Wikipedia talk:WikiProject Mathematics

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{{Maths rating}}

Do people think it would be a good idea if I had MetsBot tag all pages in Category:Mathematics with {{Maths rating|class=|importance=}}? —Mets501 (talk) 01:15, 15 October 2006 (UTC)

Could you give some background? What would be the advantage of doing that? -- Jitse Niesen (talk) 03:13, 15 October 2006 (UTC)

Eh? How can a bot give meaningful ratings? And, for all of mathematics, how can you? If the ratings are not meaningful, they shouldn't be added. This kind of useless busywork would light up every page on our watch lists, which strikes me as a spectacularly bad idea.

But I'll tell you what a bot could do that would be an interesting exercise, if you want to crawl over all the mathematics pages. Use one of the mechanical tests of readability, such as SMOG, both on the article as a whole and on the intro alone. Report back what you find. We could improve the overall quality of our writing by having short lists of easy-to-read and hard-to-read articles. Of course, better still would be to go beyond that, to teach good writing. But that a bot cannot do. --KSmrq^T 07:35, 15 October 2006 (UTC)

I'm not sure tagging all pages will be a good idea, its something like 10,000 pages most of which will probably stay unrated. For me the real use in the maths rating is identifying and grading the most important articles, I guess about 500 articles. There is some good work a bot could do. Currently only about half the articles listed in subpages of Wikipedia:WikiProject Mathematics/Wikipedia 1.0 have a rating tag, so taging these pages would help. Further as we move away from these hand compiled lists to automated lists like Wikipedia:Version 1.0 Editorial Team/Mathematics articles by quality the shear number of articles will be problematic. Hence a bot could use the field tag of the template to assemble lists for each field of mathematics.

Reply to Jitse. The mathematics article rating is part of a wider project grading much of wikipedia, WP:1.0. There are 135 participating project. The aim of WP:1.0 is to make a CD with the best of wikipedia for which they need wikiprojects to identify their best and most important articles. Grading will also help identify the better mathematics articles, and promote them to GA/FA status, find week spots in our coverage. Overall grading ties with Jimbo's talk at wikimania that we have to start changing the focus from quantity to quality. --Salix alba (talk) 08:45, 15 October 2006 (UTC)

According to Portal:maths, there are over 14,000 maths articles. I'm not sure if this is based on articles in Category:Mathematics, or List of mathematics articles, but either way, the number includes a lot of articles that are only tangentally connected wih maths. A lot would probably come under the scoep of other wikiprojects, and for that reaosn alone, it is not worth tagging every single article. IOne of the main reasons for the tagging is to try and help prioritise efforts, by highlighting important articles that need improving.

Related note: Do people think it is worth having a list (either on the wikiproject main page or a subpage) of high-importance stubs and top-importance start-class articles? (There are now no top-class stubs :-) ). Tompw 10:07, 15 October 2006 (UTC)

Tompw 10:07, 15 October 2006 (UTC)

The number 14,000 is based on all the math articles listed in the list of mathematics articles. It is true that some of them are only somewhat mathematical, as this is a general purpose encyclopedia and the distinction between what is true math and what is math-related can be blurry.

I agree with Tompw's arguments above about not tagging all math articles by a bot. Oleg Alexandrov (talk) 16:11, 15 October 2006 (UTC)

OK, no problem. I was doing it for other wikiprojects who requested it, so I figured I'd ask here. —Mets501 (talk) 19:18, 21 October 2006 (UTC)

Another empty category

There are currently no articles or subcategories in Category:Infinity paradoxes which is a subcategory of Category:Infinity. Possibly related articles are in Category:Paradoxes of naive set theory which is in Category:Basic concepts in infinite set theory which is in Category:Infinity. Does anyone want to put something in the empty category or shall we delete it? JRSpriggs 08:17, 16 October 2006 (UTC)

I say nominate for deletion. Category:Mathematics paradoxes is a reasonable upper bound, and the Category:Paradoxes of naive set theory was deliberately created to sort out those relevant to infinite cardinality. Charles Matthews 13:09, 16 October 2006 (UTC)

If I understand the rules correctly, I can add {{db-catempty}} to the category on 20 October 2006. Will that result in it being deleted? Or must I also list it somewhere? JRSpriggs 05:55, 17 October 2006 (UTC)

No, just speedy it. It's not like deleting an empty category is a big deal; if someone wants to recreate it, it takes all of five seconds. Melchoir 06:29, 17 October 2006 (UTC)

Although I have suggested (on this page) deleting a category once before, I have never gone thru the process myself. As I understand it, I would have to persuade an administrator to delete it. Do I just ask one, like User talk:Oleg Alexandrov or User talk:Arthur Rubin? JRSpriggs 07:59, 17 October 2006 (UTC)

{{db-catempty}} is actually a a speedy tag. Basically just put it on the page and wait. If no one objects it will go. WP:CSD explains more. --Salix alba (talk) 08:24, 17 October 2006 (UTC)

Apparently, someone beat me to the punch and deleted it already. I was going to add the template tonight. JRSpriggs 02:08, 20 October 2006 (UTC)

Emmy Noether

I observe that this article has (recently, I believe) become congested with umlauts. Unless, as we are not likely to, we change the spelling of Noetherian ring, this should be straightened out, with a reasonable allowance of "Noether"s for a mathematician who is usually so called in English, and who died on the faculty of Bryn Mawr College. Septentrionalis 15:36, 16 October 2006 (UTC)

And, if I may add, the German Wikipedia also spells the name de:Emmy Noether. So do German libraries, like the catalogue of the Deutsche Nationalbibliothek. And so did she herself. I'm copying this over to the talk page of the article.  --Lambiam^Talk 16:45, 16 October 2006 (UTC)

Lebesgue measure argument

I came across this article recently, and actually made some edits on it. The Lebesgue measure argument (as defined in the WP article) proves the uncountability of the reals via measure theory. As best I can tell the purpose of the argument is that it avoids the use of Cantor's diagonal argument and can be considered constructive,although I haven't actually checked whether the argument is in fact constructive. Googling on Lebesgue measure argument (verbatim) I get only two hits, from wikipedia both. Though the argument is valid and interesting (if actually constructive), does this article not violate WP:OR?

Articles may not contain any unpublished arguments, ideas, data, or theories; or any unpublished analysis or synthesis of published arguments, ideas, data, or theories that serves to advance a position.--CSTAR 17:46, 16 October 2006 (UTC)

This general idea seems to be present in the introduction to Oxtoby, John C. (1980). Measure and Category (2nd ed. ed.). Graduate Texts in Mathematics, no. 2, Springer-Verlag. {{cite book}}: |edition= has extra text (help); you could cite that as a source. —David Eppstein 18:00, 16 October 2006 (UTC)

I don't think it violates NOR, but I also don't think it's a particularly useful article as it stands. The hard part of the argument is that the measure of R as a whole is not zero, and that's not even touched in the article. When you fill everything in, I don't think it's any more "constructive" than the diagonal argument (which is pretty constructive, looked at the right way; for example, it's an intuitionistically valid proof that there's no surjection from ω onto 2^ω). The article also has a very unenlightening title. --Trovatore 18:36, 16 October 2006 (UTC)

It's not original research. It's well-known. I saw it in the first course on measure theory I ever took. I assigned it as an exercise for undergraduates when I taught a probability course at MIT. Of course, Trovatore is right about the "hard" part. Both Cantor's diagonal argument, and also his original argument for uncountability (which is three years older) are of course constructive. Michael Hardy 20:53, 16 October 2006 (UTC)

Oh---now I see that the argument given here is actually more complicated than the one I assigned. The exercise I assigned also avoided the "hard" parts, since the course assumed only first-semester calculus as a prerequisite (at MIT, first-semester calculus is about what first-year calculus is in most other places). See my comments on the talk page accompanying the article. Michael Hardy 20:57, 16 October 2006 (UTC)

Actually my question about whether this was OR concerned not so much whether the proof is OR, but whether the association of the name "Lebesgue measure argument" to the argument is actually supported in the literature. When I first came to WP over two years ago, I wouldn't have given this matter any thought -- any reasonable name would have suitable. However, with what seems the increasing trend toward WP:Wikilawyering at every junction I think this issue has to be addressed.--CSTAR 00:26, 17 October 2006 (UTC)

The name, as I mentioned, is obviously terrible. I'm not convinced the article should exist at all (a small mention in Cantor diagonal argument is probably sufficient) but if kept it should be moved to something more specific. I doubt there's a standard name in the literature, so that's not going to be much of a constraint, or much help. --Trovatore 00:30, 17 October 2006 (UTC)

I think "Lebesgue measure uncountability argument" would be sufficiently descriptive to avoid any claim that we are coining a new name, but as it stands the article is misleading as pointed out above and perhaps should get a disputed tag until the proof is completed.--agr 00:44, 17 October 2006 (UTC)

Based on the above comments, I put a Proposed AfD banner on the article.--CSTAR 02:56, 17 October 2006 (UTC)

This was deprodded by the author, so I smerged it into cardinality of the continuum, but first I prepared by moving the article to Lebesgue measure argument for uncountability of the reals, to avoid the bad redirect. There are way too many Lebesgue measure arguments to have that name reserved for this one in particular (and there's no real chance of a dab page; most of the time an argument doesn't get its own article, unless it has particular historical significance). And I put the redirect on WP:RFD.

If this is reverted we go to AfD. --Trovatore 05:56, 18 October 2006 (UTC)

Oh, just to clarify -- the redirect I put on RfD was Lebesgue measure argument, not Lebesgue measure argument for uncountability of the reals. The latter redirects where the content was merged; it should stay (though it's in my own words, so there'd be no GFDL issue in deleting it). It's the redirect Lebesgue measure argument, created by the move, that I think should be deleted. --Trovatore 06:12, 18 October 2006 (UTC)

"History of numerical approximations of π" really weird edit war---mathematicians please help

Look at the recent edit history of history of numerical approximations of π. User:DavidWBrooks has inserted this bit of wisdom into the article:

“ It has been known for millennia that π, the ratio between the circumference and radius of any circle, ”

("radius"! Sic.)

