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Open textbooks for freshmen level classes should be possible

There are free/open textbooks in mathematics, at least at the fresher level. Here are a few:
http://www.lightandmatter.com/calc/calc.pdf some physics books are at the same site
ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf
http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/FoundInfsmlCalc.pdf
http://www.mecmath.net/calc3book.pdf
http://www.opensourcemath.org/books/mauch-applied_math/applied_math.pdf
LaTeX source is available for some of them. These books mostly bridge from high school calculus to first year college vector calculus (the last one goes a bit further), but may not be aligned with a particular professor's path through the topics. There are various others at high school level, and quite a few in specialized/advanced areas, but not so many at the undergrad level. It's worth browsing through the categories at http://planetmath.org/?op=mscbrowse&from=books for slightly more advanced topics.

Indeed. At a high level of mathematics it is basically all about mathematical proofs. There is little point in doing a proof unless you give your steps. Writing out non-trivial proofs such that other people can understand them can take me weeks. Still it would be nice if the teacher explained why writing out working can be so important. When I was in school I found it hard to be motivated to do things that I didn't see a real world use for. If Fermat had bothered to write out his proof we could have saved mathematicians 300 years of head scratching: http://planetmath.org/encyclopedia/FermatsLastTheorem.html ;)

There are some great specialized online reference works that use the "cathedral" approach to good effect. As a mathematician, Planet Math comes up frequently, and it has a very well-defined, terse style which is usually much more clear than Wikipedia's mathematical muddles. On the other hand, I am always very happy when the Stanford Encyclopedia of Philosophy has relevant articles, since it takes a very scholarly approach which details the full history of the subject for millenia. When all three sources, Wikipedia, PlanetMath, and the SEP have info on the subject I want - pure bliss, I have a hope of getting it without having to dig into the original papers.

I would hope that other subject areas have similar quality online references. Maybe it is just too difficult to get specialists from many different fields to work together on one of these encyclopedias, so we will end up these large "chunks" of information available to people who know where to find them. The great thing is the internet makes these available for free.

Maybe they can pick up some caveats from the Bourbaki fraternity.
http://en.wikipedia.org/wiki/Bourbaki
An interesting article (well worth reading) is here: http://planetmath.org/encyclopedia/NicolasBourbaki.html

Not a wiki but there's http://planetmath.org/

The "meta" discussion wiki for Planetmath is AsteroidMeta. One topic of discussion I've seen is whether it should be Google-ad supported. It is qualified as a tax-exempt public charity in the U.S., and they are completely open about their finances with detailed reports.

The "meta" discussion wiki for Planetmath is AsteroidMeta. One topic of discussion I've seen is whether it should be Google-ad supported. It is qualified as a tax-exempt public charity in the U.S., and they are completely open about their finances with detailed reports.

• take up way more room than the theorem itself
• 
         
• be difficult to verify as correct by any but a handful of experts who may not be Wikipedians
• be inaccessible to most readers (the proof can be much, much more technical than the thorem statement)
• introduce copyright issues (pulling proofs out of textbooks)
• lead to arguments over proof style and proof correctness
• require mathematical experts to have greater editorial power over the content. Wikipedia has refrained from giving special powers to experts in the past.

I was someone who was once considered to be exceptional in math. Unfortunately, I made the mistake of stopping at calculus.

To regain my mastery of mathematics, I decided to take a single math problem very seriously. I figured that I would try to
understand the solution by grounding all ideas down to postulates.

I figured that this was a great way to learn mathematics anew and really get advanced. I soon learned that there were wonderful
math resources on the web. Wikipedia is really great. There's also MathWorld.com.,
PlanetMath, MathForum.org, and
Cut-The-Knot.org.

Being pretty ambitious, I chose Fermat's Last Theorem and Andrew Wiles's solution as my jump off point. I started this adventure
in 2004. Since then, because the problem is so tough, I started blogging through the different threads of the problem and I find
myself recreating the history of mathematics from the perspective of number theory.

I am not sure that this approach would work for everyone but if you are a solid problem solver, it can really make advanced
mathematics more fun. If you are interested to see what I came up with, you can check out my blog a My math blog.
I also started a general math blog.

Best of luck in learning mathematics.

