User:KarlFrei

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
 * 3) it is demonstrably wrong (and even then, it might be kept for historical interest)
 * 4) it is not in house style
 * 5) 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. ...
 * 6) (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.
 * 7) 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!
 * 8) 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?
 * 9) 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.
 * 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.


 * 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.


 * 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. &mdash; Lord_Farin (talk) 14:47, 11 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)

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.