November 2015 (UTC)


 * Ah yes, I forgot about that as it's been so long, i'll get on it and apply it all. EmperorZelos (talk) 09:11, 24 November 2015 (UTC)

Explanation
I want to add an explanation somewhere for why it must be right and left module rather than both right or left, would you recommend giving it on the definition page for tensor product or is a theorem that demonstrates the implication of dual left/right module a better alternative? EmperorZelos (talk) 11:10, 24 November 2015 (UTC)


 * It's a theorem. Therefore it lives on its own page, as is prescribed by the house style guide.


 * Note that links for both "left module" and "right module" are missing. The page is of limited use unless every concept on it can be linked back to more basic concepts (obviously until you reach the axioms).


 * Again, I recommend you browse through the website (perhaps using "random proof" and broswing the links from there) and see how other pages handle this sort of thing. "Also see" is a usual technique for directing the user towards those theorems which are directly relevant (it's counterproductive to list a large number of peripheral results in such a section). --prime mover (talk) 11:41, 24 November 2015 (UTC)


 * Also note the difference between proof categories and definition categories. I will leave it to you to correct it -- it is easy to work out what needs to be done. --prime mover (talk) 11:43, 24 November 2015 (UTC)


 * Incidentally, I presume English is not your first language? --prime mover (talk) 11:45, 24 November 2015 (UTC)


 * Unfortunately you are correct, I am swedish by birth but I try my best to improve my english constantly. I apologise for any errors I make in that department. EmperorZelos (talk) 11:46, 24 November 2015 (UTC)


 * Not a problem. Your English is better than my Swedish :-) Please accept that I may need to amend spelling and grammar as the work progresses. --prime mover (talk) 11:51, 24 November 2015 (UTC)


 * I'll teach you if you want ;-) I'll not only accept it, I will feel offended if none corrects me when they can! :-) EmperorZelos (talk) 11:52, 24 November 2015 (UTC)

I have set up the categories as appropriate. There are standard templates, as you will see when you investigate. It takes away a lot of the pain!

As you see, there are two completely different namespaces: one for definitions and one for proofs, and they are kept in separate category trees (although there are links maintained between them maintained via the various templates.

Note there is still some work to do in order to get the new pages to house style. I will attend to it. --prime mover (talk) 12:11, 24 November 2015 (UTC)

left right what?
Okey I gotta ask, what's with all the unneccisery /left, /right etc that are in the codes but seems to add absolutely muffin to appearnece? EmperorZelos (talk) 10:13, 25 November 2015 (UTC)


 * Help:FAQ/Questions about contributions/Why do we need \left and \right with every pair of parentheses?

Thank you
Thanks for the corrections you made on the Area Between Two Parallel Chords page, I fixed the sign. But I have another problem: I noticed that the formula also holds for chords that are not parallel, with the two special cases 1) the chords intersect, and 2) the centre isn't included in the area (I'm currently working on both). So my question is: do I have to just change the title and content (also how do I change the title?) or do I have to create a whole new page for Area Between Two Chords? Thanks again.


 * Click More --> Move at the top of the page.


 * A few comments:


 * a) When the chords intersect (or at least, when they cross), the area between the chords is not well-defined, so I can't immediately see how that would work


 * b) Suggest you need a couple of new diagrams. Hint: you can always adjust the size of the presentation of a file image by using "|400px" or whatever size at the end of the invocation of the file link. See other examples of pages with files in them.


 * c) Also about graphics: using GeoGebra you can get a better-looking style to your letters by hiding the automatically-generated labels and replacing them with "Text" fields which look smart when you select $\LaTeX$ mode.


 * d) I direct you to sign your posts.


 * I prefer not to do lots of long and detailed how-to notes on the grounds that one learns best when one teaches oneself. --prime mover (talk) 19:29, 8 December 2015 (UTC)

Two Questions
First off: thanks for putting up with me flooding the wiki with all of these untidy pages as I master the house style, and simultaneously my personal ability to provide a cogent exposition.

My first question: on Defining Sequence of Natural Logarithm is Strictly Decreasing, you've directed me to tidy up the page with respect to my use of the eqn template. What particular part of my use of eqn is wanting? My guess is that the implications go too "wide" and the proof possibly just not as easy to parse as it ought to be, though perhaps there is another nuance I am missing.

Second: I'm thinking of making a large list of properties of inequality of real numbers (like the list of laws for exponent combination). My impetus for this is that I think it would help one when one is going through a long string of inequalities in a proof to have a list to scroll through. The Inequalities category is good, though it is a little too general (the page I'd like to create would mostly be the elementary results, not so many more advanced theorems), and the multitude of naming options for these properties might lead to a quick read-through of names of the properties not yielding a find. Perhaps this is just me, though. Either way, is this a good idea? --Keith.U (talk) 06:54, 30 June 2016 (UTC)


 * The "tidy" template is not necessarily a directive to you so much as a reminder to admins that the work needs to be done. If you want to learn more about the niceties (it takes a while to get into it) then your best bet is to diff the pages and work out what changes were made to get from A to B (so to speak). (In particular, note the style where we leave an extra line feed between sections -- we believe this makes a page easier on the eye.) (We also note that where pages are defined as subpages, e.g. "Main Page/Subpage", then there may be a redirect to that definition, whose use is preferred.)


