# User talk:KarlFrei

Welcome to ProofWiki! Since you're new, you may want to check out the general help page. It's the best first stop to see how things are done (next to reading proofs, of course!). Please feel free to contribute to whichever area of mathematics interests you, either by adding new proofs, or fixing up existing ones. If you have any questions please feel free to contact one of the administrators, or post your question on the questions page.

Here are some useful pages to help get you started:

• Community Portal - To see what needs to be done, and keep up to date with the community.
• Recent Changes - To keep up with what's new, and what's being added.
• Check out our house style if you are keen on contributing.
• Main Page talk - This is where most of the main discussions regarding the direction of the site take place. If you have any ideas, please share them!

Cheers! prime mover (talk) 03:51, 6 September 2018 (EDT)

## New proofs of existing results

Hi

Please note that when you develop a new proof of a result which has already got a proof for it, this should be added as a separate proof. Please do not replace an existing proof.

This is specifically with reference to Inverse of Positive Real is Positive. --prime mover (talk) 08:31, 18 September 2018 (EDT)

Sorry. I did not view this as a new proof but only as a constructive version of an existing proof (basically just reorder the lines). However, I've now realized that it doesn't even work, so I've gone back to the previous version. Again, apologies! KarlFrei (talk) 08:43, 18 September 2018 (EDT)

## New proofs of existing results

Hi

Please do not just replace an existing proof with a new one, even if the new one does look better to you than the one it replaces.

I have reversed out your latest edit, and I will now do the job of re-incorporating the new one as a separate proof. --prime mover (talk) 07:41, 23 September 2018 (EDT)

... There. Like that. --prime mover (talk) 07:51, 23 September 2018 (EDT)
Sorry, I am really not trying to be difficult, but in this case I thought the replacement I did was obvious.
The old proof was essentially A -> B -> C -> D and D -> C -> B -> A (wrapped in two proofs by contradiction). I replaced this by A <-> B <-> C <-> D. Surely this kind of change is unproblematic?? What is the point of keeping the two directions separate? KarlFrei (talk) 11:14, 23 September 2018 (EDT)
Also, it seems you could at least keep in the proofs by contraposition which I put in (and which you have been allowing so far). I really don't see what is so problematic about these minor edits. KarlFrei (talk) 11:18, 23 September 2018 (EDT)
Your new proof remains in place. As for the one you changed from being a proof by contradiction to a proof by contraposition, I can't see what you're complaining about. You deleted it. --prime mover (talk) 16:14, 23 September 2018 (EDT)

Even if a proof is supposed to replace another because it is "better" in some relevant way, it is usually good to have this idea checked by other contributors. We have merged two proofs of the same result on occasion, when the differences were deemed sufficiently negligible. However it should be noted that these were exceptions. — Lord_Farin (talk) 16:43, 23 September 2018 (EDT)

