Talk:Main Page/Archive 3

The use of Q.E.D.
Although considered "traditional" in the field of mathematics, the use of "Q.E.D." is apparently, I believe, considered somewhat archaic nowadays, and a bit (perish the thought) camp. I believe (from reading around the subject and general communications) that a more common way of signalling the end of a proof is by using the symbol "$$\blacksquare$$" (whose LaTeX is \blacksquare) at the far right of the page, and for a subproof (e.g. a lemma proved in the course of proving a particular result) "$$\Box$$" (whose LaTeX is \Box).

What does anyone think? I have consistently not used Q.E.D. in any of my proofs, but then I haven't been using $$\blacksquare$$ either (mainly because it wasn't available when ProofWiki started and I never got into the habit). If we decide that the "house style" is to include Q.E.D. can we agree that it goes in a particularly fancy font, and have a template link in the ProofWiki Specific section?

Happy Hogswatch to all, btw. --Matt Westwood 07:45, 26 December 2008 (UTC)

I'm fine using either one, either the use of Q.E.D. or the squares. Since the use of Q.E.D. is on the way out we should probably use the squares, I'll make a template.

My thoughts are to have it called qed, and have a option for writing say and then it would put a white square, otherwise a black one. Thoughts? --Joe 16:51, 26 December 2008 (UTC)

No objections either way, although I've never been entirely convinced that adding QED at the end of the proof really adds anything (the proof is over, it should be obvious that it's done). That said, if we do use the square, I don't think it should be all the way to the right, I previewed a page with it and was looking for it, and I still missed seeing it the first time around. --Cynic-(talk) 17:09, 26 December 2008 (UTC)

I've noticed that too, maybe we should just have it so that it's put directly at end of last statement.

For example:

Therefore, $$x=\pi$$ $$\blacksquare$$

Thoughts?--Joe 17:18, 26 December 2008 (UTC)

Maybe put in some space before the box: Therefore, $$x=\pi\qquad\blacksquare$$

Unfortunately, putting a spacing command in (\qquad for example) at the beginning of a math section doesn't do anything, so it would be hard to have it be a template. I don't know if you can create your own functions for wiki LaTeX like you can on standard LaTeX, if you can, you could always make a \qed command. Also, don't put periods in after the box, it looks weird. --Cynic-(talk) 21:22, 26 December 2008 (UTC)

"Therefore, $$x=\pi$$ $$\blacksquare$$" works for me, although I wonder whether it might make the looknfeel more consistent to put it on a new line.

"My thoughts are to have it called qed, and have a option for writing say and then it would put a white square, otherwise a black one" works for me as well.

STOP PRESS: I've just got my hands on "Theory of Sets" by Bourbaki (happy xmas, father-in-law) and in the very second line of the mathematical exposition he uses $$\Box$$ as one of the symbols:

"The signs of a Mathematical Theory $$\mathcal{T}$$ are the following:


 * 1) The logical signs: $$\Box, \tau, \vee, \rceil$$.
 * 2) The letters."

Either this is going to have to be translated into a more "conventional" symbolism (goodness, that will take me a day or two, this book is hev-VEE) or we're just going to have to be really careful. --Matt Westwood 23:09, 26 December 2008 (UTC)

Well, as we're dealing with the geometry that's being posted, another issue has occured to me. \Box seems to be the best choice to be the symbol to represent a quadrilateral. We could always use $$\Diamond$$ (\Diamond) but I think \Box works better. Or it might not be an issue, since it would always appear before letters when representing a quadrilateral and on it's own line when representing QED. Thoughts? --Cynic-(talk) 21:13, 30 December 2008 (UTC)

Might not be a problem. The use of \Box for the end of a lemma is going to be rare because it's more usual for such sub-results to have their own pages anyway. In fact that might be a useful general recommendation, i.e. to put all lemmas on their own separate pages, what say? Then the issue wouldn't arise, we'd just have $$\blacksquare$$ for proof endings. --Matt Westwood 22:18, 30 December 2008 (UTC)

I agree. --Joe 01:07, 31 December 2008 (UTC)

WLOG and WRT
Would it be a useful idea to have a page for abbreviations?

Thus one could write WLOG for "without loss of generality" and WRT for "with respect to" and so on.

The initial thought as I started writing this post was to streamline the development of complicated proofs but the counterargument is that it may make the proof less transparent to a noobie. --Matt Westwood 10:10, 29 December 2008 (UTC)

I would make a Symbols:Abbreviations page to put them on, but I wouldn't link them, it'll just make proofs look a lot more complex. Besides, I think anyone who is interested enough to read a difficult proof knows what something like WLOG means. --Cynic-(talk) 16:54, 29 December 2008 (UTC)

I think we could even put them in as definitions. --Joe 16:57, 29 December 2008 (UTC)

I don't see how it would make proofs more complex, you'd just get WLOG appearing in blue. The only added complexity would be to the source code, and that should not really be a concern in this context.

