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Should Wikipedia Allow Mathematical Proofs?
Beetle B. writes "An argument has arisen over whether Wikipedia should allow pages that provide proofs for mathematical theorems (such as this one).
On the one hand, Wikipedia is a useful source of information and people can benefit from these proofs. On the other hand, how does one choose which proofs to include and which not to? Should Wikipedia just become a textbook that teaches mathematics? Should it just state the bare results of theorems and not provide proofs (except as external links)? Or should they take an intermediate approach and formulate a criterion for which proofs to include and which to exclude?"
I find wikipedia useful, and the math is generally well done. The biggest problem is that I hate reading math symbols in anything but latex generated documents.
As I see it, all three are essentially the same but vary in their level of details. Given that wikipedia is electronic, and can essentially (re)represent it's data in various forms, why limit the amount if information present (assuming its factually correct)? Surely the level of detail of an article should be up to the user. Perhaps a better solution in this case would be to include the proofs but make them 'rolled up' by default - IE 'click here for details'. I know 'rolling up' is possible in wikipedia; I've done it on my page there.
As a side note, its worth noting that the article submitter engaged in the discussion about the article for deletion. They voted to delete the article.
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The problem is in ensuring that the proofs are accurate. That's no trivial task, especially if too many such proofs get added to Wikipedia.
As with most things in life the best solution is probably somewhere in the middle. Hundred page proofs are not really suitable for Wikipedia and a complete ban on proofs would leave the site lacking. If it is sensible to include the proof or part of the proof then it should be included.
The maintainers of Wikipedia really needs to ask themselves what they wants it to be. Do they want it to be an encyclopedia or does it want to be the source of all knowledge. Personally I think it should aim to be the best encyclopedia going as I suspect being the one source of all knowledge is probably impossible and there is a danger the real worth of the site will be swamped by too much detail.
Wikipedia should be the starting point of learning not the start, middle and end.
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> Of course they should allow proofs.
I was thinking the same thing until I went through their arguments.
The thing for me is that
a) how do you know the proof is correct?
b) how do you organize all of the mathematics coherently?
As an alternative proposal, how about a mathpedia? Where everything
mathematical can be put and refereed by mathematicians?
--Johnny is a mathematician
That "problem" is not unique to proofs. That is the issue on Wikipedia.
In fact, it is usually a lot easier for someone to check a proof than for someone to look verify who the last prime minister of Malawi was.
They should have links to each mathematical symbol to explain what the symbol means in the current case... Trained Mathematicians are use to seeing this symbols and use them in their current focus. But the symbol can mean different things for different forms of Math. For example Pi in geometry is roughly the number 3.1415926535.... in statistics it is its own function, completely unrelated to the geometry pi.
Mathematicians seem happy to officiate their ideas so only Mathematicians can read them and leave the common man out of the loop, making math look that much harder and scarier. If wikipeadia took an approach of helping people understand the proofs vs. then just giving them but allowing someone to understand it, even on the more basic levels such as clicking on the Uppercase Sigma (looks like a big E) it should bring you to the link on summation.
Math is not actually hard it just has been formalized over thousands of years by Mathematicians to make sure their jobs stay relevant, keep the common man out of the study, and little work has been placed to opening up math for the common folk. Wikipedia has a great opportunity to break down this class structure and allow someone say in high school to lookup a College Level Proof and in time following links get a basic understanding of the proof and able to work it out. But as for the example wikipedia gave as a High School student or a college with a non math focused major it would be literarily all greek to me. And look it up I wouldn't know where to go next... Like knowing the Big E is actually sigma, If I would have guessed I would say it was an Epsilon Shaped like an E Epsilon logical connection huh...
I am actually quite tired of the "Dumbing Down" excuse to fix problems in education that are classically created complex just because some nobles wanted to seem special. Dumbing Down is saying just take this as fact and get the next step. What we can do is open up math so people can understand the details in language they understand or can jump into without having to be formally taught all the prerequisites.
Simplification is not Dumbing Down, but to dumb things down you need to simplify things. The Ven Diagram would be a Big Circle Labeled Simplification the little circle will be labeled Dumbing Down.
If something is so important that you feel the need to post it on the internet... It probably isn't that important.