OK, I see. I have put the better version of Proof 1 back in and added an official suggestion to delete it (as I believe we only need proof 2). Please do assess this suggestion (at some point). Thank you! KarlFrei (talk) 02:46, 24 September 2018 (EDT)
It is only your assessment that your version of Proof 1 is "better" than the one that was there originally. I still remain unconvinced that a proof by contraposition is inherently "better" than a proof by contradiction. Both proof forms are valid. In fact, there is no reason why both forms should not both exist on this site.
I need to understand your motivation in reducing site content. You seem insistent on changing things to match your preferences, rather than adding your own valid content. This site attempts to document all approaches.
A longer, more discursive proof is friendlier to a newer and more inexperienced mathematician than a minimalist and compact proof, however more elegant and sophisticated the latter may appear. As the philosophy of this site is to allow intellectual accessibility to all, no matter where they are on their mathematical journey, it needs to be demonstrated rigorously that shorter and less explanatory proofs are just as effective as communicating the knowledge as the longer proof before the longer form is directly replaced by a shorter one. --prime mover (talk) 03:11, 24 September 2018 (EDT)
Hang on, I started off by adding some new content! Then I just came across this missing proof of contraposition (which was also an addition, come to think of it) and started to fix it (in my opinion).
I honestly thought that I was actually making things easier for the reader. All I am doing (in my view) is removing some unnecessary steps. I would not say that this is "reducing site content". I want the proofs to be accessible. And I know that each additional logical step potentially loses some readers.
"Reducing site content" was deleting a proof to make way for another one. --prime mover (talk) 04:53, 24 September 2018 (EDT)
I am sorry. I understand what you mean now. I guess the point is that I simply did not see this as deleting a proof, but merely combining both directions into one. KarlFrei (talk) 05:07, 24 September 2018 (EDT)
The proofs I have been changing are proofs by contraposition (OK, once or twice I had to make a very minor change to turn it into one). By this I mean that the assumption which is contradicted in the proof by contradiction is not needed anywhere in the proof until the final line. Look for instance at Path in Tree is Unique/Necessary_Condition. If you compare the two versions side by side, you will see that it actually contains more lines now.
I still need to be convinced that your way is better than the original way. A proof by contradiction is NOT a proof by contraposition until it is turned into one. Whether you prefer it to be a proof by contraposition or not, does not mean it is a proof by contraposition until it is turned into it. I confess that I don't care enough for it to be an issue. --prime mover (talk) 04:53, 24 September 2018 (EDT)
We should probably end this discussion here, but allow me to explain that intuitively, I would say that a proof by contradiction is a proof which starts off with an assumption and uses it to create a contradiction. The proofs I have been editing could put this assumption right before the last line as they do not use it (except that this would look really strange). All I am trying to do is save on logical steps. KarlFrei (talk) 05:07, 24 September 2018 (EDT)
But once you delete a proof (preferring a different approach), you can't really expect the one you deleted to be restored to the version you changed it to. (I refer to Preimages All Exist iff Surjection/Proof 1.) You deleted it. You didn't want it any more. So I restored the earlier one. If you didn't want it, why complain that I put the old one back? --prime mover (talk) 04:53, 24 September 2018 (EDT)
As I wrote in the comments for my last edit there, if we must have this proof on ProofWiki, I would prefer to have the one which has fewer logical steps. KarlFrei (talk) 05:07, 24 September 2018 (EDT)
Or look at Condition for Edge to be Bridge. (old version: https://proofwiki.org/w/index.php?title=Condition_for_Edge_to_be_Bridge&oldid=342171) All I had to do here was remove the initial step "Let e be a bridge of G" for the necessary direction, and "Let e not lie on a circuit" for the sufficient direction. These are steps which do not add anything to the proof. They are not needed inside (until the end). I hope that it is OK to make these kinds of changes.
Let me also add that I am all in favor of having multiple proofs for the same result! Having a repository is great, which is why I opened an account here. I recently saw a paper with about 30 proofs of the divergence of the harmonic series, and I added my favorite to the site. But having versions of a proof which only differ in 2-3 lines does not really contribute anything, I would say.
Finally, I do realize that I wrote down Preimages_All_Exist_iff_Surjection/Proof_2 in a very brief way. I can of course expand it to use more words. Then perhaps the correspondence to Proof 1 will be clearer as well.
I have made this change now. As an afterthought, I am 100% convinced that direct proofs are easier to understand (more accessible) than proofs by contradiction OR contraposition. I would be interested to hear an opposing view. KarlFrei (talk) 04:31, 24 September 2018 (EDT)
Hope this helps, KarlFrei (talk) 04:17, 24 September 2018 (EDT)

Well yes, it is okay to amend your own proof, but in this case I can't see why you would. It says exactly the same as the one you replaced, except that it does it in words instead of symbols. As such, it possibly merits inclusion as a separate proof in its own right.

As for me, I prefer proofs using symbols to those using words, for what are to me obvious reasons, but I understand others prefer words (possibly because they do not find it easy to learn symbols, maybe because of visual impairment).

I'm not going to argue tediously over whether Proof By Contradiction is better than Proof By Contraposition, as I have other things to do.