I mention this point because there's already confusion with "iff" which I would have expected undergraduate level mathematicians to be familiar with (I had to uncorrect someone a few weeks ago who "corrected" what he thought was a spelling mistake for "if"). So I'm starting to consider linking "iff" with its def nowadays. --Matt Westwood 18:33, 29 December 2008 (UTC)

Categorization of Images
I think that we should begin to categorize the images before we get too many that get out of hand. I think this would be a good idea especially if you want to go and look to see if particular image may have been already uploaded.

Probably add a new category, called 'Images', then inside that have categories 'Geometry Images', 'Logic Images', etc. Thoughts? --Joe (talk) 01:23, 3 January 2009 (UTC)

Sounds sensible, but will need a word of instruction for those uploading images. Mind, won't most of the diagrams be geometry? --Matt Westwood 06:30, 3 January 2009 (UTC)

Yeah, number theory, for example, is not gonna need a lot of images. I think we need to make sure we have descriptive names for the images too, which would mean we meed to move some of the images we have now. --Cynic-(talk) 02:17, 4 January 2009 (UTC)

What I did with the proofs from Euclid (once I got into the swing of it) was to name the diagram after the proposition number (e.g. Euclid-I-13.png) and in the description field added the name of the link to that proposition (e.g. "Two Angles on a Straight Line make Two Right Angles"). Usually makes sense, before establishing a system for categorization of entities, to gather a load of those entities together first to see what sort of system may be needed. We have something to work with now but hopefully not too much to be too unwieldy to reorganize. --Matt Westwood 07:35, 4 January 2009 (UTC)

Definition
why defintion doesnt have categories??? i tnihk is a good idea.Maybe i can help but there is something that i dont know.. -- Gamma 23:07, 7 January 2009 (UTC)

Good question. It might make sense to have categories, but they would probably be best to have them be separate from the categories for proofs to keep things easy to find. So basically at this point, it's because categorizing them would be a lot of work. --Cynic (talk) 03:52, 8 January 2009 (UTC)

I've wondered about categorization myself, but we need to be more careful (and limiting) about what categories we use, otherwise the site's usability may suffer. --Matt Westwood 06:31, 8 January 2009 (UTC)

Definitions or Proofs?
Say there is a proof which, during the course of that proof, introduces a new term as the result of that proof.

The example currently under discussion between me and Zelmerszoetrop is Euclidean n-Space.

One way to handle this is to add "Euclidean Space" and/or "Euclidean n-Space" as an actual definition, and in it include a link to a theorem "Euclidean n-Space is a Metric Space", or merely to enter "Euclidean n-Space" as a proof, the content of which will (a) defined the term and (b) prove that it is has the purported properties that allow it to be considered as a "metric space".

I have been (fairly) consistently entering such definitions as proofs up till now (not wanting to clutter up the "Definitions" namespace with a lot of what at the time I considered to be extraneous material. Now I'm not so sure, as Zelmerszoetrop's preferred approach seems cogent and sensible.

More I think about it, more I like the way Definition:P-adic Metric has been configured. That is, we put the definition at the top, and then in a separate section any proofs (particularly the one giving rise to the definition in the first place) which are immediately relevant to that definition.

Thoughts, anyone? And if relevant, worth going through and revisiting those "definition by proof" pages and adding a new page for the definition and linking to the proof?

One downside of this (apart from the fact there's a fair amount of work to do, which for me isn't a problem) is that the possible need to split the Definitions page up into separate subcategories may become even more relevant because the number of definitions will grow significantly. --Matt Westwood 13:45, 11 January 2009 (UTC)


 * I really don't have a problem with either way of doing things, but it's certainly something we should discuss and settle on to produce a uniform style to the project. So in response to Matt Westwoods question, if we should go back to the the old "definition by proof" pages and revamp them to a new style, I would say yes, that whatever style we decide upon here should be uniform throughout the project, and I'll do my part to help with that.


 * Now, onto exactly how we should style this. The way I'm doing things, as Matt Westwood mentioned, is to create a definition page which contains as much information about a given concept as a user would need to answer questions they may have about proofs which link to that definition.  Any claims about a given topic can be proven on Theory pages and we can link to those theory pages from the definition page.


 * I understand the objection based on the enormous number of definitions that must be added, but to be frank, I think that is inevitable. At the time I'm writing, there are 1,421 proofs and 574 definitions.  I expect, or at least hope, the number of proofs will expand dramatically, and as new topics are explored (I'm just beginning the massive field of p-adic analysis, and I've gone through about 10 pages of lecture notes in topology, and hundreds of pages of notes to go), the number of definitions will necessarily grow large anyways.