-Larry

Nearly the exact same thing can be said about NCLB as a response to "failing schools." Maybe I'm biased because I teach high school. Aside from a vague notion of school failure, nobody has specific things they think schools (read: teachers) are doing wrong. Still, NCLB provides a suite of tests and measurements to detect failures. Since the tests are concrete we can point to them as specific criteria, but some of the requirements are as absurd as 3 oz of hand soap. Let me find an example:

3.1.1 The student will design and/or conduct an investigation that uses statistical methods to analyze data and communicate results. Assessment limits: * The student will design investigations stating how data will be collected and justify the method. * Types of investigations may include: simple random sampling, representative sampling, and probability simulations. * Probability simulations may include the use of spinners, number cubes, or random number generators. * In simple random sampling each member of the population is equally likely to be chosen and the members of the sample are chosen independently of each other. Sample size will be given for these investigations.

This is from Maryland's NCLB compliance test for mathematics. It looks nice but there are a few problems in the implementation. The one that sticks in my craw the most is the use of stem-and-leaf plots as a method of visualization. There are other examples like line plots. I have yet to see one of these in actual use and they're not especially interesting as a math topics. A bigger problem (that somehow doesn't stick in my craw) is the one of simple random sampling. It's called "simple" so people think it's simple but there's some subtlety to the concept. I've been at workshops where we review and revise potential test questions. The concept of simple random sampling is subtle enough that it's very easy to come up with multiple choice questions about it that have no correct answer. At the workshop that day, we brought this up and the state representatives didn't really understand and resorted to "well, write that on there and we'll review it later."

The idea of tests to measure progress is a very old one and not terrible. It's a problem of bureaucracy that leads us to tests with low validity and pointless questions. There is a completely separate group of issues surrounding special education and these tests. They don't have an analogy with DHS that I can see.

One might ask what the interest is in creating (another?) huge bureaucracy with an impossibly broad mandate in education. It's generally accepted in some circles (my bias here) that NCLB is designed to take money out of public schools by making the system more intrusive on regular classroom education and thus disruptive and distasteful. Either people object to the level of testing in favor of education and take their kids out, or the schools start "failing" and parents want to take their kids out. Fewer students = less funding and rightly so. Opting out of the system for private interests is something our government does (see: Kyoto Protocol) so why not do the same thing with your children?

The last thing I'll say about the Maryland Mathematics High School Assessment is that it's caused problems for the math sequence. That's my bias I guess is towards the math sequence leading to Calculus. Our NCLB test is tied to the Algebra I course, which used to actually be Algebra I. Now it's about 60% math and the rest is "data analysis." The course is usually taken in eighth or ninth grade. One of my colleagues who teaches Algebra I was telling me that she feels very bad about not preparing the kids for Algebra II, and she knows they're going to run into trouble there. I had some of those kids th

The power to free the knowledge of the world is within the people. The Open Source movement has shown us the way. People just have to do it. We already have a good start with wiki and for more technical stuff we have a more author centric model, but still under the gnu fdl

PlanetMath, http://planetmath.org/
PlanetPhysics, http://planetphysics.org/

The major premise of wikipedia functionality is that it can be edited by anyone, yes?

By far my biggest concern about this scandal is that your premise is actually false, and the falsity of your premise is directly related to the negative consequences of this affair in a very intimate way.

I understand that in an ideal world, anything on Wikipedia can be edited by anyone with no censorship whatsoever, and in an ideal world, two conflicting edits are resolved on the basis of the actual contributions with no regard to credentials or background or the identities of the contributors involved. Unfortunately, Wikipedia falls far short of this ideal in many important instances, and (ironically) the most serious shortcomings emerge during the most serious cases.

For example, let's look at Essjay's talk page as of today, 12:40am eastern time. This is an important article for anyone wishing to voice their opinion on the very matter that we are discussing now. Yet, despite the presence of multiple commentors on that page claiming that content is king and credentials don't matter, the simple fact is you cannot edit that page at all unless you already have an account which has been active for some amount of time, because the page is protected.

This blows a big hole in your assertion that Wikipedia can be edited by anyone at any time. I cannot edit this page at the present time, because I don't have an account, and even if I were to create an account, I would have to wait some amount of time before the account would be considered active long enough to edit that page.

Although you may like to think that an obscure user's talk page is not important enough to be considered representative of Wikipedia as a whole, the fact is that the large majority of so-called controversial pages are kept in protected status, with the result that outsiders cannot edit the page.

The sheer hypocrisy of Wikipedia's stance in this matter is astounding. It is far worse than anything I have seen in other notoriously hypocritical arenas such as presidential politics. Wikipedia is saying that, on the one hand, your (academic) credentials are actively immaterial, but on the other hand it considers your (Wikipedia account owning) credentials so essential that it won't even let you post on important matters unless you have a sufficient amount of the latter. If there is a more insidious and adversarial display of censorship to be found anywhere else in the world, I have not seen it.