I am glad to hear it. But I really do want to ask your opinion about direct proofs. Can we come to a consensus that they are easier to understand than proofs that use a contradiction (in whatever form)? I ask because you stated earlier that (paraphrasing) shorter proofs are only better if they are easier to understand.
Whether a direct proof is easier to understand than an indirect proof is unimportant. What is important is that it is different from an indirect proof. If someone has written an indirect proof, and someone else has written a direct proof, there is no reason why both should not stay. --prime mover (talk) 08:31, 24 September 2018 (EDT)
I guess I just don't understand this community's insistence on keeping everything around if possible. Possibly this is because of my Wikipedia background, which of course has almost the opposite culture. (I tried to create a page there once. Oh boy!)
We are not Wikipedia. Let me repeat that: we are very not Wikipedia. Let me add further emphasis: we are very, very, very not Wikipedia.
Unless a page is genuinely wrong, or says the same as another page, we do not delete anything. Two different proofs are not the same. And even if a page is wrong, it does not get deleted unless it cannot be corrected. See -- we even have a page on squaring the circle! --prime mover (talk) 08:31, 24 September 2018 (EDT)

However this may eventuate, the suggestion is made that rather than retread a whole body of existing proofs, maybe you might like to consider where there are gaps that may need to be filled? There are plenty of entries in the Wanted Proofs cateogory. --prime mover (talk) 04:41, 24 September 2018 (EDT)

I have seen that list :-) Unfortunately, I did not immediately spot a theorem that I had a ready proof for. Abel-Ruffini would be a huge undertaking... KarlFrei (talk) 04:57, 24 September 2018 (EDT)
I gave you the link to the wrong category: Category:Proof Wanted --prime mover (talk) 08:23, 24 September 2018 (EDT)
Oh wow. Indeed, I had not seen that page yet... KarlFrei (talk) 11:05, 24 September 2018 (EDT)

## In defence of house policy

Please excuse the extended response below.

On the contrary, I am very happy to receive a longer explanation. KarlFrei (talk) 05:14, 13 October 2018 (EDT)

Wikipedia is not like $\mathsf{Pr} \infty \mathsf{fWiki}$. As we have tried to explain, $\mathsf{Pr} \infty \mathsf{fWiki}$ prides itself (if nothing else) on its neatness and consistency of presentation. It is expected of contributors that they make an attempt to conform to the existing style of presentation. In most cases it is sufficient for them to see what is already there, and craft their pages accordingly. It is vanishingly rare for them to react negatively to having maintenance templates attached to their work. It is usually the case that they accept the changes made to their edits, and learn by example.

I really don't care about the maintenance tags. Really! Much of the focus has been on this and I probably should not have listed this in my list of rules in the first place. The point is that in the short time that I have been there, I have been told over and over again that what I am doing is wrong (for various reasons). I hope that you can understand that that is incredibly frustrating, especially as for many of these things I had no way of knowing that I was making a mistake before I made it. KarlFrei (talk) 05:14, 13 October 2018 (EDT)

Some specific cases in point seem to have got under your skin as follows:

a) The business of deleting a page which you believe there is no use for. (You know the one I mean.) For a start, we have no reason to delete this page, as it documents a point which may otherwise be completely undocumented anywhere. It is amusing and harmless. It is what it is. Your motivation for wanting to delete it is unclear. Is it as a result of your familiarity with the philosophy of Wikipedia (which has a rich tradition of deleting material which is considered superfluous, for whatever specious reasons), of which you are apparently fond? As we have discussed previously, we do not adhere to that philosophy here. This appears to be a point which causes you problems. If so, then that is something for you to come to terms with.
I came here hoping that I could make this site more beautiful and easier to understand, by adding short and nice proofs that were not yet present, and yes, also by editing proofs that could be made nicer. This is what I do all the time in my daily life. It is very difficult for me to accept that this is not wanted here. It means that if I can improve any proof, it means that we will now have $n+1$ proofs instead of $n$ proofs, where two of them are extremely alike. If anything, this almost makes the website less attractive instead of more, and harder to parse ("why are there these two alike proofs?").
Matter of opinion, matter of aesthetics. Why should your opinion of what is prettier be more important than somebody else's? --prime mover (talk) 05:31, 13 October 2018 (EDT)
Of course it's not! But that is not what this discussion is about, is it? KarlFrei (talk) 05:49, 13 October 2018 (EDT)
Sure, one can have differences of opinion about which proof is easier or clearer, but that seems to be beside the point.
And yes, I am fond of Wikipedia and the freedom there, although I am well aware that it too has many systemic problems. But the vast majority of my edits get accepted there without comment. KarlFrei (talk) 05:14, 13 October 2018 (EDT)
So what? You are once again reminded that this is not Wikipedia, and the rules are different. I might just as well say, "I am allowed to carry a gun with me everywhere I go in Nevada, and I am never challenged. But whenever I try to take it to Europe I am sanctioned." --prime mover (talk) 05:31, 13 October 2018 (EDT)
Please. I was only trying to explain what I liked about that website. I do not expect this one to become a clone of it. KarlFrei (talk) 05:49, 13 October 2018 (EDT)
b) The placing of maintenance templates on pages written by you suggesting that the page may require amendment -- along with the suggestion that you become familiar with the house rules. Reading between the lines, I suspect that you don't like the term "house rules" as it seems to offend your ideals of "freedom". We do not hold that philosophy. There are indeed rules which define the shape of pages. People are requested that the pages they write adhere to those rules. Pages which do not adhere to those rules are (in due course) rewritten and adjusted in order that the pages do adhere to those rules. In the meantime, a maintenance template is placed on such pages, to alert contributors that work needs to be done to bring the page up to the accepted standard. Now, most contributors are perfectly okay with this, and don't mind other editors tinkering with their code layout, or order of presentation, or choice of words. It is hoped that, in due course, all contributors can reach that same level of philosophical acceptance.
See above.
c) In some cases there has been reversion of edits, which of course is always contentious. This usually goes back to the nature of what $\mathsf{Pr} \infty \mathsf{fWiki}$ is. It is not like an encyclopedia, it is more like a dictionary. We are not comfortable with material being added quodlibet. Pages have structure, and specific information contained therein is categorised into specific pages. We guard against material which is of limited relevance, and make sure that all the material that is included is placed in the correct place. Specific examples: historical information goes in a Historical Notes page. Information about the pronunciation, spelling variants, etymological derivation and so on goes into a Linguistic Notes page. And (perhaps most importantly) material which consists of mathematical statements of fact need to be rigorously justified by linking to a page which contains that statement together with a proof that it holds -- and then a link to that page is to be presented on that page in an "Also see" section of the page which the contributor believes it is relevant. Hence the pressure to eliminate "Notes" sections, as they are too tempting for a contributor to fill with prattle. It is appreciated that perhaps we may have been overly-zealous in reverting your edits, but some of what you have added is actually incorrect in the context of $\mathsf{Pr} \infty \mathsf{fWiki}$ (a specific case in point being your comment about PbC being dependent upon LEM, a point which has already been analysed on $\mathsf{Pr} \infty \mathsf{fWiki}$ in some depth).
I fixed that edit as soon as I understood my mistake. The policy on Notes is fine, as I mentioned before, it should just be mentioned!
How about you just accept the fact that you're making a fuss about nothing? --prime mover (talk) 05:31, 13 October 2018 (EDT)
Wow, and here I thought we were really making progress in understanding eachother. No, I do not think that this is nothing. KarlFrei (talk) 05:49, 13 October 2018 (EDT)
d) Editing the Help pages: I have actually just checked, and the few pages I looked at on Wikipedia do allow a casual user to edit them. I'm surprised. Whatever. We don't encourage the users to do that here. Sorry. The material you added may stay, and it may not, and it may be edited, depending on how many spare brain cells one of us editors can bear on the subject. But if you have suggestions as to things that you think need to be added to the House Rules (and which are not just a petulant response to what you consider an affront to your personal right to self-expression), then place them in the appropriate Talk page.