“ is a mathematical constant, but no method of calculation was available until fairly recently. ”

Of course someone came to clean up this nonsense, but here's what he (user:Henning Makholm) wrote:

“ Unfortunately no practical system for calculating with numbers is able to express π exactly. Though this fact was only proved rigorously in recent time, it has been suspected since the earliest times ”

Is there something remotely approximating some correct statement in that? If so, what is it? (Makholm left the ratio as circumference-to-radius rather than circumference-to-diameter.) Michael Hardy 21:05, 16 October 2006 (UTC)

I think it's all rubbish. Archimedes, like any capable mathematician of his days, knew how to compute the circumference of a regular 3·2ⁿ-gon. While this method is very not practical due to slow convergence, he must have realized, when using a 96-gon to shew that π < 22/7, that he could in theory compute the value to any desired precision. Given all the fuss at some earlier time over the diagonal of a 1 by 1 square not having a rational length, the claim that "this fact [...] has been suspected since the earliest times" has to be bogus. Or was that what the forbidden fruit of the tree of knowledge of good and evil propositions was about? Lacking a definition of what it means that a system is "practical", it is hard to refute the claim about what was proved "recently" (meaning, presumably, 1882).  --Lambiam^Talk 01:24, 17 October 2006 (UTC)

Why presume 1882? That was the year when π was proved transcendental. But that's got nothing at all to do (as far as I can see at this moment) with whether any "practical system for calculating with numbers is able to express π exactly". Anyone who thinks transcendence is about "practical systems for computing exactly" should get committed forthwith to the State Hospital for the Criminally Innumerate. Michael Hardy 02:09, 17 October 2006 (UTC)

Whoa, Michael Hardy shouldn't you be now concerned that a plague of Wikilawyers will descend on that previous claim, invoking countless breaches of this, that or the other rule, policy, guideline, essay, practice or what not and cart you off to wikiprison or maybe even have you wikiexecuted?. You're a brave man, Michael Hardy! --CSTAR 02:19, 17 October 2006 (UTC)

Arbitrarily-precise approximation is different from exact computation: one wants to be able to test, e.g., inequalities of expressions involving pi, and be guaranteed of an answer in a finite time, while you can keep computing as many digits of precision as you like and not be able to tell whether something is or is not equal to zero. And there is a sense in which transcendentalness is a barrier to expressing numbers exactly in a practical computational system, but irrationality isn't: see e.g. this page describing exact representations for algebraic numbers in the LEDA system. It says "LEDA cannot deal with transcendental numbers, at least not without loss of precision - there is no number type class in LEDA that could represent π or e exactly." Of course, the inability to express these numbers in a single system is not the same as a rigorous proof that no such system can exist, and I know of no rigorous proof that it's impossible perform exact computations in the extension of the algebraics by π. So I don't think the statement in the article is quite right... —David Eppstein 06:28, 17 October 2006 (UTC)

Point taken. What I was trying to express was just that one needs to work with approximations in order to do actual computations that involve pi -- but at the same time I was trying to defuse the possible counterargument that one could manipulate symbolic expressions, or juggle around with an entire convergent series of approximations, which is as "exact" as anybody could ask for. Henning Makholm 20:39, 17 October 2006 (UTC)

This should be discussed on the article talk page, but has general interest. Let's not get sloppy about terms. We can represent π exactly in a variety of ways. For example, we can define a series or continued fraction in a finite expression or algorithm. We can also compute π to any desired number of decimal places (or other measure of error). Archimedes demonstrated one approach using polygons to find upper and lower bounds, and we have much faster ways today. Being irrational, no finite computation can give an exact decimal expansion, oddities like the Bailey-Borwein-Plouffe formula notwithstanding. Yet computations with exact rational numbers are already troublesome in, say, computational geometry with lines and planes and so on, because the denominators can grow in a nasty fashion.

As for the article, both the original insertion and its amendment are hopelessly confused, and should be removed. --KSmrq^T 12:06, 17 October 2006 (UTC)

KSmrq is of course right. However, I'm afraid I can make a qualified guess of what the article editors essentially meant. There are too many students who believe that it isn't possible to express 1/3 exact (since they won't get an exact value by pushing 1:3 on their pocket calculator:-). Exact is often identified with exactly expressed in decimal notation with a finite number of decimals. I now and then meet statements such as '1/3 isn't an exact number, but 1/4 is'. Somewhat better informed students may understand that it is possible to 'express 1/3 exactly', if you use numerals with another basis than 10. In other words, I guess that 'no practical system for calculating with numbers is able to express π exactly' essentially is meant to mean 'π is irrational'.

I don't know if it is possible to clarify things enough for eliminating this confusion among some wiki readers; but we may try to lessen it. JoergenB 20:30, 17 October 2006 (UTC)

Oh, and by the way: That some users make this sort of mistake is not sufficient reason enough to accuse them of vandalism, or to call them 'dishonest idiots', however frustrating this kind of misunderstandings may be. JoergenB 20:42, 17 October 2006 (UTC)

My point above was simply that it is possible to express algebraic irrationals exactly, by writing down an integer representation of the polynomial for which they are root together with some disambiguating information to specify which root you mean. It is also possible to express π exactly, by the notation π. But the algebraics as exactly-specified numbers have been made part of a "practical system for calculating with numbers" (namely LEDA reals) while for π we can write "π" and call it a number and compute as many digits as we like but all that isn't sufficient to perform exact computations with it. I don't think the original editor meant an explanation like that, and I agree that the best course of action is to remove the offending statement, but there is a level of explanation at which his statement makes some sense. —David Eppstein 20:43, 17 October 2006 (UTC)

As a matter of fact, something like that was what I was trying to express with "practical system for calculating with numbers". Henning Makholm 20:51, 17 October 2006 (UTC)

David Epstein wrote:

it is possible to express algebraic irrationals exactly, by writing down an integer representation of the polynomial for which they are root together with some disambiguating information to specify which root you mean

By that standard one can also say that "log₂3" expresses a number exactly. Is there some reason to limit it to algebraic numbers? If not, then the year 1882, suggested above, does not seen relevant. If it is possible to define precisely something that Henning Makholm could have meant that is actually correct, then it seems very irresponsible to write in sich a horribly vague way about such a thing, and then claim that something expressed so vaguely was proved. It can't be proved if it can't be precisely expressed. So far we're still left guessing what was meant, even after Henning Makholm's comments here. Michael Hardy 22:58, 17 October 2006 (UTC)

I see your point, David. However, I think you may be mislead by viewing some CASes (computer algebra systems), where you might do exact simplification of expressions involving algebraic roots, but not as easily with π. In the first place, there are CASes and even pocket calculators where e.g. sin π cos π is replaced automatically by exactly -1, if you wish; some CASes may do much more advanced substitutions involving π; and more to the point, already Archimedes performed exact calculations with π (see talk:history of numerical approximations of π#Intro graf). IMO, 'computable' isn't synonymous with 'computable within a present-day CAS'. JoergenB 23:18, 17 October 2006 (UTC)

I made a stupid mistake above, thinking of cos and writing sin. However, I do not think making such ridiculus mistakes make me (or anybody else) qualified for asylums. I actually do know what the elementary values of the trigonometric functions are; believe me. JoergenB 23:30, 17 October 2006 (UTC)

There is, in fact, a specific technical reason to limit things to algebraic numbers: there exist algorithms that allow a computational system to reliably determine whether two given algebraic-number representations represent equal or unequal numbers. Therefore it is possible to guarantee that the result of a test such as x ≥ y, performed as part of some larger computation, will return in a finite time: one applies the equality algorithm first, and only after it returns unequal do you need to evaluate x and y to sufficient precision to tell them apart. There are no similar equality testing algorithms known, and therefore no similar finite-time guarantees, for systems of numbers generalizing the algebraics but also allowing logs, e, or π.

Also, I wouldn't call these systems CAS. They are libraries for performing calculations with numbers as part of computer programs, similar in spirit to a standard floating point library but allowing the representation of exact algebraic numbers in place of approximate floats. But they don't do some of the other operations that a typical CAS would, such as symbolic integration.—David Eppstein 23:41, 17 October 2006 (UTC)

CAS or not CAS is a matter of opinion. In mine, the algorithms by means of which you decide whether or not two expressions for algebraic numbers stand for the same number or not, are rather typical examples for CASes; much more so than is symbolic integration. This is not very important, though. If you restrict yourself to extending the field of algebraic number with one transcendental, e.g. π, you don't really get any harder decision problems than before. If you try to incorporate e.g. all kinds of exponentiation and logarithms, you run into trouble (at least today; I don't know much about the true bounds for undecidability). This is also not very important. The most important point is this: When you use the method of exhaustion by Eudoxos in order to prove that the same constant relates diameter to circumferense and square of radius to circle area, then you are performing exact calculations with the number π, in the best of the modern meanings. This Archimedes did. (This is a rather non-trivial result; as far as I remember, the claims about the constructions in the infamous Indiana Pi Bill implied different values for these two proportions.) JoergenB 00:49, 18 October 2006 (UTC)

There are no similar equality testing algorithms known, and therefore no similar finite-time guarantees, for systems of numbers generalizing the algebraics but also allowing logs, e, or π.

Do you mean ONLY that none is known, or rather that it is known (can be proved) that none can exist? If the former, it certainly doesn't justify saying that it has been PROVED that something specific about π cannot be done. Michael Hardy 23:59, 17 October 2006 (UTC)

I don't think it has been proven uncomputable, I think it's only that none is known. So that part of the statement is I think wrong. —David Eppstein 00:29, 18 October 2006 (UTC)

There are results along those lines. Consider the expressions built up from rational numbers, π, a single variable x, sine, absolute value, addition, multiplication, and composition. The problem "is such an expression equal to zero?" is undecidable. This theorem is due to a certain Richardson and follows from Matiyasevich's theorem. The only reference I have at the moment are some lecture notes, but I probably can find more details if necessary.

Of course, it's a bit of a stretch to refer to this result as "no practical system for calculating with numbers is able to express π exactly". -- Jitse Niesen (talk) 02:17, 18 October 2006 (UTC)

This is a fun and worthwhile discussion for me; I hope I'm not alone in that view.

So far there has been some dancing around the meaning of “practical system for calculating with numbers”, which is intolerably vague.

• I would consider Macsyma, Mathematica, and Maple very practical systems for computing with numbers, as well as more general symbolic expressions.
• I would also consider the implementation of IEEE floating point arithmetic on an AMD Opteron microprocessor chip a practical system for computing with numbers.
• Both LEDA and CGAL are elaborate libraries for computational geometry which also fit the definition.
• The GNU Multi-Precision Library is another example.

David Eppstein presumably is invoking the decidability of quantifier elimination for a real closed field, which relates to Tarski's axiomatization of the reals but for a first-order theory. More concretely, this is about George Collins’ seminal work in cylindrical algebraic decomposition for semi-algebraic sets. (I would love to have Wikipedia links for the preceding sentence, but we have none of the relevant articles!) In order to have polynomial-time algorithms for arithmetic, we may restrict our attention to real roots of univariate polynomials with rational (or integer) coefficients, Q[x]; these are the real algebraic numbers. Otherwise, the time required can be far from practical. However, this theory does not allow us to introduce an arbitrary assortment of fancy functions beyond basic arithmetic.

Yet within a computer algebra system we can surely know that 4 tan⁻¹ 1 is exactly π, or that e^iπ is exactly −1. Furthermore, we can do a wide variety of calculations and comparisons with π, more than enough for most practical purposes.

In contrast, using IEEE floating-point as our standard, we cannot express 0.1 accurately! The problem is that the radix-2 expansion repeats periodically. Compare this to √2, which happens to have a periodic regular continued fraction. Or compare to e, whose continued fraction merely requires an arithmetic progression.