Moreover, even if I were to by some stroke of fortune create an account and wait the minimum amount of waiting time necessary to post on that page, I would still be attacked on the grounds of having an account that is too new for my comments to merit consideration. See for instance the comment where Netscott dismisses the opinion of Snackycakes on this very basis. Again, it is hard for me to reconcile this blatantly hostile stance with Wikipedia's official (and largely ficticious) policy of honoring contributions based solely on content.

However, on top of this (already long) rant, the absolute worst part is that Essjay is an administrator and a member of the oversight committee, and as such, he has more power on Wikipedia than all but five other people in terms of deciding which pages to protect, which users to ban, and which comments to delete. In other words, Essjay, the very user whose integrity I feel is justifiably subject to question, is in a strong position to disproportionately influence this debate about himself, not because of the merit of his contributions to the debate in question, but because of his...

credentials.

I should close by saying that I am not by any means the anti-Wikipedia zealot that this post makes me out to be. As a matter of fact, I am a founding member of PlanetMath and a strong supp

Infinity and Infinity+1 are the same thing.

For Cardinals: TRUE
For Ordinals: FALSE

"The long line is a non-paracompact Hausdorff -dimensional manifold constructed as follows. Let be the first uncountable ordinal and consider the set"
from http://planetmath.org/encyclopedia/LongLine.html

For Cardinals, there are precisely as many rational numbers as there are prime numbers.

There is a LOT more to infinity than you think.

http://planetmath.org/ should do the trick.

"many formulations of linear algebra use the maximum number of linearly independent vectors as the DEFINITION of the dimension of that vector space"

http://planetmath.org/encyclopedia/Dimension2.html

http://planetmath.org/ should do the trick.

"many formulations of linear algebra use the maximum number of linearly independent vectors as the DEFINITION of the dimension of that vector space"

http://planetmath.org/encyclopedia/Dimension2.html

For those as clueless as I was...

Zipf's law

The probability of occurrence of words or other items starts high and tapers off. Thus, a few occur very often while many others occur rarely.

Note: In the English language words like "and," "the," "to," and "of" occur often while words like "undeniable" are rare. This law applies to words in human or computer languages, operating system calls, colors in images, etc., and is the basis of many (if not, all!) compression approaches.

More precisely it is the observation that frequency of occurrence of some event (P), as a function of the rank (i) when the rank is determined by the above frequency of occurrence, is a power-law function P(i) ~ 1/i^a with the exponent a close to unity (1).

Named for Harvard linguistic professor George Kingsley Zipf.

http://www.nist.gov/dads/HTML/zipfslaw.html
http://planetmath.org/encyclopedia/ZipfsLaw.html
http://www.nslij-genetics.org/wli/zipf/

Those are so not sets. ("...there is no set of all sets...") The reason you can't just define collections to be sets all willy-nilly is that if you could, you could define the set of all sets that don't contain themselves--does it contain itself? It is true that there are sets that contain themselves. It is true that they have to be infinite.

Just as a point of information: mathematicians have defined "immanent" to be a certain generalization of the determinant (of a matrix).

http://planetmath.org/encyclopedia/Immanent.html

So there's this plane flying out of Poland, and they're trucking along and the pilot comes on the intercom as they pass near Paris. He says, "If you look to the right, you can see the Eiffel tower." So, of course, everyone looks out the right window. Not only that, but everyone on the left side gets up and moves to look out as well. In a freak coincidence, the plane hits a patch of turbulence at just that instant, begins bucking and pitching and crashes just outside the city. This is big news, obviously, and a major investigation is launched to determine the cause of such a horrific crash. They bring on aeronautical engineers, physicists, meteorologists, anybody to try to explain what happened--but no success. Finally, they bring on an electrical engineer to analyze the radio and recording equipment to try and give the team more clues. Immediately after hearing the situation explained to him, the EE puts his chin in his hand, thinks a second and says, "Aha! Of course the system became unstable, all the Poles were on the right hand side of the plane!"

Reference for an explanation (sort of).

Two other good maths encyclopedias are PlanetMath and Wikipedia both are open content, open source etc. PlanetMath is pear reviewed and at a high level.

Although I don't use (and in some cases understand :-) half the stuff they taught me, I feel like the act of trying to understand it increased my ability to understand a larger range of concepts - kind of like working out to increase muscle capacity.

Exactly.
I got my degree in math, and I doubt I'll use the fact that I not only understand, but can prove
The generalized Stokes Theorem.