One final point: the reason I deleted the material on my own talk page is because I was unhappy with the tone of some of your remarks. I'm not saying I'm a paragon of virtue myself, but your words came across as spoiling for a fight. In another forum and on another website you'd probably have got one. --prime mover (talk) 04:13, 13 October 2018 (EDT)

I apologize for the tone of some of my previous remarks. I hope that you can understand how incredibly annoying and frustrating this whole process has been for me (and not just for you). I would not wish this on anyone. I would hope that we could attract additional quality contributors to this site in the future. But this is so much harder if we don't tell them what we are looking for right at the start. Again, this is NOT about the house style or about placing "tidy" tags on pages. It is about all the rules that were secret.
I'm sure you can likewise understand that a bunch of people who have been working hard on this site for a number of years get extremely annoyed at a new contributor coming along being appallingly rude to them because of being put right about straightforward, trivial matters. It will take a lot of hard work on your behalf to satisfy some of us that your actions are in good faith and that you have not come along for the purposes of disruption. --prime mover (talk) 05:31, 13 October 2018 (EDT)
I did not realize I was being appallingly rude, and I apologize. But as I keep saying, these are not matters that are straightforward and trivial to me. Instead they seem hidden and mysterious. KarlFrei (talk) 05:49, 13 October 2018 (EDT)
Anyway, I will get back to editing for a while as you suggest. KarlFrei (talk) 05:53, 13 October 2018 (EDT)
This was especially annoying to me as the solution was very simple: just expand the help pages to explain these missing rules. This is probably what I should have done in the first place (while I still had the chance). But since the solution was so simple, and the secrecy of these rules was so annoying to me, I completely lost my composure when you waved off some of my complaints as being about "minutiae" and this site "not being perfect". KarlFrei (talk) 05:14, 13 October 2018 (EDT)
Perhaps we have things we would rather be doing than instantly jumping to your command to update the rules. Maybe it is something we will come to in due course. maybe not. --prime mover (talk) 05:31, 13 October 2018 (EDT)
Again, I only wrote this because it seems so little work, and so easy. In all the time you took in replying to me (and for which I AM grateful) it would have been just as easy to briefly clarify these things in the rules. In no way should my suggestions be interpreted as commands. Of course not! I am merely trying to make things better for the next editor, who may arrive tomorrow. That is all.
Also, I do of course understand that you have other things to do. But please don't be so dismissive of suggestions for improvement. I really do apologize for my tone previously, but I do think they are reasonable. KarlFrei (talk) 05:49, 13 October 2018 (EDT)

May I suggest that this discussion is paused until the respective tanks for tolerance to different perspective are refilled? This is tiresome.

PM, Karl is not trying to force new rules, just trying to improve the onboarding of future contributors. Don't take offence.

Karl, PM is concerned that again someone tries to change what has proven quite effective. Don't take offence.

You both want the best for ProofWiki. Please make a mental note of this fact and keep it in mind when something confuses or enrages you.

Please take a break. This discussion has time. — Lord_Farin (talk) 06:21, 13 October 2018 (EDT)

## House Rules

The following are the house rules as I currently understand them. I had to piece them together bit by bit as they are not written down anywhere that I could find. As such, there may be things I have misunderstood; I welcome corrections.

You will have seen our house style. But did you know that we also have house rules? These are probably the most important rules that new users (particularly mathematicians, who should tread very lightly on this website) should know about, but to keep things interesting, we do not mention them anywhere in Help:Editing or our house style, and we certainly don't tell new users about them when they join. Where would be the fun in that? There is plenty of time to tell them when they start not following these rules (for some strange reason).

Here they are, in order of importance.