 * Given the inevitability of a very large number of definitions on the way, it seems only reasonable to me that it be split into a variety of sub-categories. Even if the number of definitions was fixed at less than 600, I would endorse this because it is rather a pain to go through the current list and find the definition I'm looking for when writing proofs.


 * I'd be interested to hear anybody else's opinion on this: is there a better way to incorporate proofs of claims made about concepts in definitions? How to best categorize definitions? Zelmerszoetrop 14:06, 11 January 2009 (UTC)


 * I kind of like what we did with types of angles, that is to say define them on a page like Definition:Straight Angle and then link to the proofs of characteristics. And yes, since we can search the database, I don't think it's particularly problematic if it gets big.  --Cynic  (talk) 18:01, 11 January 2009 (UTC)

Heading Style
Would you mind using == Theorem == instead of = Theorem = when you're writing proofs? Just a stylistic issue, we've been mostly doing it that way across the site. Thanks! --Cynic (talk) 18:23, 11 January 2009 (UTC)

I confess I only started doing it like that because of the "Level 2 Headline" box in the boxes at the top of the edit screen. It occurs to me that sometimes one needs to use quite a deep hierarchy and then it pays to start with = Theorem =. Besides, at === and deeper you don't get dividing lines between the sections. So I wonder whether we might want to treat it as a guideline rather than a mandatory requirement. --Matt Westwood 18:36, 11 January 2009 (UTC)

Personally, I think that it the Level 1 Headline looks huge. There is also good precedent for it, since it's what wikipedia does. As usual, it would be a lot of work to standardize every page to level 1 headlines. We could make it a suggestion, but I think it would be better to only use level 1 if we're going to have several levels of subheading. And if you have a lot of subheadings, you'll probably have a Table of Contents, and we could always insert horizontal lines with after a low level subheading if it would be unclear without it. --Cynic (talk) 18:55, 11 January 2009 (UTC)

registration coming up
Hey guys, fee's for ProofWiki are coming up soon and I would really appreciate it if anyone could help out with the cost. I'm planning on switching to a different host, but that won't affect the price. The deal I'm looking at right now is about $80 for a yearly contract. I don't mind paying for it myself but of course would rather not! So if anyone can help out that would be really awesome! If you can go here.

Anyone think it would be too tacky to add a link on the main page to the donate page and see if we can get random donations? --Joe (talk) 18:07, 15 January 2009 (UTC)
 * How's that tacky? Wikipedia just had that huge "personal plea" from Wales on the main page.  I'm all for it.  Now, I'm not sure, but I might be able to throw in about $20 bucks - when do you need it by? Zelmerszoetrop 19:51, 15 January 2009 (UTC)

I can bung you a few quid via PayPal if you like. Email me with details. --Matt Westwood 21:43, 15 January 2009 (UTC)

Awesome, thanks guys! You can donate by paypal at site support. I'm hoping to do the move this weekend, but anytime at all would be fine. I'll just pay for it personally, and just add the difference after. --Joe (talk) 22:15, 15 January 2009 (UTC)


 * Sorry, my paycheck was a little less than I thought, and I'm a college student. 10 was all I could afford right now, but I just sent it out through paypal. Zelmerszoetrop 21:16, 16 January 2009 (UTC)

No need to be sorry, any at all is great help. That's 10 less I have to come up with! Thanks for donating! --Joe (talk) 01:16, 17 January 2009 (UTC)

I tried to donate, but it doesn't seem to like my cards (http://i43.tinypic.com/11hvs69.jpg). Maybe they don't want Americans donating to Canadians or something stupid like that. I sent a complaint to paypal, and I'll try again when they get back to me. Sorry :( --Cynic  (talk) 20:16, 18 January 2009 (UTC)

It's fine for Brits. --Matt Westwood 21:32, 18 January 2009 (UTC)

Worked fine for me, I'm in America. Zelmerszoetrop 00:08, 19 January 2009 (UTC)

Lemmas and Such
Do we want the boxes around the lemmas, or should we set them off some other way? --Cynic (talk) 18:08, 19 January 2009 (UTC)


 * I'm really not sure. I haven't even been consistent myself; I wrote both Fundamental Theorem of Finite Abelian Groups and the still incomplete Classification of Compact One-Manifolds, the latter of which uses the convention on Existence of Non-Measurable Sets and the former of which uses bold type.  It's something that needs settling.
 * In fact, I'm not even sure we should be putting lemmas on pages at all - maybe they should have independent pages of their own. Since this is an issue that applies to all articles that need lemmas and not just this, I'm moving this discussion to the main page. Zelmerszoetrop 18:50, 19 January 2009 (UTC)