The moral is, if we are too sloppy to define our terms, we're sunk. But I repeat myself. --KSmrq^T 15:19, 18 October 2006 (UTC)

Citation guidelines proposal

Since the discussions seem to have abated for some time now, I am asking the Mathematics and Physics WikiProjects if they support the new citation guidelines that I (and others) have devised. The point of the guidelines is to establish an appropriate, sensible standard for referencing articles in our fields so that we are less likely to run into objections (such as those that have come up recently) when we try to write technical articles that others then tell us are impropoerly sourced. I think these guidelines are now well thought out enough that they can be added to the main pages of the two WikiProjects and perhaps linked from WP:CITE. I should also note that they seem to have attracted some encouragement from outside the WikiProjects, on their talk page, mine, and on WP:CITE.

One outstanding issue is where to move the page. I don't have any great ideas. Wikipedia:WikiProjects Mathematics and Physics/Citation guidelines is too cumbersome. We could just leave it under physics as Wikipedia:WikiProject Physics/Citation guidelines or be BOLD and put it at Wikipedia:Scientific citation guidelines (presumably this would mean we would have to engage with the rest of the community to ensure there is consensus). I submit we should go with Wikipedia:WikiProject Physics/Citation guidelines and once we have consensus here go to Wikipedia:WikiProject Biology and Wikipedia:WikiProject Chemistry (and wherever else seems appropriate) to solicit their opinions, and then move it out of the physics WikiProject. We could even eventually go ask the wider Wikipedia community what they think at WP:CITE but I think that should be left as a longer term project. –Joke 22:14, 16 October 2006 (UTC)

Since there doesn't seem to be any objection to this proposal, I have gone ahead and moved it to Wikipedia:Scientific citation guidelines and added links on the pages of the relevant WikiProjects and on WP:CITE. –Joke 03:52, 26 October 2006 (UTC)

Support

1. I already offered my support on the talk page of WikiProject Physics, but also with my mathematician's hat on I support this.  --Lambiam^Talk 01:28, 17 October 2006 (UTC)
2. I like this proposal and support any step that moves it forward to wider acceptance. —David Eppstein 01:58, 17 October 2006 (UTC)
3. CMummert 02:55, 18 October 2006 (UTC)
4. I've also left a more detailed comment on the guidelines talkpage. --Salix alba (talk) 07:30, 18 October 2006 (UTC)
5. Support - excellent draft guideline - clearly written, pragmatic, comprehensive without becoming verbose. Gandalf61 08:10, 18 October 2006 (UTC)
6. This seems the best way to proceed. I would suggest also posting at Wikipedia:WikiProject Science. Tompw 15:13, 18 October 2006 (UTC)

Object

Neutral/Comment

1. I generally support it, but I find the statement "articles that link to [eponymous articles] may choose not to cite the original papers, depending on the context" too vague. I would prefer if such cases were handled just like links from a summmary to a sub-article. This would reduce the "dense referencing" and facilitate maintenance, since the sub-article is the best place to discuss and maintain attribution. — Sebastian (talk) 05:35, 19 October 2006 (UTC)
2. I agree with the text. I questioned some details on the talk page. The only problem I have is that I'm not convinced that it's a good idea to have separate citation guidelines. My impression is that most Wikipedia editors would agree with it, but that the so-called inline citation squad, having strong opinions on this topic, are very vocal at WP:CITE and (for some reason I don't quite fathom) at WP:GA. However, they are not in the maths or physics WikiProjects (or if they are, they haven't come out of the closet yet). -- Jitse Niesen (talk) 14:55, 21 October 2006 (UTC)

Hamiltonian, anyone?

If your expertise allows you to contribute in a meaningful way to articles involving Hamiltonians and their applications, please take a look at Wikipedia talk:WikiProject Physics#Hamiltonian articles.  --Lambiam^Talk 01:35, 17 October 2006 (UTC)

Constructibility

In Talk:Borel algebra the following question is proposed by User:Leocat:

Can someone tell me how to construct an isomorphism between such Polish spaces as the unit ball in L^2[0,1] and the real line with the natural topology?

Now by Kuratowski's theorem, both objects are uncountable polish spaces and hence Borel isomorphic, so "there exists" an isomorphism. My guess is that this isomorphism is constructible, but I don't know enough about constructive mathematics to know for sure.

If anybody knows the answer to this question, you can post it there.--CSTAR 02:30, 19 October 2006 (UTC)

Well, it depends on what you mean by "constructive". There's no single agreed definition of that term. (By the way, be careful of substituting "constructible" for "constructive"; "constructible" has another constellation of meanings.)

Here's one partial answer: The arguments I know for the existence of such an isomorphism certainly use excluded middle. Basically you show that there's a Borel injection from Cantor space into any Polish space, and you show there's a Borel injection from any Polish space into Baire space, and you show there's a Borel injection from Baire space into Cantor space, and then you chase around the triangle using the Schroeder-Bernstein construction. It's the last part that uses excluded middle; you have to distinguish whether a point is or is not in the range of an injection, and without using excluded middle, it's going to be tough to prove that it either is or isn't. --Trovatore 03:47, 19 October 2006 (UTC)

Convention for definitions: Use := or \equiv?

Problem and options

In WP:MSM I didn't see anything about which infix to use for definitions. Some use $\equiv$, but I find this very misleading, since it already has two other meanings: equivalence (hence its Latex code) and identity. I would therefore advocate := or the equal sign with "def" underneath. (Sorry, I don't know the Latex code for that.) — Sebastian (talk) 04:58, 19 October 2006 (UTC)

We also have a (carefully hidden) page of conventions, but this convention is not among them. I agree that the “triple equal” is not a good choice. The “colon equal” is my preference; I also like to use it for algorithms (where I save bare “equal” for equality tests). I have not found a decent way to stack something over or under an equality or arrow within the tragically limited abilities of texvc, Wikipedia’s TeX engine. Unicode provides a single character for “Assign” (“≔”, U+2254, &#x2254;), a single character for “triangleq” (“≜”, U+225C, &#x225C;), and one for “equal to by definition” (“≝”, U+225D, &#x225D;). I cannot recommend any of these characters at present, because they will not display well (if at all) for many of our readers. Displayed in a larger font size for clarity, here are the choices mentioned:

=

≔

≜

≝

≡

Although in LaTeX itself we could use \overset{\mathrm{def}}{=}, and blahtex supports this, texvc does not. Thus the two character sequence “colon equal” (“:=”) is left as the only viable choice. However, as always, no matter what convention you adopt, please do not leave readers guessing; tell them explicitly if there is any reasonable chance of misunderstanding. --KSmrq^T 06:58, 19 October 2006 (UTC)

How about $=_{\operatorname {def} }$? JRSpriggs 08:55, 19 October 2006 (UTC)

Discussion: "=" vs ":=" vs words

I feel strongly that we should not need the := type symbol here. If something is a definition we should say so in words. The proper use for := is for assignment in computer science. As far as I'm concerned, := is up there with iff as technical language we should always avoid. since it makes articles impenetrable. Charles Matthews 09:00, 19 October 2006 (UTC)

I agree that words are preferable, for the reasons given by Charles Matthews. JPD (talk) 10:23, 19 October 2006 (UTC)

I would agree. Despite having seen := used in some decent books in recent years, I feel quite strongly that in maths, as opposed to computing, the use of := is a bit of a neologism, and the words should be perfectly clear. Similarly, I assume we agree that we should not use inverted A and E for "for all" and "there exists"? Madmath789 11:10, 19 October 2006 (UTC)

Ditto, but := is not so heinous a notation as is being insinuated here ;) Dysprosia 11:18, 19 October 2006 (UTC)

To clarify my position, in most cases I also would use something like “Let x be the reciprocal of y” rather than “x := ¹⁄_y.” I believe, so far, I’ve not needed the latter for Wikipedia. However, situations can arise where it is helpful to adopt a distinct notation. Rather than take a fixed position banning it, perhaps we might strongly discourage it, but offer a notation should the need arise. Frankly, given the fact that current technical limitations preclude any really satisfactory symbol, I think most editors will choose to write around the problem, as we prefer. Our style guide already says the following:

• Careful thought should be given to each formula included, and words should be used instead if possible.

Beyond that, if Wikipedia intends to let anyone edit, then we might also want to begin to teach writing skills. A typical mathematical education teaches neither English composition nor technical writing for a broad audience. --KSmrq^T 13:00, 19 October 2006 (UTC)

So it seems we have a consensus that we don't want $\equiv$ for definitions. This Google search shows that we have less than 60 occurrences, so it is practically feasible to weed out the wrong ones.

Many of these cases may indeed be better expressed with words. But I would not completely rule out ":=". Trying to express every definition in words can get clumsy. E.g. I can't think of a way to rephrase "... where $c\;$ is the speed of light and $\gamma \ :=1/{\sqrt {...}}$ is called the Lorentz factor" withouth distracting at least some readers. Moreover, readers who are unfamiliar with ":=" can enter it in the search field (although unfortunately they can't enter a single colon). — Sebastian (talk) 18:02, 19 October 2006 (UTC)

Users can enter it in the search box, but this will lead them to = (which redirects to Equals sign), since an initial colon is discarded; see Wikipedia:Naming conventions (technical restrictions)#Colon.  --Lambiam^Talk 10:05, 20 October 2006 (UTC)

If its just a list of constants then a simple $\gamma \ =1/{\sqrt {...}}$ seems fine. For more complicated definitions then words are more approprate. --Salix alba (talk) 18:52, 19 October 2006 (UTC)

Re the issue with gamma and the Lorentz factor - you can say "... where $c\;$ is the speed of light and $\gamma$ is the Lorentz factor (defined as $1/{\sqrt {...}}$)." This avoids any equivalency symbology. It actually puts the information in its proper place, as it makes the full Lorentz factor a parenthetical item for those who do not know the Lorentz factor, whereas those who do can just glaze over it. --Carl ^{(talk|contribs)} 02:29, 28 October 2006 (UTC)

Good idea - if you like you can add it to the table below, or I'll do so later. — Sebastian (talk) 02:46, 28 October 2006 (UTC)

In writing about mathematics and physics, I've never found a problem using an equals sign and then stating, in words, whether what you're writing is a definition. I think anything else is just a gimmick. –Joke 18:54, 19 October 2006 (UTC)

I don't think your Google search finds pages in which ≡ is typed as a unicode character. In fact I haven't been able to figure out how to get Google to search for that character. And the WP search box only gives me ≡ which is uninformative. So finding all instances of ≡ in WP pages may be problematic. —David Eppstein 21:33, 19 October 2006 (UTC)

David, thanks for thinking of this. I still hope that there are not too many Unicode ≡ instances since it doesn't seem possible to use this in a formula. <math>a ≡ b</math> at least yields: Failed to parse (syntax error): {\displaystyle a ≡ b} .