1. There is to be no deletion of any material. Ever. (You can tell because the pages Help:Editing and our house style conspicuously do NOT contain the word "delete" or "deleting". How much more obvious could we have made this policy?)
2. There is to be no change in any material on this website UNLESS
1. it is demonstrably wrong (and even then, it might be kept for historical interest)
2. it is not in house style
3. it is an expansion of existing material, by adding links to related concepts or definitions, adding missing steps in proofs, adding explanations for steps in proofs, etc. etc. ... Now added to the editing help pages
3. (As a logical consequence of rules 1 and 2 above:) Editing on this website means expanding and expanding only. Even though we have a general disclaimer stating that your text may be "altered" and even "edited mercilessly", that is just a bit of a joke we put in for laughs. If you write anything on this website, and it is mathematically correct and in the proper style, you may rest assured that it will still be there in a hundred years. Never mind if your proof is circuitous or hard to follow or understandable only by you: if we cannot show it is actually wrong, it will stay. Go for it! On the other hand, if you happen to see any step in a proof that can be simplified, then unfortunately the only way to get the simpler step on ProofWiki is to write a completely separate proof with this change, no matter how small and easy the change or how big the proof. Do not under any circumstances attempt to "improve" any proof by editing it. (Mathematicians beware!) If you come here hoping to possibly make some proofs easier, you can do that by adding a new version, but the version which you think is bad MUST stay on the website next to it, because we say so. This should now be clearer on the editing help pages as well, though the motivation remains murky.
4. Regarding the house style. This is another one of our little jokes. Even though that page says "House style", these are actually also RULES, and they are very strict and incredibly important. Following them is almost as important, if not more important, than actually contributing mathematics to this website. Ignore them and you will keep getting reminders until you either start obeying the rules (because they must be obeyed) or give up in disgust or possibly get blocked. As for questioning any rules, you are free to try it, and good luck!
5. Introducing a general "Notes" section is discouraged. Again, you could have seen this by noticing that there is no section "Notes" on the help pages, only more specific sections like "Historical notes". Must we spell out everything? Added to editing help pages
6. Discussing this list of rules is discouraged. See here.

To summarize: even though the main page of ProofWiki has collaboration as a primary goal, this should not be taken to mean that you can just start and change text as you would on that nasty Wikipedia. Remember the mantra: editing means expanding or putting articles into house style.