 * Use of eqn: the o column is designed to hold the equation operator, i.e. equals, less than etc. The ll column is designed to be used for the \implies. Thus the first line goes in l, o, r columns. The subsequent lines also go into  l, o, r columns, but with implies in ll</tt> column.


 * As for the rest, feel free to carry on as you are. Pages of lists of simple results seem to work, people like them, it makes it easier (see the extreme case of Trigonometric Identities, which grew organically (like a fungus) so yes, if you like.


 * Finally, note that it is pure coincidence that I happen to be going through chapter 1.2.2 of Knuth at exactly the same time as you are working through this section on real powers yourself. So I'm likely to restructure things and rename pages according to what seems to make sense a lot, and so beware the same is likely to happen to pages of yours. As you probably know from the computer industry, naming things is one of the two most difficult things in computer science (the other one being cache management and off-by-one errors) and so it sometimes takes a while to get the name of a page "right". --prime mover (talk) 07:16, 30 June 2016 (UTC)


 * A small clarifying question on the final note of your first paragraph there. Are you saying that when referencing definitions, I should use the subpage that most accurately defines the notion I am using? E.g., if I'm talking about a complex sequence, should my definition of "sequence" point to the very general page for sequences, to the more specific page for complex sequences, or something in between? --Keith.U (talk) 07:31, 30 June 2016 (UTC)


 * I'm conflicted on that. If e.g. you have 2 definitions for a concept and they're both visible on the main page, might as well use the main page version. But if you specifically need to reference just one of them e.g. to avoid circularity arguments, then you may find it's better to be particularly careful.


 * If there are different definitions depending on context, e.g. continuity/topology, metric spaces, complex plane, etc. use the specific one for that context. If you are in a metric space context and the definition for metric spaces is not there but you have a topology context, consider adding that definition in the context you need it. Same applies to complex and real -- in analysis it derails the reader suddenly to explain a proof in real analysis by referencing a proof from topology.


 * If you have a particular reason to do it a particular way and can argue your case, then go for it. --prime mover (talk) 09:54, 30 June 2016 (UTC)

Riemann Integral
I have, at User:Keith.U/Sandbox/Riemann Integral, typed up a second popular definition of the Riemann integral (the one more devoid of Darboux). This is the approach that Bartle/Sherbert take, and I've seen it in a few other analysis texts. This proof is indeed equivalent to the definition given at Definition: Riemann Integrable. The page referenced, however, has a good quantity of related development on it, and the talk page leads me to believe that perhaps the approach I've prepared has been attempted before, and maybe exists somewhere else on the site. I'm gonna wait for your advice before I proceed with adding it to the main page, if at all. --Keith.U (talk) 14:54, 11 July 2016 (UTC)


 * This is another area where a more comprehensive job needs to be done. While I'm fairly happy with the definition of indefinite integral as the antiderivative, I've never been quite okay with what we've got for definite integral. So if you want to take on the job, always in the same overall philosophy of alternative definitions, linked by equivalence proof, then knock yourself out, as you krazy kidz say nowadays. $\odot_\smile \odot$ --prime mover (talk) 15:36, 11 July 2016 (UTC)

House Style
I do feel bad that I'm still not getting the hang of it. I see you've mentioned "sorting out links". Does this refer to linking to definitions? If so, I can see from some recent revisions that I'm probably not doing so enough. I'll fix that. Otherwise, what in particular is wanting? --Keith.U (talk) 15:48, 11 July 2016 (UTC)


 * a) We depart significantly from the Wikipedia way of linking only once per page to a concept. In we try wherever possible to make every occurrence of a term, however often it is referenced, into a wikilink.  In that way, the reader can instantly click on any definition or result on which a proof depends and go straight to the appropriate page, without having to hunt back up the page to wherever the concept was first raised.


 * b) Whenever we have a concept defined in several different contexts, e.g. Continuity in the context of Topology, Metric Spaces, Real Numbers etc., we make very sure that we link to the specific one. If a particular context has not been covered, but instead only exists in its topological context, for example, when it is to be used in the context of real analysis, then it is highly recommended that the missing definition is added. In that way a reader is not derailed by suddenly having to switch mental gears from the context of real analysis to that of topology (or group theory, as happens in at least one place in some of the Exponential work ongoing). This is undesirable because often students on Real Analysis may not have encountered either Topology or Group Theory, and links to such pages are incomprehensible without considerable investigation.  We want our proofs to be clear and as obvious as possible to follow.