Salix and Joke: I think you're missing my point. Of course it is possible to write just an equal sign, but you're losing information: The colon tells the reader unobtrusively: "Don't worry about what this $\gamma$ is all about and whether you've seen it before - it is just a definition." And I agree, it is not a problem for anybody who writes English reasonably well to state in words whether it's a definition. But how do you actually do this in a case like the above without overemphasizing a side issue and breaking the flow of thought? — Sebastian (talk) 22:52, 19 October 2006 (UTC)

It only tells the reader something if it is an established convention, the conversation here indicates its not. Picking a random maths book its full of statements like If A=..., let A=...., We define A=.... and where A=.... The preceding words are enough to unambiguiously tell the readers whats happening. Any other notation will break the flow of the text, making the reason pause to think, 'whats this new notation i've not seen before'. Personally I think we should follow KISS principle and minimise inroducing unnecessary notation.--Salix alba (talk) 23:26, 19 October 2006 (UTC)

Scroll up. Math tags are for LaTeX code, not Unicode characters. $a\equiv b$ uses \equiv, not ≡. Dysprosia 06:48, 20 October 2006 (UTC)

I'd like to make the point that it is particularly important for WP to highlight definitions on pages. Not to sneak them into notation. There is indeed a kind of format issue with the typical 'where' construction after a formula. That, I think, is a separate and useful discussion. Mathematicians can take it to be the syntax "let x be an A, y a B, ...", preceding a statement. In science it certainly is frequently done with a trailing "where c is the speed of light ...". These context-establishing things matter quite a lot. But I really don't see that the := assignment is a good thing in there. For one thing it comes from the wrong programming paradigm (functional programming rules ...). Charles Matthews 08:59, 20 October 2006 (UTC)

I would expect to see mutable variables with assignment in an imperative programming language, whereas a functional programming language would limit its bindings to “let” constructs and function calls. Did you misspeak, or did I misunderstand? --KSmrq^T 11:54, 20 October 2006 (UTC)

An unambiguous indication that something is a definition has its merits, but I agree that we should avoid conventions that are insufficiently established and may be unnecessarily puzzling to our readers. I've seen maths books using ":=" for definitions, but then somewhere in the introductory parts there will be a section on notation explaining the use. I don't think I ever saw this use in a physics textbook. We should go with a simple "=" sign, making sure the context establishes the definitional nature. Would that some unclarity there was the worst problem in the understandability of our maths articles...  --Lambiam^Talk 10:19, 20 October 2006 (UTC)

For a whole week, none of the seven people who found it particularly important to highlight definitions made any contribution to actually achieve this. Since

• my main concern is eliminating the ambiguous use of "\equiv";
• we have over 300 articles with several occurrences each and
• editing the text to highlight definitions takes a lot of time for each occurrence (at least for me)

it seems that simply replacing "\equiv" with ":=" wherever applicable would be the most sensitive thing to do for now. (Replacing it with just "=" is not good since it would delete information, and other alternatives were even less favored.) I am volunteering to do that. After that, I will be done, and the proponents of prose can edit these occurrences at their leisure. Let me know what you think. — Sebastian (talk) 21:00, 27 October 2006 (UTC)

Well, as I wrote earlier (see below): "We can also simply fix such things as we encounter them. In terms of best use of time to increase the quality of maths articles, it is (in my opinion) more effective to work on some stub articles or other pages that have been flagged as needing attention." And I do not only work on maths articles. I'd say this edit qualifies, though. Further, as I explained before, I'm opposed to using := for definitions. The large majority of readers will not be familiar with this meaning.  --Lambiam^Talk 23:50, 27 October 2006 (UTC)

I'm sorry! I didn't mean to include you. I am really grateful for your active contribution to the table, too.

You may be right that many readers of Wikipedia may not be familiar with ":=" - but this is irrelevant in this context. What is relevant is the difference

$Difference=C_{equiv}+K_{equiv}-K_{colon}$

where

$C_{equiv}$ is the number of people who are confused by the ambiguous use of "≡"

$K_{equiv}$ is the number of people who don't know "≡" but notice it to the extend that it hurts their understanding of a formula;

$K_{colon}$ is the number of people who don't know ":=" and notice it to the extend that it hurts their understanding of a formula;

I may be wrong, but I believe this difference is positive. I believe that $K_{equiv}\approx K_{colon}$ (the added bar is no less conspicuous than the added two dots, and the discussion here showed that "≡" isn't that popular either). And $C_{equiv}>0$, because it includes at least me.  ;-) — Sebastian (talk) 00:40, 28 October 2006 (UTC)

Speaking for the others, when they commented they may well have had the impression that you were requesting input on how to deal with this in WP:MSM, rather than attempting to press-gang them into a task force. As to your exercise in linear programming, aren't you overlooking the quantity $C_{colon}$? $Difference=C_{equiv}+K_{equiv}-C_{colon}-K_{colon}$. You may say it isn't ambiguous, but there is also the meaning of assignment in Pascal and other programming languages. And I've seen it used for denoting substitutions. How many places are there where ≡ is actually ambiguous? In any case, I suggest that you do not ignore the judgement of several editors that := is not appropriate.  --Lambiam^Talk 01:25, 28 October 2006 (UTC)

I don't know how you come up with the accusation that I'm press-ganging anyone. In the contrary, I have been volunteering my time to fix something that bothers me. I would like to do this in a simple way, as described above. It is others who demand it to be done in a much more work intensive way, which practically makes it impossible for me to do it alone.

Re $C_{colon}$: I assumed it to be 0, but it actually is less than 0. The connection with assignment is not confusing but helpful.

Re How many places are there where ≡ is actually ambiguous?: In every place, by definition. There are three contradicting definitions for "≡" (including "is identical"), and it takes always some extra bit of information to distinguish between them.

Re ":= is not appropriate": This is your same absolute statement again, where we need a relative comparison. We have to choose one option. If I understand you correctly, your preferred option seems to be to leave everything as is until we run out of "stub articles and other pages that have been flagged as needing attention" - which will be when pigs fly. I don't think that is any more "appropriate". — Sebastian (talk) 02:02, 28 October 2006 (UTC)

We've got someone who wants to make a productive contribution to Wikipedia mathematics articles. Can we not find a way to put that to good use? If substitution of ":=" for "\equiv" or "≡" is undesirable, which appears to be the consensus, then what task would be helpful? The claim that "we have over 300 articles with several occurrences each", if accurate, could be converted into a list of those articles. That list could be linked here. Interested parties could work through it, eliminating items as they are fixed. What I have found in working through a similar list, the blahtex problem article list, is that often an article that exhibits one dubious construction accompanies it with other problems. This is the wisdom behind the suggestion to fix things as we encounter them.

And please, help my frayed nerves and stop abusing TeX. It is wrong to write

$K_{equiv}\approx K_{colon}.\,\!$

A correct form is

$K_{\mathrm {equiv} }\approx K_{\mathrm {colon} }.\,\!$

This is not just a matter of italics; without proper markup TeX thinks you mean to multiply the single-letter variables c, o, l, o, n, and uses the wrong font and the wrong spacing. Compare

$fifty\,\!$

versus

${\mathit {fifty}}\,\!$

for appearance. --KSmrq^T 03:15, 28 October 2006 (UTC)

Thank you, KSmrq! This list exists already: /equivlist (described in next section). After David Eppstein was busy again, today, we might just have touched 300. And I'll take your point about \mathrm. I found that it is already in Help:Math, but hidden in the Rendering section. I thought it was just used to define types. Maybe this could be written a bit more explicitly? — Sebastian (talk) 03:26, 28 October 2006 (UTC)

I also strongly agree that ":=" is awfully ugly, and that it should be always avoided in math. Oleg Alexandrov (talk) 03:27, 28 October 2006 (UTC)

To KSmrq: I have not used "\mathrm{...}" because I do not know what it means. What does it mean? I have used "\operatorname{...}" in some similar situations. JRSpriggs 06:04, 28 October 2006 (UTC)

operatorname and mathrm are different, IIRC because the former adjusts the spacing for operators (compare entering $|$ and $\mid$ into TeX), the other does not. Dysprosia 06:46, 28 October 2006 (UTC)

Conclusion and /equivlist

Thanks Charles, Salix and Lambian – you make enough good points to elevate your preferred solution (of banning ":=") within my margin of error close to my preferred solution (of allowing it where it helps). The reason I'm not entirely swayed is that I absolutely disagree with Lambian’s last remark: It is precisely because our math articles are often hard to understand that readers need help. Even small things can provide a straw for struggling readers to cling to. But I acknowledge that there is a tradeoff, and which solution actually provides more help is a moot judgment call. — Sebastian (talk) 20:57, 21 October 2006 (UTC)

I'm slightly puzzled by your absolute disagreement with my last remark. I only meant to express dissatisfaction with the complete lack of understandability of several of our maths articles. While small things might make the difference of the proverbial straw for some articles, too many articles are like a heavy block of concrete dropped on the camel. So all this lamentation – which you should also see in light of my expressed opinion that ":=" is more problematic – is saying is this: I wish possible ambiguity of "=" was the worst problem we have. I don't think you want to claim that it is actually the worst, or that you wish for worse problems.  --Lambiam^Talk 03:09, 22 October 2006 (UTC)

No worries! If that statement wasn't meant as an argument against any small improvements then I agree with it, of course. — Sebastian (talk) 17:06, 22 October 2006 (UTC)

OK, now that we agree that we don't want to use \equiv for definitions, we need to do two things:

1. Add the policy to WP:MSM. I'm fine with the policy proposed by Charles, Salix and Lambian, but I wouldn't want to be the one who adds it to MSM.
2. Change "\equiv" to "=" where it means definition. Here's a list of all articles that contain "\equiv": /equivlist. Let's work with this together: Whoever cleared an article, just deletes its line from the list. I'll begin with articles that I understand. — Sebastian (talk) 17:06, 22 October 2006 (UTC)

Dang! I just realized that my original query yields far too few results. Unfortunately, searching for "math" does not, as I thought, yield all pages that contain the <math> tag, but only those that contain the word "math" in plain text - which are mostly entries like "J. Math. Pures Appl.". Replacing "math" in the query with "function" already yields 408 results. Does anyone have an idea how to filter all mathematical articles in a Google (or other) search? — Sebastian (talk) 18:05, 22 October 2006 (UTC)

I replaced the list with the result of this query, which gives us a few too many articles, but at least we won't miss any. — Sebastian (talk) 19:25, 22 October 2006 (UTC)

We can also simply fix such things as we encounter them. In terms of best use of time to increase the quality of maths articles, it is (in my opinion) more effective to work on some stub articles or other pages that have been flagged as needing attention.  --Lambiam^Talk 20:57, 22 October 2006 (UTC)

Policy for WP:MSM

So, what exactly should we add to WP:MSM? How about the following:

• For definitions, do not use "\equiv". If something is a definition try to say so in words. If that isn't possible, use ":=".

We could also recommend "$=_{def}\,$" , as used in Implementation of mathematics in set theory. — Sebastian (talk) 23:18, 24 October 2006 (UTC)

That article specifically discusses how to "define" (actually "implement") things in terms of other things, and then you want these "definitions" to stand out. Because they aren't truly definitions in the usual sense, we should not take them as examples.