I feel compelled to respond. First and foremost, there have been experiences with people who replaced entire proofs with their own, completely unrelated, "better" variants. Seeing as the amount of space on this webserver is not a limiting factor, adding a second proof makes sense.
Additionally, a reasonable number of proofs have been added based on actual source works. Even if the proofs are substandard, it is then good to see how things can be approached differently (or "the old-fashioned way"). Especially in logic there is always the trap of circularity to be avoided as an extra consideration.
A next point to take in is the varying level of our readership, for which different levels of sophistication can be put in place.
Just to put this out here, the changes I have been making (or wanting to make) were pretty clearly (in my view, of course) simplifications and attempts to make the material more accessible, not less. Mostly it has only been a matter of removing some unnecessary logical steps. KarlFrei (talk) 11:12, 12 October 2018 (EDT)
Considering all this, it rapidly becomes a philosophical question if "simplifying" or "clarifying" a proof is a material change or just a small improvement. Hence we tend to err on the side of caution, but I would very much like there to be room for reasonable discussion. I hope to have nuanced the motivation "because we say so" mentioned under 3.
I will try to start such a discussion soon. KarlFrei (talk) 11:12, 12 October 2018 (EDT)
Regarding 4, I don't see why it should concern you too much. If you focus on adding mathematics, what is going wrong? Sure there will be maintenance templates added, but I struggle to see how this is a problem. Maybe you can elaborate.
This comment grew out of frustration with the various things that I was not allowed to do here. At times it seems that this website is focused on following the rules rather than creating a repository of beautiful and fully explainable proofs. KarlFrei (talk) 11:12, 12 October 2018 (EDT)
Regarding 5, this has grown by experience. If we insist on drawing the comparison with Wikipedia: for mathematical articles I often find that they are useless by their clutter and idiosyncrasy, and their lack of overall consistency. Everybody works besides each other. While this is surely a collaboration model, it doesn't make the layout a support for the content and as such makes the page less successful in its goal: getting information across to its readers.
As for 6 I can understand both parties. On the one hand the eager new contributor, feeling limited by seemingly arbitrary rules, frustrated by the inflexibility of the seasoned contributor. On the other hand the seasoned contributor having honed a workflow which has proven itself hundreds if not thousands of times, feeling challenged by a new perspective which still has to prove its merits, frustrated by the eager new contributor seemingly simply propagating their own idiosyncrasies.
I would hope that they spent the time to get acquainted with each other, managed to find a way forward.
And for those areas where finding a way forward without prior consensus endangers overall site consistency, the established standards should be properly identified in the Help section, so that everyone is aware of both the standards and their motivation. — Lord_Farin (talk) 14:47, 11 October 2018 (EDT)
I have begun editing the help pages. I have attempted to do this in a neutral fashion while still conveying the rules as they currently are. KarlFrei (talk) 11:12, 12 October 2018 (EDT)
Well that was a bit of a mistake on our behalf. I was convinced I'd protected those pages to be admin-only. --prime mover (talk) 12:30, 12 October 2018 (EDT)
Sorry, but to the best of my knowledge we would have decided that it would be against policy for anyone to edit help pages without a certain amount of discussion beforehand. And at this stage, I am not sure that it is a good idea to open the Help pages to general editing. If everyone else thinks differently, then yeah, let's open them again -- but there's a danger there in people amending the rules just to suit themselves with no idea of the history of the site. --prime mover (talk) 12:43, 12 October 2018 (EDT)
In any event, I am glad to see that you did not (yet?) revert my edits. I hope that my edits, which are apparently the last I will make on those pages, were at least helpful. I truly made an effort to write them down correctly, much as I disagree with them.
I had to sit on my hands. --prime mover (talk) 16:30, 12 October 2018 (EDT)
But why? Did I not write them down correctly? Do you actively want people to run afoul of these rules before telling them about them? I am trying to help here. If I have misrepresented anything, please do correct my mistakes to make the rules clearer, but please don't just remove them. KarlFrei (talk) 02:59, 13 October 2018 (EDT)
Regarding people just amending the rules, to me it is very hard to see how that could ever happen - but that discussion is moot now anyway. KarlFrei (talk) 14:20, 12 October 2018 (EDT)
Well, you just did it. I'm not sure, but I don't think L_F was actually inviting you to edit stuff. --prime mover (talk) 16:30, 12 October 2018 (EDT)
I am sorry, I was unclear. You wrote above "amending the rules just to suit themselves" and that is what I was referring to. I thought you meant "changing the rules to be more to their liking". This is just about the last thing I did. Regarding the lack of invitation, I didn't realize that I needed one! Perhaps I still have too much of a Wikipedia mindset (to be specific, be bold). KarlFrei (talk) 02:59, 13 October 2018 (EDT)
Mmm, tough call. On the one hand I think it is vital that discussion takes place before help pages are amended. In this regard I agree with PM. On the other hand I think that it might increase the maintenance burden on the admins.
I don't believe there should be much of a maintenance burden concerning the help pages. They should not need to be subject to constant maintenance, as it is not expected that they be changed much. --prime mover (talk) 04:13, 13 October 2018 (EDT)
But thinking more about it, if the talk pages are used for discussion, even of "add the following section between X and Y: <snip>", it should not be too much of a problem.
I guess the problem with "be bold" is that, as mentioned before, we have had "bold" users in the past who created an incredible pile of intertwined circular crap, and when confronted with it, left. And this has made us a bit more cautious (perhaps overcautious). — Lord_Farin (talk) 03:57, 13 October 2018 (EDT)

Have you thought of working in your sandbox? I have had countless conversations, mostly on similar grounds. There are things that I do not agree with, but not using sandbox was actually a mistake. Firstly, since it is a personal section, there is much less attention paid to what happens over there. Under some basic rules one can experiment with incomplete messy proofs almost uninterrupted. You could follow a very simplistic plan: write down or copy the proof on one day, work on style the second day, add all relevant hyperlinks on the third day and so on. Secondly, spending more time on one proof will allow to understand it better, and so enable the contributor to improve it by making it more approachable to a casual reader. At the same time, this tactic builds the character of a contributor, allowing him to evolve from a chaotic artist to a more mature content creator.