This also seems like a good time and place to discuss a single convention on cases, as seen in Ostrowski's Theorem. Zelmerszoetrop 18:55, 19 January 2009 (UTC)

What's the difference between a lemma and a theorem anyway? Surely just a matter of degree? Is it that a lemma is (generally) used in only one proof? There's another page where there's a lemma proved (Lagrange's Theorem I believe) which has a far wider application than just on that page (and in fact has been proved elsewhere anyway). I need to tidy that up. --Matt Westwood 19:03, 19 January 2009 (UTC)


 * There is no formal distinction. A lemma is frequently something which is not used on its own, but rather to prove one or more theorems, while a theorem is something that can be independently applied.  The boundaries are fuzzy.  I like the idea of putting the more important lemmas on seperate pages, and it wouldn't be hard to do. At the same time, take a look at the page that started this discussion, Existence of Non-Measurable Sets.  The lemmas used here are truly specific to the proof, and would be unlikely to be used in anything else.  Breaking the lemma onto a seperate page only serves to obscure the proof and break the flow of thought of the end user.  It would be possible to write these lemmas as integrated parts of the proof, rather than stand-alone lemmas as they are now on that page, but I think that also serves to obscure the proof as having distinct parts.  Zelmerszoetrop 19:59, 19 January 2009 (UTC)


 * I agree that major lemmas should have their own pages. One thing we could do cor minor lemmas is give them a subpage.  For example, we could have Existence of Non-Measurable Sets/Lemma 1 so they are separate for purposes of reading but grouped for purposes of finding them.  I will also note that I find the boxes unpleasant to read from.  --Cynic  (talk) 03:40, 20 January 2009 (UTC)

Another style issue: bold or italic?
When introducing a definition, should it be bold or italic? For example, "a definition is a series of words saying something", as opposed to "a definition is a precise way of confusing people." I've been using italic for some time now (just seemed natural). However, User:Zelmerszoetrop has been using bold for, it seems, better effect. I don't personally like the look of bold, but I admit it does seem to stand out more.

Would it be appropriate to ask for a style ruling while we only have approx 500 defs to go back and alter? --Matt Westwood 20:26, 21 January 2009 (UTC)

Well spotted. In papers I usually use italics, but it seemed natural here to use bold definitions. They stand out better; also this is consistent with Wikipedia's use. lasserempe 08:57, 16 February 2009 (UTC)

Uncategorized Pages
Hey, I'm not sure where to put some of these! --Joe (talk) 18:22, 23 January 2009 (UTC)
 * Took care of it. Zelmerszoetrop 18:53, 23 January 2009 (UTC)

Group Examples
I was looking at Category:Group Examples, and I was thinking, why not consolidate this category?

My suggestion is three-fold:


 * 1) Make definition pages for a variety of important groups, ie, $$\mathbb{Z}_n, A_n, GL(F,n)$$, etc.
 * 2) Make two pages, one for finite/discrete groups, and another for continuous/lie groups, which contains proofs of group structure for each of the important examples
 * 3) Remove material that is unneeded or extraneous.  Some of the groups in the category are not likely to be used again in other proofs; I'm talking for the moment about this one, this one, and especially this this last one, since it lacks an operation description.

This way, when we begin Lie theory and other really advanced topics, we have a single page to refer to for these proofs. Any or all of these propositions can be enacted separately. Any thoughts? Zelmerszoetrop 02:43, 27 January 2009 (UTC)

I agree with all of these suggestions, although depending on how many "important" groups there are and how much information each one needs, we might want to combine some or all of them onto a Definition:Important Groups page. However, as I consider this, it may be a horrible idea. --Cynic (talk) 04:51, 27 January 2009 (UTC)

I'm very, very, VERY anti the idea of removing stuff, unless it's utterly wrong. Whether a proof is used as a step in another proof is irrelevant. "Let's not bother with a page for FLT, it's not used anywhere else ..."

The "important groups" should have their own special pages, agreed. And there's nothing to stop a separate category being set up called "Lie Groups" or whatever. If they're important enough to have books entitled "Lie Groups" then they're important enough for a category. And as for discrete groups, I anticipate a category (once we get out that far) for all the work contributing towards the Classification project (the one that culminated in the Monster back in the 80's).

However you want to play it, just don't remove any stuff, okay? Not until you've shown that this last one is most definitely not a group! --Matt Westwood 06:35, 27 January 2009 (UTC)

We definitely should not delete a proof, unless it is wrong or is a repeat; even if it has no applications. Saying that I totally understand where you coming from and your points make sense, especially when trying to link areas of math together. I think maybe the best apporach would be though categorization as Matt suggested. --Joe (talk) 10:11, 27 January 2009 (UTC)

Metric Spaces and Analysis
I'm setting myself a new project: to go through all the basic results in Category:Analysis and recraft them in the context of the general metric space.