However, the fact that I can read a sentence consisting of primarily goofy symbols even worse that the linked one makes me an excellent Perl programmer ;-)

Here's a link related to the Miller-Rabin for those with better maths than my good self. Whoever finally works out primes will be a famous man or woman.

Oh, and Slashdot is for people whose first response is "Cool!", not "Why?"

Have you actually every read Nyquist's theorem? It does in fact say that you capture all of the information about anything at a frequency of half the sample rate or below. You can reconstruct the signal exactly by using sinc functions as suggested.

Although the theorem itself is mathematically rigorous, the reason it works is reasonably obvious. By limiting the frequency of something, you limit how much the signal can possibly change in between samples -- that's what frequency is measuring.

The key point that you may not have realised is that the frequency limit is a limit on frequency components, i.e. on the frequency of the sinusoidal signals which combine to make the actual signal. It's this frequency that's limited.

So, it's true that if you have a really crazy waveform repeating at 20KHz, you won't be able to reconstruct it with samples at 40KHz. However, that's because the really crazy waveform actually has higher-frequency components.

Your ear (AFAIK) wouldn't hear those higher-frequency components, so, Nyquist's theorem is directly relevant to audio data formats.

Best reference I could find: PlanetMath

The link you've provided is broken - the item is here.

Some years ago Murray and Herrnstein published the book "The Bell Curve: Intelligence and Class Structure in American Life" in which they inferred from their data that blacks naturally had a lower IQ than whites. Their data certainly seemed to support this claim. However, they were suspect to Simpson's Paradox, in that if they had further stratified their data by social class, then their data may very well have suggested their claim was false. So since many minorities live in poverty or near-poverty, the IQ scores for their races were subsequently lowered. I am naturally very skeptical of studies such as these for the very same reason. As for the study in this post, they would have to have raised children from birth in uniform conditions in order to avoid any biases that culture might induce. Since this is not likely to be the case for this study, I have a hard time believing their conclusions. I would be much more prone to believe that children who are raised in a similar manner as girls in the US are worse at math --- whatever that means --- than those who are raised like boys. Barbie dolls or Legos, which one helps a child develop spacial reasoning? Which one is traditional given to boys, to girls? Now if you'll excuse me, I am late. I am meeting my friends and we are going to play that wonderful game "Jump to Conclusions".

No, of course not. Everything has errors. I actually agree with you that there are no true experts in any field. Any human is at best an imperfect approximation of an expert and prone to making errors.

The place where I disagree with you is in our response to imperfection. You seem to say, no one is an expert so therefore we shouldn't even try to give preference to those with more knowledge. Let every man fend for himself with his own thinking skills. I on the other hand believe that relying on an imperfect approximation to an expert is still better than relying on a random user. For lack of a better word, I will continue to refer to such approximations as "experts" in what follows.

You've already given several high profile examples where the experts were wrong. But you seem to discount the innumerable mundane instances where the experts are right. I believe the benefits of the countless instances where experts are right outweigh the problems caused when they are wrong. The barrier to challenging an expert should be higher than it is now. If it is simply mob rule, then there will always be more uneducated users who believe 1+1 always equals 2 than there are experts to correct them. Critical thinking is not going to solve this problem, because the number of experts is so small that random statistical noise guarantees they will be outnumbered by the wrongheader thinkers.

Getting back to your question, you did ask for alternatives to Wikipedia. I've helped to develop a site called PlanetMath which is a mathematics oriented wiki with a different contributions model that requires discussion as a precondition for making changes and gives the author more control over article edits. I would not go so far as to claim that it is accurate or authoritative, or that it has better quality material than wikipedia (in fact I doubt that these statements are true). However I do believe that the review process makes it more sustainable in the long term. Time will tell, and meanwhile the benefits of having alternatives far outweighs the temporary costs of duplication of effort.

I'm sure there are similar such websites for other fields of study, but I am not intimately familiar with them since my area of expertise is mathematics.

Broken slashdot link quoting Planet Math

I don't know about math.com, but there's

http://planetmath.org/

"PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible. PlanetMath's content is created collaboratively: the main feature is the mathematics encyclopedia with entries written and reviewed by members. The entries are contributed under the terms of the GNU Free Documentation License (FDL) in order to preserve the rights of both the authors and readers in a sensible way."

This proves the existence of negative numbers in the same way that (1+i)(1-i) = 2 proves the existence of imaginary numbers. I.e. it does not. Unless you have additive inverses or subtraction a^2-ab+b^2 does not parse.

Additive inverses and subtraction are defined for integers and real numbers, but not for p-adic numbers .