Improvement of already existent sections is tricky. I feel that many articles could be improved, but this is not always possible. Probably, the author of a given article should have the ruling decision. However, majority of articles have been done by very few contributors, so this issue is largely monopolised. Note that the same authors have been around for many years if not for a decade, and knowing the difficulty of managing such a project with very little support, you have to give credit where credit is due. Maybe you could start filling up one of the non-existent major branches of mathematics? That way you would have much more control over quality, and gain experience needed to manage larger data structures. Julius (talk) 16:29, 11 October 2018 (EDT)

Shouldn't authorship of pages not matter on a wiki? KarlFrei (talk) 11:12, 12 October 2018 (EDT)
There is a message posted up at the bottom of the Edit page: "If you do not want your writing to be edited mercilessly and redistributed at will, then do not submit it here." Thus there is no sense of actual "ownership" of a page, and indeed, if you object to having templates placed on the page indicating that it needs work to be brought up to house style, or whatever, then you may well have difficulty coming to terms with the way this place works. --prime mover (talk) 12:47, 12 October 2018 (EDT)

## Proofs by Contraposition

The below was collected in response to the statement made on my talk page that "Whether you prefer it to be a proof by contraposition or not, does not mean it is a proof by contraposition until it is turned into it."

## Source 1

There is a useful rule of thumb, when you have a proof by contradiction, to see whether it is "really" a proof by contrapositive.

In a proof of by contrapositive, you prove P→Q by assuming ¬Q and reasoning until you obtain ¬P.

In a "genuine" proof by contradiction, you assume both P and ¬Q, and deduce some other contradiction R∧¬R.

So, at then end of your proof, ask yourself: Is the "contradiction" just that I have deduced ¬P, when the implication was P→Q? Did I never use P as an assumption? If both answers are "yes" then your proof is a proof by contraposition, and you can rephrase it in that way.

For example, here is a proof by "contradiction":

   Proposition: Assume A⊆B. If x∉B then x∉A.


Proof. We proceed by contradiction. Assume x∉B and x∈A. Then, since A⊆B, we have x∈B. This is a contradiction, so the proof is complete.

That proof can be directly rephrased into a proof by contrapositive:

   Proposition: Assume A⊆B. If x∉B then x∉A.


Proof. We proceed by contraposition. Assume x∈A. Then, since A⊆B, we have x∈B. This is what we wanted to prove, so the proof is complete.

Proof by contradiction can be applied to a much broader class of statements than proof by contraposition, which only works for implications. But there are proofs of implications by contradiction that cannot be directly rephrased into proofs by contraposition.

   Proposition: If x is a multiple of 6 then x is a multiple of 2.


Proof. We proceed by contradiction. Let xbe a number that is a multiple of 6 but not a multiple of 2. Then x=6y for some y. We can rewrite this equation as 1⋅x=2⋅(3y). Because the right hand side is a multiple of 2, so is the left hand side. Then, because 2 is prime, and 1⋅x is a multiple of 2, either x is a multiple of 2 or 1 is a multiple of 2. Since we have assumed that x is not a multiple of 2, we see that 1 must be a multiple of 2. But that is impossible: we know 1 is not a multiple of 2. So we have a contradiction: 1 is a multiple of 2 and 1 is not a multiple of 2. The proof is complete.

Of course that proposition can be proved directly as well: the point is just that the proof given is genuinely a proof by contradiction, rather than a proof by contraposition. The key benefit of proof by contradiction is that you can stop when you find any contradiction, not only a contradiction directly involving the hypotheses.

It's not the same.

If P and Q are statements about instances that (a priori independently) are true for some instances and false for others then proving P⇒Q is the same as proving the contrapositive ¬Q ⇒¬P. Both mean the same thing: The set of instances for which P is true is contained in the set of instances where Q is true.

Proving a statement A by contradiction is something else: You add ¬A to your list of axioms, and using the rules of logic arrive at a contradiction, e.g., at 1=0. Then you say: My axiom system was fine before adding ¬A. Since this addition has spoiled it, in reality A has to be true.

## Source 2

When coming to prove P⇒Q, we can either:

1. Prove directly, that is assume P and show Q;
2. Prove by contradiction, that is assume P and ¬Q and derive contradiction; or
3. Prove the contrapositive, that is assume ¬Q and show ¬P.