It is different when definitions are introduced in the course of a discursive account. There are many ways of doing this:

• We denote g_1(x) + g_2(x) + ... + g_n(x) by f_n(x). Then the sum f(x) is the limit lim f_n(x).
• Letting f_n(x) stand for g_1(x) + g_2(x) + ... + g_n(x), we now define the sum f(x) to be the limit lim f_n(x).
• Define f_n(x) = g_1(x) + g_2(x) + ... + g_n(x). Then the sum f(x) is simply the limit lim f_n(x).
• Let a sequence of functions f_n be defined by f_n(x) = g_1(x) + g_2(x) + ... + g_n(x). Then the sum f(x) is the limit lim f_n(x).
• We can then define the sum f(x) by f(x) = lim f_n(x), the limit of a sequence of approximations, where f_n(x) = g_1(x) + g_2(x) + ... + g_n(x).

In most cases the problem is not so much the ambiguity of a form like X = Y by itself, but the lack of appropriate text connecting the formulas. I don't want to outlaw the use of =_def, but it shouldn't be encouraged by our style manual.

Perhaps we can collect some bad examples and show how to fix them. One candidate I'm nominating is in Geometric mean, where it is not obvious (I think) to the mathematically unwashed that in the first maths display the l.h.s. is not a definiend. Here no new symbol is being introduced. Contrast this with Polylogarithm. Here the opposite confusion might be possible: a reader stumbling upon the article might think that the two expressions connected by an equality sign are just two different, already understood, ways of saying the same thing. An easy fix is to amend the first line to read: "... is the special function Li_s that is defined by:". Or one could write: "The polylogarithm (also known as Jonquiere's function) Li_s is a special function that is defined by:".  --Lambiam^Talk 02:40, 25 October 2006 (UTC)

Good idea! Here's the seed for such a table, taken from my recent edits. I never claimed I was good at writing math prose, so I'm sure you'll have some ideas for improvements. Please don't hesitate to edit them directly in the table. I added a column "Found in" so we can easily take the improvement to the article. — Sebastian (talk) 20:24, 25 October 2006 (UTC)

Old version of table deleted - see new version below.

Hamilton's characteristic function <math>W</math> is often defined as... seems to mean that Hamilton's characteristic function has different definitions and that the often used one is ... This is not the same meaning as the initial sentence, which is that Hamilton's characteristic function, which is defined as ..., is also often used. Isnt't it right? pom 20:56, 25 October 2006 (UTC)

Good point! The word "often" is a bit overused in that article (and maybe Wikipedia in general) anyway. For the reader, much more relevant than how often something is used is what it's used for, and if it helps him/her solve his/her problem. I'll take a closer look at that article and see what I can do. — Sebastian (talk) 21:41, 25 October 2006 (UTC)

I suggest: "another action function, Hamilton's characteristic function $W$, is often introduced. It is defined as ...". I find LaTeX notation hard to read and have produced a typeset version below, in which I've tentatively applied this suggestion (but not in the actual article). By the way, all the examples use(d) \equiv, but many articles using = for definitions would also improve by similar changes.  --Lambiam^Talk 22:24, 25 October 2006 (UTC)

Yes, I like your proposed wording. But since I just rewrote the whole section it would doesn't fit exactly anymore. Please feel free to change as you see fit. I also prefer the typeset, so we don't need to keep my old table. I agree with your point that these changes have a wider applicability - all the more reason to put something like this table in the style guide. — Sebastian (talk) 22:45, 25 October 2006 (UTC)

Before	After	Found in	Comments
where $\mathbf {n} \equiv (n_{1},n_{2},\cdots ,n_{d})$	where we define $\mathbf {n} =(n_{1},n_{2},\cdots ,n_{d})$ and $\mathbf {k} =(k_{1},k_{2},\cdots ,k_{d})$	Discrete Fourier transform
where the division $\mathbf {n} /\mathbf {N} \equiv (n_{1}/N_{1},\cdots ,n_{d}/N_{d})$ is performed element-wise	where the division $\mathbf {n} /\mathbf {N}$ is defined as $\mathbf {n} /\mathbf {N} =(n_{1}/N_{1},\cdots ,n_{d}/N_{d})$ to be performed element-wise	Discrete Fourier transform
The action is defined as the integral of the Lagrangian $L$ for an input evolution between the two time points ${\mathcal {S}}[\mathbf {q} (t)]\equiv \int _{t_{1}}^{t_{2}}L[\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t]\,\mathrm {d} t$	The action ${\mathcal {S}}[\mathbf {q} (t)]$is defined as the integral of the Lagrangian $L$ for an input evolution between the two time points ${\mathcal {S}}[\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L[\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t]\,\mathrm {d} t$	Action (physics)	Copy defined term in sentence for clarity.
another action function is often defined:Hamilton's characteristic function $W\equiv S-Et$.	another action function, Hamilton's characteristic function $W$, is often introduced. It is defined as $W=S-Et$.	Action (physics)	Move "defined" closer to equation.
the final and initial positions, $x_{1}\equiv x(t_{1})$ and $x_{2}\equiv x(t_{2})$, are specified in advance.	the final and initial positions are specified in advance as $x_{1}=x(t_{1})$ and $x_{2}=x(t_{2})$.	Action (physics)	use "specified ... as" to indicate definition.
The difference between these two evolutions is infinitesimally small at all times: $\varepsilon (t)\equiv x_{\mathrm {per} }(t)-x_{\mathrm {true} }(t)$	The difference between these two evolutions, which we will call $\varepsilon (t)$, is infinitesimally small at all times:  $\varepsilon (t)=x_{\mathrm {per} }(t)-x_{\mathrm {true} }(t)$	Action (physics)	The original equation served two purposes: Defining $\varepsilon (t)$ and showing which term is small. Explain these two verbatim.

Related problems encountered in listed articles

Optimal stopping uses ≡ to state the distribution of a random variable. That is, they write X ≡ D where X is a variable and D is a distribution. Is that one of the uses of ≡ we should be avoiding, or is it ok? Maybe it should be a membership symbol rather than either = or ≡? —David Eppstein 21:31, 22 October 2006 (UTC)

I believe X ≡ D is very much a non-standard use and should therefore be avoided. What is wrong with saying in words that X is a random variable with distribution D? Independence has to be stated in words anyway; we have no notation for that.  --Lambiam^Talk 01:14, 25 October 2006 (UTC)

Thanks for the suggestion — I agree that saying it words works better than notation here, and have made that change. The rest of the article could still use some work, but that's beyond the scope of what I want to do with it tonight. —David Eppstein 02:51, 25 October 2006 (UTC)

Powers

Any consensus on the policy on fractional powers? We write the squareroot sign for powers of one half, but what about cube roots? Do we put the squareroot with the 3 above, or do we put ^1/3? And the others? yandman 09:57, 19 October 2006 (UTC)

Any reason not to use <math>\sqrt[3]{n}</math>? —David Eppstein 15:05, 19 October 2006 (UTC)

Where do we draw the line? ${\sqrt[{20}]{n}}$ looks a bit silly to me. yandman 15:31, 19 October 2006 (UTC)

Single digits or single letters only seems like a reasonable rule of thumb to me. —David Eppstein 15:44, 19 October 2006 (UTC)

(Edit conflict). Well, I think it's better than $n^{\frac {1}{20}}$, personally. However, $n^{\frac {3}{20}}$ looks better than ${\sqrt[{\frac {20}{3}}]{n}}$. So, I would be inclined to say use the "root" notation when the root is an interger, and use the "exponent" notation otherwise. Whatever the ourcome of this discussion, I think it (the outcome) should be added to Wikipedia:Manual of Style (mathematics). Tompw 15:56, 19 October 2006 (UTC)

The integer root rule works badly for formulas like O(n^1/32,582,658). As usual, it's an area where common sense and rules of thumb may be more appropriate than strict guidelines... —David Eppstein 16:04, 19 October 2006 (UTC)

I have to say, I don't think I have ever seen the notation ${\sqrt[{3}]{n}}$ in a book above introductory college textbooks. In journals it is very common to use the superscript even for square roots when it would simplify notation. (e.g. a lot of people prefer

${\frac {4\pi ^{6}A^{1/2}g_{s}^{2}}{\Gamma (1/4)n}}$

to

${\frac {4\pi ^{6}{\sqrt {A}}g_{s}^{2}}{\Gamma (1/4)n}}$,

and for long formulae you would definitely use parentheses and an exponent instead of a very large square root sign.) Using an exponent has the added benefit that simple formulae with exponents will not render to .png for people (such as myself) who have their math tags set to render to text for simple formulae. –Joke 00:05, 21 October 2006 (UTC)

One I recently edited here was

$c={\sqrt[{3}]{\frac {2\pi ^{2}}{3}}}\approx 1.874.$

in no-three-in-line problem. You could inline it as π^2/3(2/3)^1/3 (or some variation of the same with frac instead of slashes) but I think all the /3's make it confusing, and using the cube root sign makes it very clear visually that everything in the expression has a fractional exponent. On the other hand, I prefer your first formula to your second because the fractional exponent is formatted more similarly to all the other exponents. —David Eppstein 00:14, 21 October 2006 (UTC)

One more advanced instance of radicals that I've seen is in field theory. It is, of course, common to write $\mathbb {Q} ({\sqrt {2}})$ to denote the field obtained by adjoining a number whose square is 2. Also common, though, is to write $\mathbb {Q} ({\sqrt[{3}]{2}})$ to denote the field obtained by adjoining a number whose cube is 2. It would be improper to write $\mathbb {Q} (2^{\frac {1}{3}})$, both because of convention, and because the exponential notation (for whatever reason, possibly convention) suggests a sort of deterministic choice, especially when its argument (i.e. 2) is real. $\mathbb {Q} ({\sqrt[{3}]{2}})$ is an abstract field, and could just as easily be $\mathbb {Q} (2^{\frac {1}{3}}e^{\frac {2\pi i}{3}})$, where now I've deliberately used the exponential notation to single out particular complex numbers. Worse, of course, and not only in the context of field theory, is the fact that the power functions are of course not one-to-one, so that their inverses are multi-valued, and so using fractional powers only makes sense in the presence of a convention as to the specification of a particular value (like when we take square roots of positive real numbers).

However, since this doesn't seem to be a discussion of whether to use one symbol or the other but rather when, I would say that it's as much a matter of audience as of aesthetics. Certainly the radicals should be avoided for roots which consist of more than a single character or for all but diminutive radicands (i.e. ${\sqrt[{2}]{2}},{\sqrt[{5}]{n}},{\sqrt[{n}]{m}}$, but not ${\sqrt[{20}]{2,535,099,102,664}}$) but on the other hand, in articles which are expected to see traffic by novices, radicals may be preferred. Fractional powers constitute a mild form of mathematical jargon and certainly represent a reasonably sophisticated idea that, say, students below college might not be comfortable with. Conversely, of course, in professional-level articles, we should probably avoid radicals unless (as in field theory) their use is conventional.