It would be good if someone were to follow along behind me with a shovel and a bucket and clean up. I'm bound to make mistakes, I haven't actually tried this before. --Matt Westwood 07:34, 2 February 2009 (UTC)

Lighten the load?
Having spend what feels like 10 hours today in meetings with various company directors and assorted bigwigs today wrangling over resourcing and funding, I feel somewhat emotionally and mentally shattered, and my work head got lost on the way home, so I'm wearing my happy-head for a change ... and I thought: what about a category (oh all right, a page, then) for jokes? Each one can be linked directly to the theorem or definition to which it is relates. Come on, you know how good an idea it is ... --Matt Westwood 22:57, 4 February 2009 (UTC)

works for me. --Cynic (talk) 02:07, 5 February 2009 (UTC)

sounds good, how about Jokes? --Joe (talk) 03:17, 5 February 2009 (UTC)

Not sure how the category works as such, you might have to adjust how I've done it. --Matt Westwood 07:22, 5 February 2009 (UTC)

Our License
I recently received the following email regarding the ProofWiki license:

Recently I had the idea of building almost exactly a website like Proofwiki, and the I Googled around a bit to find that the idea is there... I really think the idea is wonderful, can be very useful, and can get very far. I'm a professional mathematician and would like to convince some people to start working on adding things to Proofwiki.

I write to you because I have a suggestion that in my opinion is very important. I have seen that the license for pages at Proofwiki is a Creative Commons License which does not allow commercial uses. I think this can be a major drawback for the following reasons: In the fututre, if the site grows, it is possible that people want to use it to include parts in mathematical works, or to do all kinds of compilations of material from the site, and this will not be allowed by this license if one wants, say, to sell a book containing text from the site. One of the uses I had thought for a site like this is to contain complete proofs of large theorems, which would need help from many mathematicians, and I think this kind of license can be a disadvantage when getting that kind of disinterested help. Besides, as the site grows, it will be impossible to change the license, as then one would have to ask all previous contributors. Wikipedia is a wonderful site, the people who work there have learned a lot on how to make it work, and I think they have good reasons to publish everything under the GFDL license.

Would it be possible to open a discussion on Proofwiki about this?

Please forward this message to any admin of Proofwiki, and anyone who may be interested.

The idea of gathering complete proofs is a really nice one. I will like to see how Proofwiki evolves. Let me know what you think about this!

Originally I had no particular feelings toward any license, but I have been thinking of changing to the GNU license for a while now. So if nobody objects I'll switch to this license. It is the one that Wikipedia uses and it works well for them. --Joe (talk) 00:06, 11 February 2009 (UTC)

100% in favour. --Matt Westwood 06:23, 11 February 2009 (UTC)

That was a fast answer! --Cañizo 10:42, 11 February 2009 (UTC)

Awesome! --Joe (talk) 11:10, 11 February 2009 (UTC)

I am not certain in how far the GFDL is appropriate for the uses mentioned - as far as I understand, any derivative work would itself need to be published under the GFDL; in the case of printed work, it would also need to contain a copy of the license itself. Or am I mixing things up? This being said, I have no problems with either of these licenses being used. lasserempe 09:05, 16 February 2009 (UTC)

If I understand correctly, a book including material under the GFDL would have to be published under the same license and include a copy of it as you say, but the point is that someone may want to sell printed copies of it. Also, someone may want to get paid for writing derivative works on the material of the site, say, mathematicians who are paid for writing something new but want to include some known things from this site. Maybe this would legally be a "commercial use". I'm not sure on the legal issues, but the GFDL would avoid these possible problems.--Cañizo 02:24, 17 February 2009 (UTC)

Isn't the GFDL the license were under right now? --Joe (talk) 02:53, 17 February 2009 (UTC)

Delimiters for writing math
A suggestion: given that writing math is very common in ProofWiki, it would make sense to have a shortcut delimiter instead of. The one used in LaTeX, a dollar sign, is shorter and would save a lot of typing. Cañizo 23:40, 11 February 2009 (UTC)

There's a button at the top of the edit pane which puts them in - it's the $$\sqrt n$$ one.--Matt Westwood 21:26, 12 February 2009 (UTC)

I may be able to add a key combination that will instert $$$$ into the code when you press it. --Joe (talk) 12:47, 12 February 2009 (UTC)