This proves the existence of negative numbers in the same way that (1+i)(1-i) = 2 proves the existence of imaginary numbers. I.e. it does not. Unless you have additive inverses or subtraction a^2-ab+b^2 does not parse.

Additive inverses and subtraction are defined for integers and real numbers, but not for p-adic numbers .

> What, so how do you express transitivity?

I think you mean, how do you express that A is a subset (or superset) of B. "Transitivity" is a more abstract concept. Both of these operations are transitive, but so are many others. (Equality, inequality relations, etc.)

You can see some set theory notation here. There's an operator that looks sort of like an underlined capital C that says "<left thing> is a subset of <right thing>"

Similarly there are open source content sites like planetmath.org. I think there are similar sites in other discplines worth sending along.

I'd also think about toolsets that might be of use in the third world like cad software and the like.

If you have disk space (or get the CD working), collections of art and photographs would be good too. Toss in a copy of the Gimp.

Finally, music generation software would probably be very popular.

If they want to toss together a bunch of math definitions, they should be more honest that they are just creating a reference. Yet PlanetMath is already doing this, with the Free Encyclopedia of Mathematics.

In general a textbook requires a high degree of cohesion and singular vision; this may not be compatible with a commons-based project style at all.

Another cool fact about the golden ratio (phi) is that it deserves the name `most irrational number': for any real number it is possible to construct a sequence of fractions that converges to it (using continued fractions). The series of `best' approximations (i.e. the approximation closest to phi that has denominator smaller than a given upper bound) for phi is 1, 1 + 1/1, 1 + 1/(1 + 1/1), 1 + 1/(1 + 1/(1 + 1/1)), and so on. Because all denominators are 1, the error in these estimates shrinks only very slowly. To approximate the golden ratio within a given error, you need fractions with large numerator and denominator; transcendental numbers such as pi can (perhaps surprisingly) be approximated much better by fractions.

Well, you remember somewhat incorrectly: mergesort is naturally stable, given implementation details, of course, whereas making quicksort stable makes the "quick" part of the name a misnomer. Heapsort you remember correctly.

One reference

He's talking about math.

I disagree. I would rather show a student a formal proof than to show them that this formula works for this particular triangle. This method of showing an example is not an acceptable proof in Math, and students should know what make acceptable proofs and what do NOT constitute acceptable proofs.

Of the many proofs of the Pythagorean Theorem, I believe Proof #4 on this page would be the easiest proof.

I also think students need to learn proper proof writing, which includes proof by contradiction, proof my mathematical induction, proof by contrapositive, as well as the regular If P, then Q proof.

they cheat by using a construct known as a "limit" in order to make it infinitely close to infinity but not quite. It lets you do all kind of cheaty types of things like measure the volume under a curve that gets infinitely steep as it approaches a particular point (asymptote), add up a series numbers that get infinitely small as you go along (sum of an infinite series), make .999999 repeating actually equal 1, and prove that Achilles does actually catch the turtle.

Usually wikipedia is extremely good, but the articles (some of them linked in comments here on slashdot) look liked they just cut and past crap out of a graduate text book without explanation.

And that's probably even more true of PlanetMath.

Theoretically, it's possible to weed through the hierarchy of definitions in either resource and figure out what was meant. Practically, you usually have to have advanced training in the subject to be able to put it all together.

But, really, you can't blame either site too much for using technical language this way. They're not making the concepts deliberately unintelligable: the vocabulary is there because it's the most convenient generally-accepted means of accurately describing the concepts involved.

It's comparatively rare that the general public, or even the general scientific public, takes an interest in conceptually advanced theorems such as this one. That's why there's little in the way of such resources.

I'm sorry if I appeared defensive. I know PlanetMath has many shortcomings (perhaps better than anyone else) and am genuinely interested in finding out how to make it better.

You are right, PlanetMath is particularly lacking in statistics entries. And yes, more entries need diagrams, and in general, entries need to be more extensive. We're getting better, though. And there are plenty of examples of entries that are better on PlanetMath than MathWorld =)

As to the ownership process, you can actually set all your entries as world-editable, a-la Wiki, with PlanetMath's ACL system. I know I have done a poor job advertising this, and it should be a basic preference asked of users when they first sign up. For details see this documentation.

You also raise the objection of not liking how PlanetMath entries are "decomposed" (i.e., main entry, then proof, example, result, etc). This does not have to be the case--- people can just write one monolithic entry and put as much as they want in it. But it turns out that it is rare that someone has enough time to do this, and people are more productive when they can create and manage their own small, well-defined piece of content. It lowers the barrier to contribution, which is very important for a 100% volunteer effort. And in fact, it is potentially "invisible"--- in the future, it could be a preference to simply view all attachments as concatenated with the main entry.