Ryan Reich 21:32, 21 October 2006 (UTC)

I definitely agree. Out of habit I might have used the formula

$c=\left[{\frac {2\pi ^{2}}{3}}\right]^{1/3}\approx 1.874,$

but I think either looks great, especially compared to the inline formula you produced. –Joke 00:35, 21 October 2006 (UTC)

Note: it is generally a good idea to use linear notation in sub- and superscripts ($x^{a/b}$, not $x^{\frac {a}{b}}$). Particularly when the formulas are rendered in low resolution as they are here. Fredrik Johansson 22:45, 21 October 2006 (UTC)

Use of the radical forces texvc to produce a PNG, which looks bad inline. Using wiki markup, we can write x^a⁄_b. --KSmrq^T 23:59, 21 October 2006 (UTC)

English composition

Put it before them briefly so they will read it, clearly so they will appreciate it, picturesquely so they will remember it and, above all, accurately so they will be guided by its light.
— attributed to Joseph Pulitzer

Wikipedia mathematics editors are brilliant and well-educated, naturally. Yet many have never studied the art of readable writing, especially for the general public. I’d like to offer a few suggestions. With your approval, they may later find their way into our Manual of Style.

I begin by quoting two well-known mathematicians.

The first rule of style is to have something to say. The second rule of style is to control yourself when, by chance, you have two things to say; say first one, then the other, not both at the same time.
— George Pólya

[T]he problem is to communicate an idea. To do so, and to do it clearly, you must have something to say, and you must have someone to say it to, you must organize what you want to say, and you must arrange it in the order you want it said in, you must write it, rewrite it, and re-rewrite it several times, and you must be willing to think hard about and work hard on mechanical details such as diction, notation, and punctuation. That’s all there is to it.
— Paul Halmos

When I give a lecture or write a paper, I consider myself lucky if I can convey one idea clearly, so that my audience pays attention, understands, remembers, and is inspired. This is more difficult than it sounds! Both mathematicians quoted above agree. Thus the heart of good technical writing is our first guideline:

• Know precisely what you want to say.

Halmos next says to know your audience, and again I agree; yet for Wikipedia the audience can include university faculty, the general public, and youngsters. Readability studies suggest several ways to help. Two basic guidelines, with broad empirical support, are:

• Avoid long sentences with complicated structure.
• Avoid unfamiliar words with many syllables.

And more technically,

• Minimize adjectives, adverbs, and passive verbs.

These studies also emphasize the value of structure, as do both our mathematicians. Structure occurs on three levels: sentence, paragraph, and article. All three should be clear, logical, and memorable. And I have just illustrated the next suggestion:

• Use twos and threes for organization.

Examples of twos include if–then and either–or. More generally, balanced structure and parallel structure help the reader. This is less useful at the paragraph level; but we can suggest the following.

• Give each paragraph a clear topic, preferably in its first or last sentence.

At the article level, the order and content of sections should never leave the reader disoriented. Work for a natural flow, a sense of inevitability. We want readers to know where they’ve been and where they’re going.

Pay particular attention to the introduction, especially the first paragraph. The first sentence should both engage readers, and orient them to what is to come. It need not summarize the article.

All of the suggestions so far apply to any kind of writing. I have a few personal touchstones for mathematics. It is natural to include theorems and proofs, but I also try to incorporate:

• Motivation
• Intuition
  ———
• Examples
• Counterexamples
  ———
• Pictures
• Connections

Finally, I do my best to sneak in a little humor. Some may damn this as “unencyclopedic”, but the best teachers have always done so. We all know, when we’re honest with ourselves, that when we laugh, we learn. With that in mind, I end with another quotation.

I hope that posterity will judge me kindly, not only as to the things which I have explained, but also to those which I have intentionally omitted so as to leave to others the pleasure of discovery.
— René Descartes

Perhaps another time I can add links to writing resources. Meanwhile, take what you can of value from these suggestions, and help make Wikipedia better. --KSmrq^T 16:04, 19 October 2006 (UTC)

Sour comment: you know when Samuel Johnson said that if you were particularly proud of a piece of writing, you should cross it out? Here on WP you needn't bother. Someone else will surely edit it out for you. Charles Matthews 16:20, 19 October 2006 (UTC)

That's partly what motivated me to write this. I'm hoping to elevate the awareness of editors, both in their own writing and in critiquing others. I have no illusions that all those who read this handful of suggestions will become great technical writers overnight, or perhaps ever. Still, it may begin to help. Halmos himself said, “The ability to communicate effectively, the power to be intelligible, is congenital, I believe, or, in any event, it is so early acquired that by the time someone reads my wisdom on the subject he is likely to be invariant under it.” Yet he tried. Perhaps those drawn to improve Wikipedia will also wish to improve themselves, and maybe they can. We can hope. --KSmrq^T 19:11, 19 October 2006 (UTC)

"Minimize adjectives, adverbs, and passive verbs.". Umm... no. Without adjectives and adverbs, a sentence contains only nouns and verbs, which would make it less readable. Maybe what you meant was "avoid excessive adjectives and adverbs", which is something I completely agree with. (This is not the same as minimisation. Any sentence can have *all* its adverbs and adjectives removed - the ultimate in minimisation - and remain grammatically correct.)

Also, what is wrong with the passive voice? I use the passive voice occasionally, whenever I feel that the object of a sentence is the important part, rather than the subject. I hope I don't come across as overly-critical here, as I agree with the broad thrust of your comments.

"Always have a quotation handy; it saves original thought". Tompw 22:13, 20 October 2006 (UTC)

This is Wikipedia; original thought is prohibited. ;-)

Readability studies disagree with your objections. Here is one survey you may find enlightening. I also refer you to Strunk & White's acclaimed guide, The Elements of Style. Among their guidelines are these, supporting the one in question.

• Use the active voice.
• Write with nouns and verbs.
• Avoid the use of qualifiers.
• Avoid fancy words.

Those who have been force-fed an excess of Strunk & White may appreciate Lanham's amusing Style: An Anti-Textbook (ISBN 978-0-300-01720-5).

Your objection shows you think about how you write. English is a second language for many of our editors; yet even our native-speakers will not become good writers unless they, too, begin to think about their writing. --KSmrq^T 11:58, 21 October 2006 (UTC)

Simenon apparently used to draft his books by locking himself in a room for 72 hours, to get a draft. When he had recovered from that, he went through crossing out all the adjectives and adverbs he could find ... Charles Matthews 15:13, 22 October 2006 (UTC)

Sounds painful. I think that providing people think about what they write, don't write in the same way they speak, and read through what they have written, then few stylistic problems will crop up. (Also, British English and American English do have different styles, and I'm British. I know US manuals on style seem to regard the passive voice as an abomination. The UK and US are two nations divided by a common language...) Tompw 16:06, 22 October 2006 (UTC)

This is not a question of taste, but of readability. If you want to write as readably as possible, you must train yourself to use active voice. That is what readability studies tell us, whether we like it or not. Lest we think only Yanks and Brits have something to say, I quote a German author and a French author, both of some stature:

A writer is somebody for whom writing is more difficult than it is for other people.— Thomas Mann

Those who write clearly have readers; those who write obscurely have commentators.— Albert Camus

While I would not ask James Joyce to write like Ernest Hemingway, I’ll wager The Old Man and the Sea gets read cover-to-cover more often than Ulysses. --KSmrq^T 05:07, 23 October 2006 (UTC)

Eigendecomposition

I've added "eigendecomposition" as a synonym for "spectral decomposition" in the spectral theorem article: I'm almost completely sure that's right, but my maths is a bit rusty these days -- could someone more up-to-date double-check this, please? -- The Anome 11:59, 20 October 2006 (UTC)

Aaagh: Google says 84K of hits for eigendecomposition. That's already far too many ... Charles Matthews 15:43, 20 October 2006 (UTC)

Erdős number tags

Apart from the fact that I think it's annoying to be told Atiyah has Erdős number 4, as if this was on the same level as a Fields Medal: I think we should point out clearly that any information here should be verifiable. Apart from a complete list of collaborators of Erdős, it is going to be hard to verify numbers at all; certainly the only assertion you'd responsibly get is ≤ 3 and so on. Charles Matthews 19:01, 20 October 2006 (UTC)

Mea culpa; I supported keeping the categories. But I see no reason to mention the number in the text. I believe we can verify 1 and 2 easily, and larger numbers with more difficulty. Not that I'm volunteering to do it! Note date and location of birth can also be hard to track down, and we often manage that anyway. Shall we say every Erdős number tag should be accompanied by a certificate of authenticity on the talk page? That puts the burden on those who wish to add these categories. --KSmrq^T 19:32, 20 October 2006 (UTC)

Cats should only be added if the reason is obvious when looking at the article (and I suppose the talk page). Note the hard part, unless we link to an Erdős number site, is verifying that Atiyah is not EN 3...Septentrionalis 19:38, 20 October 2006 (UTC)

For numbers one and two, one can look at the Erdős number project data, and for greater numbers (with the appropriate subscription?) you can use MathSciNet, assuming all the relevant pubs are in their database (a reasonable assumption for mathematicians, not so much for other kinds of scientist). So I don't see the difficulty of finding verifiable data as being much of an obstacle. —David Eppstein 20:21, 20 October 2006 (UTC)

PS Atiyah should be 3 not 4, according to MathSciNet:

Michael Francis Atiyah   	 coauthored with   	 Laurel A. Smith   	 MR0343269 (49 #8013)
Laurel A. Smith 	coauthored with 	Persi W. Diaconis 	MR0954495 (89m:60163)
Persi W. Diaconis 	coauthored with 	Paul Erdös1 	MR2126886 (2005m:60011)

—David Eppstein 20:22, 20 October 2006 (UTC)

Questions is mathscinet a reliable source? It can provide an upper bound on the Erdős number, but not necessarily the exact EN. Also it becomes on the bounds of original research to use the database, as its not a traditional publication. I know some of the people adding these cats are using mathscinet for their info (see my talk page). Still despite these reservations if we ate going to have the numbers its better to get the most accurate number possible. I like KSmrq's solution, maybe we should workup a policy on how to handle these. These cats are going to become an annoying waste of time.--Salix alba (talk) 21:20, 20 October 2006 (UTC)

It's original research to enter two names on a web query form and report the result of that form? It does take some checking afterwards to make sure the papers it returns are real joint publications, but the chain it gives you is readily verifiable, often without further need of their database. —David Eppstein 22:03, 20 October 2006 (UTC)

I guess that Archimedes must have been a lousy mathematician, because he did not have an Erdős number (sarcasm). JRSpriggs 06:38, 21 October 2006 (UTC)

The problem with MathSciNet is that it is not just a web query form. I mean accesibility. Well, one may pay for a subscription; as far as I can understand $2226 will do. I found nothing about such commercial databases in WP:VER policy. Do we really consider it verifiable? I guess we do not. So, inserting info about Erdos number would require a chain of publications as indicated above. As for reliability, MathSciNet should be considered on the top; it is likely to give you the best (often exact) easily verifiable result. Erdos number caracteristic seems to be interesting and notable enough to be mentioned in the bios, at least for EN<=5 mathematicians (and it is not considered as a coeficient related to notability of this mathematician!). Essentially, I agree with David. --Beaumont (@) 09:50, 21 October 2006 (UTC)

The strongest evidence that Archimedes, Euler, and Gauss were not great mathematicians is that none of them won a Fields Medal. (Tongue firmly in cheek.) :D

Relax, no one is obliged to research and incorporate an Erdős number category for any mathematician. It has not yet joined “use massive numbers of inline citations” as a criterion for Good or Featured articles.