If you do that, put the cursor automatically in the middle after adding the text... one would need to do it anyway. I still think that using the dollar sign would be nice: consistent with LaTeX, to which most mathematicians are already used. And I don't see a good reason not to do it, as the dollar sign will be otherwise very rarely used in Proofwiki (unlike in Wikipedia). Also, external editors like Emacs could be used in LaTeX mode with no modification. Cañizo 14:01, 12 February 2009 (UTC)

I completely agree that it would be a lot better to use dollar signs. I certain that it can be done, but not quite sure on how to do it. I would have to write a new extension for the mediawiki software. Hopefully if I get some free time this weekend I'll look into it and see what I can do. If anyone has any ideas of how this can be done please let me know. --Joe (talk) 15:38, 12 February 2009 (UTC)

whoops, added sig to my post above. --Matt Westwood 21:26, 12 February 2009 (UTC)

I'm still working on the key combination, but I was thinking; would it be better(faster) if we made two templates {$} and {/$} that were essentially $$ and $$ ? --Joe (talk) 15:37, 14 February 2009 (UTC)

I can't see the point. Isn't the "square root of n" button above the edit pane good enough?

Oh, and incidentally, I was reading a LaTeX page the other day (can't remember where) that says the dollar delimiters are deprecated nowadays. I would argue against it, as it then limits the portability of our formulas to other math sites (particularly MathHelpForum may be a site for which this one is a useful resource) which has the usual math delimiters. --Matt Westwood 17:42, 14 February 2009 (UTC)

References inside a proof
Can one refer to a previous equation inside a proof by using some kind of wiki syntax? Say, for example, "from equation (7) we deduce that...", as one does in LaTeX. Cañizo 00:14, 12 February 2009 (UTC)

I asked about that another time - apparently not. Best is to make your own labels and refer to them explicitly. Never mind ... --Matt Westwood 06:20, 12 February 2009 (UTC)

Details in Proofs / Subpages?
I am copying below some comments I made regarding details in the proof of the Riemann Removable Singularities Theorem; I believe this point is worth discussing more generally.


 * There is always a tension in mathematical writing between the level of detail and keeping proofs understandable. The idea of giving every single detail makes it essentially impossible to present more complicated results in a way that anyone will ever read. Indeed, people likely to read these proofs would be expected to fill in minor details themselves - I do not think that any complex analysis textbook would give the details mentioned above.


 * (Another issue is the complete formulization of mathematical proofs, which is also a worthwhile undertaking. However, I believe it complements rather than replaces the idea of writing proofs that have the ideas clearly presented and are readable by humans.)


 * Of course, electronic media have other opportunities than print, and it could be possible to exploit this. What I mean is that, within a proof, if there is a statement "clearly ...", this statement could have a link that provides additional details. Thus, a reader who understands this point (or is willing to accept it at least for the time being) can read on easily, while those who would like to see the details can follow the link. Also, if someone writes the overall proof, then such little details can be filled in by others, without encumbering the full flow of the argument.


 * I am not entirely sure how easy it would be to implement this with wiki technology, without creating a plethora of additional pages. Are subpages (and sub-sub-pages etc.) supported? I believe this is a discussion worth having, and perhaps this comment should be moved to another, more public thread. If ProofWiki intends to grow to include much of classical mathematics (not to mention contemporary results), these questions should be addressed sooner rather than later. lasserempe 09:02, 16 February 2009 (UTC)

I have this argument regularly with my colleagues at work. IMO if you're going to write to communicate you've got to explain stuff, or at least provide a reference to an explanation. In the proof in question, the bit which is supposed to be taken as obvious may well be obvious, but it still needs to be specified somewhere. And then all you need is a little blue link to the proof. The whole point of ProofWiki is that it is possible to provide every single step, every definition, every explanation, because all you need to do is link to it.

As for the section:
 * As for the mathematics in question, by the definition of the $$o$$-Notation, the claim means that $$|f(z)|/|z-z_0|\to 0$$ as $$z\to z_0$$, if $$f$$ is bounded near $$z_0$$. This is trivial because the product of a bounded sequence and a null sequence is again a null sequence.

All very well, but it's trivial only if (a) you know what the $$o$$-notation is (it's not defined here yet) and (b) you understand what the product of a bounded sequence and a null sequence actually is. None of this is explained anywhere.