Simple is very hard to define. For instance, the prime number theorem has an analytic and elementary proof. The elementary proof has many unmotivated steps that leave you scratching your head asking "why?". The analytic proof uses more complex concepts, but applies them in a more straightforwards manner.
Inventions have long since reached their limit, and I see no hope for further development.-- Frontinus, 1st cent. AD
I do not understand the problem. A wikiproof site, just like wikiquote, could be a nice solution.
Existing articles are not 'polluted' with proofs and can link to the relevant wikiproof article. The wikiproof site can implement specific features that are usefull for mathematical proofs.
Reemi
While I am sure that it is easy to argue that proofs should be included (I don't really mind either way), as a Wikipedia administrator I know that one of the hardest things to do is to find a source for something, especially something as specific as a proof. I don't mind the extra information that a proof provides, but it is a manhole up from which crackpot theories may crawl, looking more authoritative because they have a mathematical proof which might not even be valid.
The problem is verification, that editors are not violating the policy against original research (another barrier to crackpot theories). The idea of verification in Wikipedia is that if you look something up in Wikipedia, you should be able to find it elsewhere - and Wikipedia should provide a citation of that source to make it easy to check.
As long as they can be cited to some particular source, and don't otherwise disrupt the flow of the articles to which they are added, I think proofs are fine - there's no reason I can think of to exclude them. If they are used randomly or someone makes up their own proof, however, that is unverifiable original research that is much more likely to lead to errors.
I don't want to exclude information - I want the information there to be reliable.
Why not choose one proof and show that in Wikipedia. Maybe the shortest or the one that will server the widest audience. Save the rest for one of the Wikibooks on mathematics. A good choice might be The Book of Mathematical Proofs
"Should Wikipedia just become a textbook that teaches mathematics?"
Wikipedia should become whatever people want it to be. Who knows in advance what that is?
With the approval of the author of a well-known open-source program, I posted information about how to use the program. Next day that contribution was gone, removed by someone who said that Wikipedia should not become a place for software manuals. But my explanation was the clearest, most complete available at the time; the author of the software did not want to spend time re-writing his own manual.
The problem is not to decide which kinds of content to include in Wikipedia. Wikipedia does not have that problem of paper encyclopedias, paper and printing cost. More pages in Wikipedia are almost free. The only problem Wikipedia has with more content is organizing the content so that it is easy for the reader to make use of what he or she wants, and easy to ignore the rest.
The problem with Wikipedia is not with content, it is a social problem. There are many, many people with some kind of anger problem. Such people don't have many friends. But although they reject and discourage other people, they are still human and need to socialize. So, they spend time with open social groups like Wikipedia. They are there with the hidden and not-so-hidden purpose of having targets for their anger.
Angry people have plenty of free time because other people usually don't want to talk with them. Angry people have the time to dominate social groups, and destroy them. Wikipedia's problem is how to recognize angry, destructive contributors and how deal with their anger.
Hence the Memory Alpha wikia. Perhaps someone should create a Wikia for mathematical theorems and proofs?
...and the people interested in them generally have journal subscriptions and such to access details. I think a decent criteria to start with is if the proof takes less than a page, and uses high school level mathematics. University students or faculty have access to the university's subscriptions so a cite on Wikipedia suffices.
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I am a Physics undergrad student (so IAAP :P), and I've found Wikipedia to be an excellent source for mathematical information. The reason for this is the depth of information available. If, for example, I've forgotten a certain equation, my first port of call is Wikipedia:
The first section usually gives a concise overview.
The second gives the equations.
The third gives the derivation.
This is exactly what Wikipedia should be, in my opinion. I can get as much or as little information as I require, and I can't see any reason for intentionally removing or leaving out relevant data. I'm all for keeping articles free from pointless clutter, but derivations aren't pointless.
I thought Wikipedia was about "Free Access To All Human Knowledge", not "Free Access To A Good Percentage of Quite a Lot of Human Knowledge, But Some Things You'll Just Have To Accept, OK?".
But the basic principle, that the Wikipedia should host as many proofs as anyone cares to type up, seems basically right. Of course, all of it should be in MathML!
How would you know there aren't enough experts checking a certain information? Of course, IF YOU DELETE IT then you made sure there isn't anyone reading it and checking it.