I understand that you don't have much time, and honestly, I don't expect this of anyone who is competent. But someone like you could help PlanetMath out a lot simply by filing corrections (like "this needs a diagram"!). Most authors are very good about integrating suggestions this way. This is a consequence of the ownership system that you may not have anticipated.

As for the name, PlanetMath was started when MathWorld was offline for a year, so its not exactly like there was any competition for this "motif". Oh well!

Thanks again for taking the time to talk to me.

I'm sorry for taking awhile to respond to your questions.

I really don't mean to sound like I'm bashing PlanetMath. I really do appreciate the site, and hope to see it grow and expand. I believe it will. I think it's a great resource.

And I didn't recommend it just because it's an open alternative to Mathworld in the sense of being a clone. It does have statistics content that Mathworld lacks. It's different enough to be treated differently.

I am aware of the fact that it is organized by subject and alphabetically. I don't mean to imply that PlanetMath is completely disorganized or incomplete to the point of being unusable, just that there are features of it that reduce its usability or usefulness.

Just as an example, compare the Mathworld entry and the PlanetMath entry for the normal distribution. I use this just an example, but it is particularly apt for statistics.

The first thing you will note is that the Mathworld entry is much longer and covers content in much more depth. Not only is more content covered, but more is explained. The PlanetMath entry is much like a glossary entry, the Mathworld entry is more explanatory.

The other feature to note is that the Mathworld entry contains a number of figures, which are extremely helpful in understanding distributions. The PlanetMath entry lacks any figures.

There are also citations given the Mathworld entry, and none in the PlanetMath entry.

There are also oddities before you even get to the entry on the normal distribution in PlanetMath. For example, note that in browsing the N section in order to find the normal distribution, there is an entry comprising a proof that the normal distribution is a distribution. It seems to me that this should be integrated into the entry on the normal distribution, not exist as a separate entry.

These are just examples, but I believe they are fairly representative of issues I have come across when using PlanetMath.

I really don't want PlanetMath to become a Mathworld clone (I already feel a little uncomfortable about the name of the site), but I do think PlanetMath could use some improvements in certain ways.

Again, though--I think it's a great site, and I generally never search Mathworld without searching PlanetMath also, because there are many times Planetmath has information Mathworld lacks.

Why haven't I contributed? I guess one reason is just that I'm extremely busy. Another, I suppose, is that I'm intimidated by the "ownership" process. And there is a part in the back of my mind that is concerned about whether or not if I contribute, the material will be lost, either due to financial failures, loss of interest, or lawsuits by Mathworld over the similarity of the site.

Perhaps those aren't really good reasons not to contribute. I probably will contribute at some point. But I still do have minor (but notable) concerns about how the site operates.

I'm sorry for taking awhile to respond to your questions.

I really don't mean to sound like I'm bashing PlanetMath. I really do appreciate the site, and hope to see it grow and expand. I believe it will. I think it's a great resource.

And I didn't recommend it just because it's an open alternative to Mathworld in the sense of being a clone. It does have statistics content that Mathworld lacks. It's different enough to be treated differently.

I am aware of the fact that it is organized by subject and alphabetically. I don't mean to imply that PlanetMath is completely disorganized or incomplete to the point of being unusable, just that there are features of it that reduce its usability or usefulness.

Just as an example, compare the Mathworld entry and the PlanetMath entry for the normal distribution. I use this just an example, but it is particularly apt for statistics.

The first thing you will note is that the Mathworld entry is much longer and covers content in much more depth. Not only is more content covered, but more is explained. The PlanetMath entry is much like a glossary entry, the Mathworld entry is more explanatory.

The other feature to note is that the Mathworld entry contains a number of figures, which are extremely helpful in understanding distributions. The PlanetMath entry lacks any figures.

There are also citations given the Mathworld entry, and none in the PlanetMath entry.

There are also oddities before you even get to the entry on the normal distribution in PlanetMath. For example, note that in browsing the N section in order to find the normal distribution, there is an entry comprising a proof that the normal distribution is a distribution. It seems to me that this should be integrated into the entry on the normal distribution, not exist as a separate entry.

These are just examples, but I believe they are fairly representative of issues I have come across when using PlanetMath.

I really don't want PlanetMath to become a Mathworld clone (I already feel a little uncomfortable about the name of the site), but I do think PlanetMath could use some improvements in certain ways.