Either this site is to be truly and usefully educational, in which case it should be possible to explain any proof to anyone merely by means of a logical flow of explanations and definitions, or it's just an exercise in showing off to each other how clever we are. If it's to turn into the latter, then please delete my account and bar me from any more edits or I'll be tempted to petty vandalism. --Matt Westwood 18:59, 16 February 2009 (UTC)

For the record, subpages are easy enough. Just take a look at the archives of this page, for example: Talk:Main Page/Archive 1 and Talk:Main Page/Archive 2. Just put a slash after the name of the page you want to create a subpage of and then put the name of the subpage. Whether we want to do that is a separate question. --Cynic (talk) 19:53, 16 February 2009 (UTC)

I'm sorry, Matt, but I think your comment about 'showing off how clever we are' trivializes and polemicizes a serious issue. I can similarly say, if you want to have every proof broken down into the smallest details (on its main page, I add), then you can count me out, because it will not be of any use to anyone once the proofs get more complex. I have read as many articles where the idea of the proof was buried within the details as those where the argument was so vague that a non-specialist would not be able to follow without reading scores of previous papers by the authors. Neither of these is desirable, and finding the right balance for the intended audience is never easy.

You talk about explaining the proof, and I find that this is precisely the tension here: what level of detail makes a good explanation? It depends to some degree on your readership.

Because of this, I do really think that the digital format has great advantages here. Details can be provided via a link, without interrupting the flow of the explanation. And if someone comes along, reads a proof and thinks "this step was not really clear to me, so it might not be clear to others", there is the choice of whether to add it to the full proof, or link it to a subpage. But I do not want to start doing this unless there is a consensus that this is a good idea. lasserempe 22:33, 16 February 2009 (UTC)

It's the way it's been done so far. Okay, so you don't want to bother with the fine detail, okay, so don't bother with it, but at least recognise the fact that there's something there which needs to be proven, and leave a link for it. I don't believe we need every detail of every proof all on the same page, that's the whole point of this format. But every statement needs a proof and/or a definition, or it's not a proof.

I don't see that "a plethora of additional pages" is a problem in wiki format like it is in books. There are in fact few books that are worth a cat's cuss because the author hasn't bothered to remember that what is obvious to him is not necessarily obvious to the readers (that appallingly unreadable Hocking and Young for example), OTOH the works of Stewart, Sutherland and Knuth are limpid and lucid because they do explain stuff. So exactly how many pages do you want to limit this site to? Do you want me to delete half the pages here? --Matt Westwood 22:43, 16 February 2009 (UTC)

I think the level of explicit detail on a proof should be related to the result you are proving. One should completely write out ideas which are new to the result, or which are part of the main difficulty, and make a decision on where to stop for the rest. Here, if you're proving a result on complex analysis, one can probably assume that the reader knows about sequences, as this is the most common, but one should certainly have a link to the "o" notation page. Also, I don't see any problem on having lots, I mean lots, of pages for small results. How about a page called "the product of a bounded sequence times a null sequence is a null sequence"? Not even a subpage, but a page by itself. That result is assumed to be much simpler, and the proof should correspondingly assume less from the reader. --Cañizo 01:45, 17 February 2009 (UTC)

I agree with Cañizo. We don't need to try and limit ourselves. If a result can be proven it should have its own page. --Joe (talk) 01:52, 17 February 2009 (UTC)

I think we should try and stay away from subpages, except for special cases. Say if you have a HUGE proof, and want to break it up into sections. --Joe (talk) 12:55, 17 February 2009 (UTC)

Indexing
It would be useful to have, for each definition / theorem, easy access to the articles that use it. As ProofWiki grows, inevitably some statements / definitions will be adjusted, and it would be important to be able to check that such changes do not mess up dependent articles, imo. lasserempe 09:02, 16 February 2009 (UTC)

Just click the "What Links Here" button at the top of the last (toolbox) category on the left side toolbar. --Cynic (talk) 19:53, 16 February 2009 (UTC)

Doh! Thanks ... lasserempe 22:08, 16 February 2009 (UTC)

Naming Conventions
It seems that e.g. searching for "Continuous Function" does not bring up "Definition:Continuous Function" as a search result, presumably because there is no space after the colon. Would it not make sense to change this? lasserempe 14:52, 16 February 2009 (UTC)

In order for the search to return the definition, you need to click off the tick box for the definition namespace. In your user preferences you can select which namespace(s) you would like to search by default. --Joe (talk) 18:02, 16 February 2009 (UTC)

About the naming conventions: one could redirect the page "Continuous function" to the page "Definition:Continuous function" to avoid some hassles (searching as you mention, or linking to it when mentioning continuous functions), and it would lead to no confusion. Actually, maybe one could do without the "Definition:" prefix as long as pages are categorized and one clearly indicates in the page that it is a definition (as with a Definition heading). Usually, names of pages for definitions have no verb (something), while theorems are usually statements (something is something else), so one can usually tell them apart by looking at the title.--Cañizo 02:07, 17 February 2009 (UTC)

The "Definition" indicates a different namespace from the original one. Since the site was created with the goal of proofs in mind, we left the main namespace reserved for just proofs. Since some people thought it would be a good idea to be able to link to internal definitions, we created the definition namespace just for the purpose of stating definition. Separating the two allows us to keep track of how many proofs/definitions we have separately. --Joe (talk) 03:02, 17 February 2009 (UTC)

Use of subset symbols
I have just noticed that here the convention seems to be that $$\subset$$ denotes the "strict subset" relation. I believe that it is by now almost standard in mathematical writing to use this symbol to denote the regular subset relation, so I see a big potential for confusion here if we are not careful. Perhaps we should agree to only use $$\subsetneq$$ and $$\subseteq$$; in this case no ambiguity can arise.