So if you have something like a mathematical proof, and noone modifies it, is that a sign that nobody understands it, or that it's correct? I would guess the latter, but even if not, I would not go on deleting it just because I sustepct something. Who am I to delete stuff that smarter people than me have written?
Or do you mean to say that the basis/policy on which Wikipedia works is admins who are ignorant about topic X will delete articles about topic X?
"The agriculture ministry is not in charge of Gundam" - Japanese ministry official.
As a parent post said, it may not be wise to do so as it blurs the purpose of Wikipedia. If I were involved in Wikimedia, I'd create a wiki solely to hold proofs and explanations, and reference them from the Wikipedia article.
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In my limited observation of the phenomenon, the consensus has generally been reached among mathematical WP editors that the proofs do not belong in the main article about the "Foo function", and they are often not notable as articles themselves (i.e. "Proof of the foo function" pages). As a result, attaching relevant proofs to an article as a subpage has become something of a pattern. I've seen it well done in some of the General Relativity articles (it functions nicely as a sort of appendix for the article where all of the relevant proofs are collected). Anyways, this problem has been solved before with dictionary definitions. (i.e. moved to http://wiktionary.org/) It seems to me like a similar solution would work here. In fact now that I look, it seems that someone has proposed such a project, although not targeted at solving this particular issue. It seems to have not gotten very far though.
"Wikipedia is not a venue for publishing, publicizing or promoting original research." Either a proof is published elsewhere, in which case it should be referenced, or it has not, in which case it is original research, and should not be on Wikipedia.
You are correct in thinking that "computer engineering" and "software engineering" are not scientific disciplines, because they aren't. They are also not computer science. A software engineer is to a computer scientist what a mechanical engineer is to a physicist.
The lines seem to be blurred when it comes to computer science because, more so than with any other scientific discipline, great computer scientists have a tendency to also be great engineers. As Fred Brooks wrote in The Mythical Man Month: For the human makers of things, the incompletenesses and inconsistencies of our ideas become clear only during implementation. Thus it is that writing, experimentation, "working out" are essential disciplines for the theoretician. There is very little separating the science from the engineering when the medium is information and logic, so computer scientists have the luxury of taking their science through to an actual concrete implementation very quickly and by themselves.
A physicist, on the other hand, would usually require an enormous amount of education in material properties, state of the art in manufacturing technologies, and/or a massive amount of infrastructure to provide power etc. to engineer an actual implementation that tests his theories. For physics, and most other sciences, application of theory requires a non-trivial and entirely different set of skills and knowledge than it takes to develop theory, which is why there is a much more distinctive break between the science and engineering in physics, biology, chemistry, etc. than there is with computer science, where a program might not only serve as the definition and description of a theory, but also as a concrete implementation.
Even I don't understand wikipedia's articles on math sometimes.(and I have a degree in math) I had one of my professors tell the following joke...
"Wikipedia is proof that math majors can't find jobs."
Wikipedia articles on math/physics topics really need to develop a whole new format. One thing I would like to see is more casual articles on math topics. Sure, I can almost every popular mathematical proof on wikipedia....but wikipedia is a general knowledge database.
The proofs should DEFINITELY be on the same page, but a lot more care should be taken to make the articles more approachable. I used to use wikipedia in conjunction with my textbook...and several times I wound up preferring the textbook. This wasn't on instructional topics, but on rather general topics. The wikipedia article was simply to confusing, and too technical.
Basically, remember that wikipedia articles DO have an instructional quality. Most mathematicians aren't reading the wikipedia article on the "twin prime conjecture". Encyclopedia articles aren't written for people who know everything about the topic, they are written for people who need information.
**(BTW...this comment is written in the same manner as most of the articles. It has all the essential information, but in a very impractical format)**
When it comes down to it, no one really cares about proofs. Proofs are very long. Proofs are very terse. If I want the proof, I"ll look up the real article from whatever journal or proceedings it was published in. Wikipedia is for quick overviews of subjects. Not for detailed technical discussions. It's an encyclopedia, not a science journal.
Then giving the quality of the math articles on Wikipedia, I can't imagine this would make the articles better. What I mean is, that while the articles are never wrong (or at least obviously wrong), they're always way like 10 times harder to understand they have to be. Seriously. Any undergraduate math book is way clearer.