For those who want to add these categories, the most verifiable “certificate of authenticity” would be the chain of publications. It is irrelevant whether MathSciNet or some other method is used to assist in finding the chain. I would recommend affixing any such certificate near the top of the talk page, to make it easy to find. --KSmrq^T 12:24, 21 October 2006 (UTC)

Re paying for the service: instead, you could walk into the library of most public universities and use the computers there. —David Eppstein 15:29, 21 October 2006 (UTC)

Errr ... you assuming we are all in the USA, or something? I think those who add such a category do owe us a list of intermediate people, and the best way is to add it to the page itself, by the category, and commented out. Charles Matthews 19:02, 21 October 2006 (UTC)

Err, there are no university libraries in other countries? Or (like some private ones in the US) they don't let you in without some kind of affiliation? In any case I agree that it's appropriate to provide a chain of intermediates; when the EN is mentioned in the text, it would be appropriate to do so there, but your suggestion of commented out next to the cat makes sense too. The few times I've changed these recently I've put the chain in the edit summary, but I guess that's not as easy to find. —David Eppstein 19:09, 21 October 2006 (UTC)

Name of theorem?

I was fiddling with some formulas, and seem to have stumbled over the following theorem: given any topological space X and any homomorphism $g:X\to X$, there exists a measure $\mu$ such that it is preserved by the pushforward $g_{*}\mu =\mu$ (aka the direct image functor on the category of measurable spaces(?)); equivalently, there is always a measure such that g is a measure-preserving map, and furthermore, this measure is unique. This theorem is little more than a fancy-pants version of the Frobenius-Perron theorem, and the measure is more or less the Haar measure. I was wondering if this theorem has a name? Is it in textbooks? Or is it supposed to be a nameless corollary of the theorem that defines the Haar measure? Thanks. linas 03:35, 21 October 2006 (UTC)

I find uniqueness hard to believe. JRSpriggs 06:40, 21 October 2006 (UTC)

I question existence, since Haar measure depends on having a group structure. --KSmrq^T 12:29, 21 October 2006 (UTC)

Existence is true if X is a compact space, in which case the measure μ can be taken to be a probability measure. This is just the compactness of the space of Borel probability measures in the weak* topology and the fact the group of integers Z is amenable (actually this is equivalent to a fixed point theorem for continuous affine mappings on compact convex sets. See Dunford Schwartz, although I don't have it in front of me so I don't know the exact formulation.) However, in general uniqueness is false even for compact X and imposing the additional requirement that the measure μ is a probability measure. Uniqueness is a special property called unique ergodicity. For non-compact X, existence is also false without some additional assumption on X.--CSTAR 16:08, 21 October 2006 (UTC)

Thank you CSTAR, this pointer is just what I needed. (My X was indeed compact, and my g ergodic. I'm not sure what other additional errors assumptions I might have accidentally made along the way.) KSmrq, I'm looking at dynamical systems, so the hand-waving physics argument for existence is that physical systems always have a ground state, and, for systems in thermodynamic equilibrium (i.e. ergodic), so that all symmetries are broken, the ground state is unique. I'm grappling with general formulations, but this is new territory to me. I assume "Dunford Schwartz" is the book "Linear Operators" from 1958. I presume newer books on operator theory will have similar content. linas 22:17, 21 October 2006 (UTC)

Dunford and Schwartz vols 1 and 2 (vol 3 is much less interesting), though dated, are unsurpassed as general references in functional analysis.--CSTAR 22:30, 21 October 2006 (UTC)

Someone creted an artcle just today, for at least half of what I was looking for: the Krylov-Bogolyubov theorem. linas 23:48, 26 October 2006 (UTC)

Archiving automatically (well almost)

As some of you already know, Wikipedia talk:WikiProject Physics recently started using User:Werdnabot to automatically archive its talk sections when ten days have passed since the last new comment. Perhaps we should start to think about whether we want to follow their example. JRSpriggs 11:51, 22 October 2006 (UTC)

Seems a good idea to me. --Salix alba (talk) 19:11, 22 October 2006 (UTC)

Rather than base it on elapsed time alone, I suggest that archiving of old material should only be done when a certain page size is reached. There's no point in archiving a question which hasn't been answered after 10 days, if it's the only thing there. StuRat 04:16, 24 October 2006 (UTC)

Please read the instructions at User:Werdnabot/Archiver/Howto. You will see that the options are very limited. We can change the time interval between the last signed message in a section and the time of archiving, but we cannot control the size of what is taken or what is left. Our choice is reduced, but we avoid having to do all the archiving manually. JRSpriggs 09:05, 24 October 2006 (UTC)

OK. I tried to turn Werdnabot on for this page. It should run every six hours and archive sections 12 days old or older (last edit). I have never done this before, so I am not sure whether or how well it will work. JRSpriggs 09:28, 27 October 2006 (UTC)

By the way, there is one way in which it is not automatic. When we get to November 12, someone will have to create the new archive file for November 2006 and edit the code for Werdnabot invokation to reflect the new file name. The same every month thereafter. JRSpriggs 09:32, 27 October 2006 (UTC)

General Comment about Math articles from a non-mathematician

I think your readership might be better served by providing more background explanation and examples of advanced math concepts designed for a lay audience than your current pages do. Since Wolfram Mathworld already does an excellent job of rigorous textbook style explanations with all of the relevant equations why not just link to them for this content and give Wikipedia readers a simplified plain English version with some real-world applications (along with the graphs suggested above, and perhaps historical development and relevance and maybe some nice pictures of engineering applications etc.) to get them started? --—The preceding unsigned comment was added by 67.174.240.33 (talk) 22 October, 2006

It is probably true that a lot of math articles could be made a lot more accessible than they are to lay audiences. But being accessible to the extent the material allows is not the same as lobotomizing all technical or rigorous content. And I think it's often possible to do a lot better than mathworld in terms of depth and rigor and correctness, and to have content in a single place that's useful for readers at all levels. —David Eppstein 01:25, 23 October 2006 (UTC)

I hope wikipedia is strong enough to be self-contained and not to rely on third party's stuff. --Beaumont (@) 09:10, 23 October 2006 (UTC)

Several points. I'm all for history. MathWorld was invented, basically, to promote the kind of mathematics where formulae are central. It has then branched out. We on the other hand have always taken the whole range of mathematics as our remit. Some doesn't have obvious engineering aspects. In other cases, for example cryptography, we have _both_ the mathematical articles on finite fields, say, _and_ articles dedicated to cryptographic applications. Charles Matthews 09:18, 23 October 2006 (UTC)

Yeah, there is room on Wikipedia for all kinds of articles at all kinds of levels. But indeed, making articles more acessible is a great goal. Adding a more elementary intro, putting a picture here and there, making more connections between math and physics or other applications are very good things, and we are aware of that. Oleg Alexandrov (talk) 14:47, 23 October 2006 (UTC)

I'm aware of a number of articles where the introductory material has been made harder, in some supposed trade-off with accuracy or a more 'professional' feel. It would be interesing to compile a list where the intro is unnecessarily off-putting, and where the article also ought to be of general interest. Charles Matthews 16:03, 23 October 2006 (UTC)

I would really request that any such edits which complicates introductions be reported here. I think there is a consensus over here that introductions must be kept as simple as possible, and we definintely don't want people obfuscating introductions. Oleg Alexandrov (talk) 03:26, 24 October 2006 (UTC)

You mean, like the "articles that are too technical" list here? —David Eppstein 16:37, 23 October 2006 (UTC)

That page is a long list of whinges. I took one at random: D-separation. The list says 'needs more context'. It's obvious when you look at it that it's a technical thing about Bayesian networks, and sometimes technical stuff is irreducible. No, not what I meant. I meant examples of the ratchet at work, where the user-friendly sentences get shredded because some expert decides they are holding up the parade. Charles Matthews 16:47, 23 October 2006 (UTC)

In our defence, writing an intro that is accessible but still correct in all important respects is not as easy as you might think. Look at what an anon editor has created recently in Trigonometry (the Overview section) to see an example of how not to do it. Gandalf61 16:16, 23 October 2006 (UTC)

So long as Wikipedia let's anyone edit, we will have the burden of reverting and explaining why. How many editors have read WP:MSM, which advises writing broadly accessible intros? I also think we could benefit from providing a simple readability measurement tool, as many of today's word processors do. It could help take discussions out of the realm of opinion and stylistic preferences, making them more quantitative. We are fighting a tradition of professional writing that is often unreadable, and we can hardly blame people for imitating what they have seen, attempting to "sound professional". Ironic that, since empirical evidence suggests that more readable papers are more influential.

I don't trust the automatic tools enough to make them a straight-jacket requirement. I would not say, "The intro must be written at a 9th grade reading level." For one thing, there are aspects of readability that cannot be captured by counting words and syllables. Still, if an edit changes a passage from 10th grade to 16th, we can use such a measure to help train the writer.

Train we must, perpetually, if Wikipedia wishes to be a professional quality encyclopedia. As the readability improves across all our articles, it will set an example that may help. However, even that can never substitute for awareness and deliberate attention to the features that make for readability. --KSmrq^T 17:58, 23 October 2006 (UTC)

The article titled uses of trigonometry, which I originated and which is still mostly my material, is an example of the sort of thing requested here. On the other hand, some of the statistics articles tell you what a concept is used for without ever saying what it is. Those would be greatly improved by more technical material. Michael Hardy 20:33, 23 October 2006 (UTC)

One problem I see in intros is that scientists/mathematicians like to start right off with "the most general case", which often involves a complicated formula. Sometime later they reduce that down to the common formula which everybody uses. For ease of reading, the simple case, with a real world example, should appear first, and the general case/derivation should be at the end. For example, I worked on the weighted mean article, and added an example, but it still has the technical "gobbledygook" (like the discussion of variance) up front, which makes this seemingly simple topic seem complicated. (I just moved some of the complex portion to the end, but I'm worried that this edit will be reverted.) StuRat 03:49, 24 October 2006 (UTC)

Yes, each new editor must be "re-educated". It may help to repeatedly cite WP:MSM. Here is a relevant excerpt:

---

Suggested structure of a mathematics article

Probably the hardest part of writing a mathematical article (actually, any article) is the difficulty of addressing the level of mathematical knowledge on the part of the reader. For example, when writing about a field, do we assume that the reader already knows group theory? A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds.

Article introduction

The article should start with an introductory paragraph (or two), which describes the subject in general terms. Name the field(s) of mathematics this concept belongs to and describe the mathematical context in which the term appears. Write the article title in bold. Include the historical motivation, provide some names and dates, etc. Here is an example.