IMESVHO using $$\subset$$ to include the $$=$$ possibility is even stupider than trying to clean your teeth with a chainsaw. By all means use $$\subsetneq$$ when in doubt, but I'm continuing to use $$\subseteq$$ consistently to mean what it obviously means and $$\subset$$ to mean $$\subseteq \land \ne$$.--Matt Westwood 18:47, 16 February 2009 (UTC)

I agree with Matt, using $$\subset$$ to include equality is just silly. --Cynic (talk) 19:53, 16 February 2009 (UTC)


 * Why is it silly? It is only a convention, one way or another. As a research mathematician, I cannot remember the last time I have seen an article that uses $$\subset$$ to denote strict inclusion. (The LaTeX code for it also gives this away ...) In my experience, using $$\subseteq$$ is more common in schoolbooks than at university / research level, but it may also simply be that different authors use different conventions.


 * Here is a semi-objective argument against the use of $$\subset$$ as excluding equality: the "proper subset" relation includes two requirements, namely being a subset and not being equal. Hence it is used if it is to be emphasized that equality is not allowed, and the $$\subsetneq$$ symbol makes this very clear. Also, "subset" is a relation that gets used much more frequently than "proper subset", and hence one might argue that it should have the simpler symbol.


 * In any case, my point is that there is a clear ambiguity in usage. This is also noted on Wikipedia, who have taken the route I suggested above. If we are really interested in creating a good database of mathematical proofs, we should avoid possible ambiguities, and banishing the $$\subset$$-symbol from these pages seems like the best way to achieve this. lasserempe 22:08, 16 February 2009 (UTC)

By that argument, $$<$$ should be used for "less than or equal to" as its use (except for all that pointless Analysis epsilon-delta rubbish) is more common than "less than". --Matt Westwood 22:19, 16 February 2009 (UTC)


 * I tend to use both of these symbols rather evenly, I would say. And referring to the foundations of modern analysis as "rubbish" is perhaps not helpful ... anyway, I did say that it was only a "semi-objective" point. The thing is, I could be persuaded that using $$\lneq$$ everywhere instead of $$<$$ would be a sensible thing to do, but there isn't really a pressing need, given that there are no conflicting uses. I am not intent on picking a fight; I just want to point out that these uses exist, that there are good reasons and precedents for both of them, and that, if anything, the inclusive use of $$\subset$$ is more prevalent in professional mathematics. (I just openened a random Annals of Mathematics paper, to confirm my suspicions, and indeed the symbol is used in this way there.) Mathematics is all about clarity, and there is an unambiguous solution available - why not use it? lasserempe 22:43, 16 February 2009 (UTC)

Just shows how utterly stupid modern mathematicians are then. --Matt Westwood 22:47, 16 February 2009 (UTC)


 * Nice to see that you are interested in civilized discourse. I thought this was a very worthy project, and the energy invested by you and others is laudable. However, if the input of professional mathematicians is not desired on here, then I will invest my time in other things. Nice seeing you. lasserempe 22:54, 16 February 2009 (UTC)

I personally think that the person who is writing the proof gets choice on which symbols they prefer to use (so long as they are within reason). If there is ambiguity with a symbol then it should definitely be clearly stated how it will be used. If somebody wants modify the page later then so be it.

Also, I don't believe that every step has to be expressed/explained at the time of writing a proof. We can always come back and link to uncreated pages/definitions; we don't necessary have to build from the ground up.

Another point (and definitely the most important), I believe that everyone is entitled to their opinion, but should be respectful towards others. People shouldn't be persecuted for what they think is right/wrong. Everyone deserves the right to have their opinion heard. We don't need to be losing members over silly arguments, when these problems can be solved by peaceful negotiation. --Joe (talk) 00:05, 17 February 2009 (UTC)

I agree with Joe, the person making a page should determine which symbols are used. That said, more clarity has never hurt anyone, so using the most precise symbols available is never bad practice. -Andrew

Oh all right I take that back then. I just think that a lot of the "modern conventions" of mathematical symbology are utterly, utterly stupid. And I have absolutely no respect for convention when it's stupid. This site has the potential to make a difference. --Matt Westwood 06:29, 17 February 2009 (UTC)