 * c) A minor point, following on from b): such definitions usually have a forward slash in their page names, indicating they are subpages. We have (generally) made an effort to create a non-forward-slash redirect version of such a page. For example: Definition:Continuity/Real Function/Interval has a redirect Definition:Continuous on Interval which is to be used instead.  This is because if we come to refactor the database so as to move the definitions of (for example) continuity around, as has happened in the past, we need only change the single redirect page to point to the new destination, and hence less work for muggins.


 * d) I also suggest that you remove the extra spaces you seem to include in your wikilinks. While you may think it does not matter, and while it may look better, we insist on consistency. If all links are consistent, then global changes throughout the database by global search-and-replace can be done reliably. We have no desire to change the internal formatting of our source code, and personal preferences are unfortuntately to defer to our house style. This is in accordance with every successful software engineering business in the world. --prime mover (talk) 16:04, 11 July 2016 (UTC)


 * Will do. I've seen what happens to users who don't follow markup style :) --Keith.U (talk) 17:20, 11 July 2016 (UTC)


 * Some of them throw their teddies out of the pram, indeed. --prime mover (talk) 17:33, 11 July 2016 (UTC)


 * I've also noticed that some indented display-style equations will receive a space between the colon and the dollar sign, and some will not. What is the house style on this? --Keith.U (talk) 02:06, 14 July 2016 (UTC)


 * Crucifixion for the sinners who get it wrong. --prime mover (talk) 10:20, 14 July 2016 (UTC)


 * Okay. I am trying. --Keith.U (talk) 12:00, 14 July 2016 (UTC)


 * As I have pointed out before, just because there's a "tidy" template on pages you wrote does not mean you have to feel you, personally, have to tidy them. --prime mover (talk) 14:04, 14 July 2016 (UTC)

For your leisure
See Template:Subpage. It removes the need to tediously type the page name in case of transclusion set-ups. Enjoy. &mdash; Lord_Farin (talk) 13:59, 3 August 2016 (UTC)


 * Promising, but it has a hardwired heading level of 3. Is it worth defaulting to 2 but using a 3rd parameter or "level = " for subhead level? Or something? Back in a bit, emergency --prime mover (talk) 18:29, 3 August 2016 (UTC)


 * I find myself almost exclusively using 3, because I rarely refactor proofs. I'll take a look to see if I can make it configurable. &mdash; Lord_Farin (talk) 18:32, 3 August 2016 (UTC)


 * Several templates: "Subproof" template (level 2, which is "Proof n") with all the trimmings (not even any parameter required, "Corollary" (level 3, optional parameter n), and "Lemma" (again level 3) maybe? That takes care of most options. Optional title and/or subpage name, maybe. Oh, and "Definition" (level 3 of course). --prime mover (talk) 18:42, 3 August 2016 (UTC)

Ok I got it. Only thing we need to settle is whether 2 or 3 will be the default. What do you prefer? (It's only minor anyway, since we ought to use subst: to make it a one-time call.) &mdash; Lord_Farin (talk) 18:48, 3 August 2016 (UTC)


 * Since the only level 2s are proofs, probably best to make it 3. Everything's 3 except for "Proof" subpages. --prime mover (talk) 18:50, 3 August 2016 (UTC)


 * Done and tested. It all works. Plus I got to know that MW does not support self-recursing templates, so I had to write the Template:Header switch manually. Boo! &mdash; Lord_Farin (talk) 18:53, 3 August 2016 (UTC)


 * It leaves a big mess on the page:

Proof 1

... which could be considered a drawback. --prime mover (talk) 19:00, 3 August 2016 (UTC)
 * ... but then I'm using the subst in Subproof and perhaps I'm doing something too complicated.

Hmm... Somehow nesting of subst only works twice, which is surprising, and not so nice. I guess we can't have Template:Subproof in a meaningful way, then. Or it should be a blunt copy of Template:Subpage instead of transclusion. &mdash; Lord_Farin (talk) 19:13, 3 August 2016 (UTC)


 * The latter was what I did. --prime mover (talk) 19:17, 3 August 2016 (UTC)

To continue on the use of subst:</tt>, I think the last edit I did to Template:Subpage ought to have fixed the problem in using it for defining Subproof. But as the latter works as-is right now, it might not be worth the effort.

I wonder whether it makes sense to use e.g. the new Template:Defof only with subst:</tt>. I think it might make editing easier for those not intimately familiar with the site's technical mechanics. What do you think? &mdash; Lord_Farin (talk) 14:45, 14 August 2016 (EDT)


 * no doubt --prime mover (talk) 16:02, 14 August 2016 (EDT)

Simplifying the proof Element has Idempotent Power in Finite Semigroup
Hi, I think I have spotted an unnecessary step in that proof. As it seems that you authored it, could you have a look at my proposed simplification and maybe implement it if you agree? (I didn't do it directly for fear that I was missing something.) --A3nm (talk) 11:29, 3 September 2016 (EDT)