Again, though--I think it's a great site, and I generally never search Mathworld without searching PlanetMath also, because there are many times Planetmath has information Mathworld lacks.

Why haven't I contributed? I guess one reason is just that I'm extremely busy. Another, I suppose, is that I'm intimidated by the "ownership" process. And there is a part in the back of my mind that is concerned about whether or not if I contribute, the material will be lost, either due to financial failures, loss of interest, or lawsuits by Mathworld over the similarity of the site.

Perhaps those aren't really good reasons not to contribute. I probably will contribute at some point. But I still do have minor (but notable) concerns about how the site operates.

I'm sorry for taking awhile to respond to your questions.

I really don't mean to sound like I'm bashing PlanetMath. I really do appreciate the site, and hope to see it grow and expand. I believe it will. I think it's a great resource.

And I didn't recommend it just because it's an open alternative to Mathworld in the sense of being a clone. It does have statistics content that Mathworld lacks. It's different enough to be treated differently.

I am aware of the fact that it is organized by subject and alphabetically. I don't mean to imply that PlanetMath is completely disorganized or incomplete to the point of being unusable, just that there are features of it that reduce its usability or usefulness.

Just as an example, compare the Mathworld entry and the PlanetMath entry for the normal distribution. I use this just an example, but it is particularly apt for statistics.

The first thing you will note is that the Mathworld entry is much longer and covers content in much more depth. Not only is more content covered, but more is explained. The PlanetMath entry is much like a glossary entry, the Mathworld entry is more explanatory.

The other feature to note is that the Mathworld entry contains a number of figures, which are extremely helpful in understanding distributions. The PlanetMath entry lacks any figures.

There are also citations given the Mathworld entry, and none in the PlanetMath entry.

There are also oddities before you even get to the entry on the normal distribution in PlanetMath. For example, note that in browsing the N section in order to find the normal distribution, there is an entry comprising a proof that the normal distribution is a distribution. It seems to me that this should be integrated into the entry on the normal distribution, not exist as a separate entry.

These are just examples, but I believe they are fairly representative of issues I have come across when using PlanetMath.

I really don't want PlanetMath to become a Mathworld clone (I already feel a little uncomfortable about the name of the site), but I do think PlanetMath could use some improvements in certain ways.

Again, though--I think it's a great site, and I generally never search Mathworld without searching PlanetMath also, because there are many times Planetmath has information Mathworld lacks.

Why haven't I contributed? I guess one reason is just that I'm extremely busy. Another, I suppose, is that I'm intimidated by the "ownership" process. And there is a part in the back of my mind that is concerned about whether or not if I contribute, the material will be lost, either due to financial failures, loss of interest, or lawsuits by Mathworld over the similarity of the site.

Perhaps those aren't really good reasons not to contribute. I probably will contribute at some point. But I still do have minor (but notable) concerns about how the site operates.

I am a statistician of sorts (my training isn't in statistics per se, but that's what I do research on), and I'm sorry to say that I'm not aware of any good online statistics references.

There are some sites that come close.

Mathworld, for example, has some excellent reference material on statistics, but beyond some very basic or introductory material, it tends to become sparse quickly. It's typical of much of what's out there: lots of material on mathematics, but not statistics in particular. I also have ethical objections to Wolfram, and so feel uncomfortable supporting any site hosted by his company.

PlanetMath: is a good alternative to Mathworld, filling in some material that Mathworld lacks. It has the benefit of being open. However, PlanetMath suffers from the problem of being extremely disorganized. Many of the entries seem incomplete or lacking in depth. Finally, like Mathworld, it doesn't treat statistics as much as other branches of math.

HyperStat is a good online resource for introductory statistics. I've actually referred to it a couple of times in my research when I can't remember exactly what some formula is, and don't trust my memory of it. It covers introductory material in depth, but doesn't go into fundamentals or intermediate or advanced material. It's also sort of commercial, disorganized, and poorly designed.

Statsoft Electronic Textbook covers more advanced material, but doesn't seem to provide much explanation or background. It's really more a guide to doing analyses in STATISTICA than anything else.

Finally, I've noticed the Statistics Glossary more and more, but it really is a glossary more than an explanatory reference. It also doesn't get further than very introductory topics.

In short, there is a huge niche for a comprehensive, open, in depth statistics resource ala Mathworld or PlanetMath. Perhaps PlanetMath will become more organized and complete. I've thought about contributing to PlanetMath, but I don't feel completely comfortable with it.

Not true, It has been proven, at least if by "sorting" we mean "comparison sorting" where we only know how to compare two objects (the statement is provably false by the existence of faster algorithms for versions of the problem where we get more information). This is a fundamental result of computer science and is a typical (and important!) topic of study in algorithms classes. See PlanetMath's article on the subject for a proof.

(Incidentally, I tried to track down a citation for the origin of the proof but failed. Anybody know to whom this result is due?)

You can find a technical description Here, however.

Wiki's seem good, but they miss one important aspect, structure to the documents. Details about plants neetly fall in to a number of catagories Latin/Botanical name, Common name, growing habit, etc. What I'd like to do is take wiki type concept but add more structure to the data. This could help with searching. Also some fields such as height have numeric values and it would be great to search for plants with a specific height.

Anyone come across such ideas or software which could do such a thing?

BTW I'm suprised how down most slashdotters are on colaborative documents. There are some really good colaborative encyclopedia around wikipedia Planet Math. So whats wrong with OpenContent!

The parent is true, but the reasoning given is somewhat weak.

I think that all of the confusion that appears to crop up all the time around this consideration is that highschools rarely, if at all, give a rigourous definition of the reals. Often I see them defined as simply the set of infinite decimal expansions. While it is possible to define the reals in terms of infinite decimal expansions, it is necessary to actually define every number with trailing 9's to be equal to a number with trailing 0's and otherwise each digit being equal, but the one right before the trailing starts incremented by one. (i.e. 5.7499999... = 5.75000...) Thus, every number of the form n*(10^m) where n, m are integers, gets two decimal representations by definition.

This however, is not considered by many to be the most elegant definition of the reals available. The simplest way to express what the real numbers are, is as the unique ordered field satisfying the least upper bound property, but this is not very constructive. To actually talk about real numbers, you need some construction that meets these requirements. Most mathematicians use either Cauchy sequences of rationals or Dedekind cuts of rationals to define the real numbers.

Please see PlanetMath's entry for Real Number for a definition in terms of Cauchy sequences, and their entry on Dedekind Cuts has a very nice, thorough explanation and definition in terms of Dedekind cuts. Both of these are more elegant and easier to reason about than digit strings if you actually want to prove some property about the reals.

If you have any questions about this topic, or would like me to clarify or expand on something, reply to this, and I'll try to respond. Alternately, I'm available on irc.freenode.org or irc.slashnet.org as Cale. On FreeNode, I'm usually idling in #math.

The parent is true, but the reasoning given is somewhat weak.

I think that all of the confusion that appears to crop up all the time around this consideration is that highschools rarely, if at all, give a rigourous definition of the reals. Often I see them defined as simply the set of infinite decimal expansions. While it is possible to define the reals in terms of infinite decimal expansions, it is necessary to actually define every number with trailing 9's to be equal to a number with trailing 0's and otherwise each digit being equal, but the one right before the trailing starts incremented by one. (i.e. 5.7499999... = 5.75000...) Thus, every number of the form n*(10^m) where n, m are integers, gets two decimal representations by definition.

This however, is not considered by many to be the most elegant definition of the reals available. The simplest way to express what the real numbers are, is as the unique ordered field satisfying the least upper bound property, but this is not very constructive. To actually talk about real numbers, you need some construction that meets these requirements. Most mathematicians use either Cauchy sequences of rationals or Dedekind cuts of rationals to define the real numbers.

Please see PlanetMath's entry for Real Number for a definition in terms of Cauchy sequences, and their entry on Dedekind Cuts has a very nice, thorough explanation and definition in terms of Dedekind cuts. Both of these are more elegant and easier to reason about than digit strings if you actually want to prove some property about the reals.

If you have any questions about this topic, or would like me to clarify or expand on something, reply to this, and I'll try to respond. Alternately, I'm available on irc.freenode.org or irc.slashnet.org as Cale. On FreeNode, I'm usually idling in #math.

A great one is oddmuse. It is a single perl script you put in a directory and it sets everything up. It is a wiki, but also has a journal idea. You can put text, latex and images. Quite nice and very simple to "install".

A couple others along that idea... check out Wikipedia. They have software there, more difficult to install, but a very nice look to it. MySQL based. You can also do Latex (or some subset) of it.

The other one I will mention is Noosphere (which runs the Planetmath site. This I have found a little more difficult to enter stuff into, but it has many nice features.

All of them allow you to grab previous versions of the HTML document and they track who made what changes. Also, you can see my bent is toward more mathematical ones.

For what you say, I would grab the oddmuse and try that. Very easy to install.