In topology and related branches of mathematics, a continuous function is, loosely speaking, a function from one topological space to another which preserves open sets. Originally, the idea of continuity was a generalization of the informal idea of smoothness, or lack of discontinuity. The first statement of the idea of continuity was by Euler in 1784, relating to plane curves. Other mathematicians, including Bolzano and Cauchy, then refined and extended the idea of continuity. Continuous functions are the raison d'être of topology itself.

It is a good idea to also have an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate, as appropriate. For example,

In the case of real numbers, a continuous function corresponds to a graph that you can draw without lifting your pen from the paper; that is, without any gaps or jumps.

The informal introduction should clearly state that it is informal, and that it is only stated to introduce the formal and correct approach. If a physical or geometric analogy or diagram will help, use one: many of the readers may be non-mathematical scientists.

It is quite helpful to have a section for motivation or applications, which can illuminate the use of the mathematical idea and its connections to other areas of mathematics.

---

We could improve the manual (how many readers will understand raison d'être?), but the message seems clear enough: First inform and engage the general reader, then dive into the technical details.

This does address Michael Hardy's point, somewhat. We do not omit technical details, we merely postpone them. In fact, WP:MSM is explicit:

---

There should be an exact definition, in mathematical terms; often in a Definition(s) section, for example:

Let S and T be topological spaces, and let f be a function from S to T. Then f is called continuous if, for every open set O in T, the preimage f ⁻¹(O) is an open set in S.

---

I'm uncomfortable with linking "if" to if and only if and linking "for every", and with omitting links for topological space and open set; and not every mathematical topic demands or admits a definition. Quibbles aside, the call for content is clear. We don't want to be an auto mechanic who is courteous and friendly, but who never does the job. --KSmrq^T 13:22, 24 October 2006 (UTC)

The point about postponing technical details rather than omitting them is well made. In response to StuRat, however, I must point out that we are writing an encyclopedia, not a textbook. It is appropriate to start with an informal introduction which covers in informal terms even the general cases. JPD (talk) 14:01, 24 October 2006 (UTC)

More User:WAREL clones, and blocking

Current (possibly incomplete) list:

User:218.133.184.53

User:64.213.188.94

User:DEWEY

User:DYLAN LENNON

User:JLISP

User:KLIP

User:KOJIN

User:MACHIDA

User:MORI

User:POP JAM

User:SADTW

User:Suslin

User:TANAKA

User:TELL ME that

User:WATARU

Should we block more of these? So far, I've only been blocking them if the edits are incorrect, but a number of them are still live. Was there any ArbCom action taken against him? I've lost track. — Arthur Rubin | (talk) 21:34, 25 October 2006 (UTC)

There has never been an ArbCom action, and none is necessary in my opinion. I think the clones can be blocked on sight. -- Jitse Niesen (talk) 01:24, 26 October 2006 (UTC)

Yes I think Jitse is correct. The only question I would have is how do we know they are clones? Paul August ☎ 03:22, 26 October 2006 (UTC)

I blocked User:TELL ME that indefinitely after another edit to perfect number. -- Jitse Niesen (talk) 06:13, 26 October 2006 (UTC)

Project directory

Hello. The WikiProject Council has recently updated the Wikipedia:WikiProject Council/Directory. This new directory includes a variety of categories and subcategories which will, with luck, potentially draw new members to the projects who are interested in those specific subjects. Please review the directory and make any changes to the entries for your project that you see fit. There is also a directory of portals, at User:B2T2/Portal, listing all the existing portals. Feel free to add any of them to the portals or comments section of your entries in the directory. The three columns regarding assessment, peer review, and collaboration are included in the directory for both the use of the projects themselves and for that of others. Having such departments will allow a project to more quickly and easily identify its most important articles and its articles in greatest need of improvement. If you have not already done so, please consider whether your project would benefit from having departments which deal in these matters. It is my hope that all the changes to the directory can be finished by the first of next month. Please feel free to make any changes you see fit to the entries for your project before then. If you should have any questions regarding this matter, please do not hesitate to contact me. Thank you. B2T2 00:20, 26 October 2006 (UTC)

mathematician-stub

The various mathematician-stub templates are currently being discussed at Wikipedia:Stub types for deletion/Log/2006/October/19 Affected templates {{mathbiostub}}, {{mathbio-stub}}, {{math-bio-stub}}, {{mathematician-stub}}. --Salix alba (talk) 11:32, 26 October 2006 (UTC)

0.999...

Briefly browsing the archives suggests that this has not been discussed here before (and correct me if I am wrong). It seems like a good idea to bring it up now, seeing that said article has recently been the main page featured article and all.

Long ago, KSmrq has rewritten the article (which was then named Proof that 0.999... = 1) to look something like this (I'll refer to that as the "old" version). It stayed in that form for quite a while, until this edit, where Melchoir has begun a massive rewrite, ultimately resulting in something like this (I'll refer to that as the "new" version). In the meantime, the article has been, with overwhelming support, moved to the new title 0.999... to more faithfully represent its new (and old) content (as can be seen in this archive).

KSmrq has strongly opposed the move and the rewrite, and very frequently criticizes the new version and the editors who have worked on it. Needless to say, I have the greatest admiration for KSmrq's opinion, but happen to personally disagree with him on this matter (I accept some parts of the criticism, though, and believe these should be worked on on a case-by-case basis). I also got the impression that there are not many other editors who agree with him. In my opinion, while the fact that this article has become featured in its current incarnation obviously proves nothing, it supports this impression.

I therefore invite everyone here to share your opinions on the matter, with hope to finally settle this matter once and for all. I'll emphasize that it is not necessarily my wish to see consensus supporting the new version (which, again, is more to my taste), but rather to see consensus supporting some version, and having the article become as good as it can be as a result.

Those with some extra time on their hands could also skim through the extremely numerous reactions to the article (in Talk:0.999... and Talk:0.999.../Arguments) from the last two days, and see if they give them any ideas for possible changes to improve the article.

Since Talk:0.999... is a mess right now, I suggest that replies are made on this page. -- Meni Rosenfeld (talk) 16:42, 26 October 2006 (UTC)

Well, the feeling I get most strongly from the reactions is this: We have to figure out a way to get readers who doubt the validity of digit manipulations to skip the 0.999...#Digit manipulation section, or at the very least not get stuck on it. In general, the sections should better describe their relationships to each other. At the same time, it would be useful to create new articles, or improve existing ones, to describe the foundations of decimal arithmetic for all those skeptics.

In general, I'm thrilled to discuss problems on a case-by-case basis. There's a very indirect lesson from the FAC; some of the supporters praised it for having great writing. While that's certainly a welcome sentiment, it isn't ultimately any more helpful than KSmrq blasting the article for having terrible writing. We should focus on specific, actionable issues if we want to understand each other, let alone generate progress and consensus. Melchoir 19:16, 26 October 2006 (UTC)

We should just redirect this page to 1 (number), you know. Merge or not? Charles Matthews 09:29, 27 October 2006 (UTC)

Seems the mailing list discussion was right: you do need tags like <irony>.Charles Matthews

Heh, I often use <sarcasm> tags just to be sure - it is often too easy to get confused about the intention of others. Sorry for misunderstanding. Any other thoughts, though? -- Meni Rosenfeld (talk) 17:01, 27 October 2006 (UTC)

0.9999... should redirect to 0.999.... Oh, it does. Well, scope for ... Charles Matthews 18:49, 27 October 2006 (UTC)

Are you really saying that there is nothing in this article which a general reader might want to know, or benefit from knowing? Obviously, I disagree, and I trust many others will, too. -- Meni Rosenfeld (talk) 10:12, 27 October 2006 (UTC)

My opinion is that its best to do nothing and move on. We have a slightly kooky feature article, which explains a frequently asked question about the reals. Redirecting to 1 (number) is just silly, as the article is actually about any decimal ending in 999... A merge is even sillier as it would give far to much attention to just one aspect of 1.

There plently else we could be doing, Addition is I feel close to FA status, Derivative and Integral could take some work to secure their GA status and Gottfried Leibniz needs some seriuos work to sort out an unusual citation system. --Salix alba (talk) 11:31, 27 October 2006 (UTC)

If this article should be merged at all, the target should be Recurring decimal. - Fredrik Johansson 11:49, 27 October 2006 (UTC)

Please forgive me if I say a few words, but not few enough.

When BradBeattie created it on 2005-05-06, it was a sad little stub that evolved over the next weeks into a minor service article focused on the proof. Starting on 2005-10-27, I (KSmrq) began to work it over to more effectively confront the issues raised in the thousands of posts on the topic scattered around the web. A few other users helped with tweaks and vandalism reversion, of course. (We even got a visit by WAREL, lucky us.) Inevitably, on 2006-03-26, someone felt compelled to insert a pathetic infinite sum proof, which I had deliberated avoided for reasons I detailed at various times on the talk pages. That was the beginning of an accelerating downward slide. Concern for the readers who needed the article was shoved aside as more and more and more pet proofs and passions were stuffed in. I used the talk pages at great length to explain why that was counterproductive. Then, stunningly, on 2006-06-29, Melchoir began slapping on OR tags, adamantly rebuffing everyone who visited the talk page to nudge him out of such extremism. As the end of summer approached and sensible editors took vacations, Melchoir launched an all-out assault, beginning on 2006-08-23 and continuing with twenty or more edits a day. After making a few protests that drew retaliatory threats, I wrote it off as a bad investment of my time and took the article off my watch list.

Today the article has

• a few meaningless pictures
• section after rambling section arranged in no particular order
• explanations of no benefit to those who really need them
• topics better left to their own articles (such as Construction of real numbers)
• a blizzard of 63 (!) odd, redundant in-line citations, including this series:

  27. Griffiths & Hilton §24.2 "Sequences" p.386

  28. Griffiths & Hilton pp.388, 393

  29. Griffiths & Hilton pp.395

  30. Griffiths & Hilton pp.viii, 395

• a total of 49 (!) references of dubious utility

This article in its present state does not represent the best of our mathematics community, nor the best of Wikipedia. To the contrary, I find it an embarrassment to both.

I believe it was helpful to have a small service article on the proof, and when Melchoir goaded a naive editor into nominating the article for deletion (before taking it over), large numbers of other mathematics editors agreed with that view. I think it would be helpful to recreate a small article under the original “Proof…” title, with Melchoir prohibited from editing it. Then this abomination can drift into well-deserved oblivion.

Honestly, this is a minor backwater of mathematics. I am happy to have played a pivotal role in moving the proofs away from endless ineffective parroting of "geometric series" and the like, at least temporarily. I am happy to have raised awareness of the role of standard real numbers. I am not happy with what has happened since. I am unwilling to engage in more fruitless debates with Melchoir (or his surrogates). I will not participate in a revert war. And, frankly, I'm inclined never to see this topic ever again, a view I suspect is widely shared!

I am more concerned with the bigger questions implicit in this debacle. However, I have already exhausted my patience, and likely yours as well, so I'll stop here. --KSmrq^T 19:23, 27 October 2006 (UTC)

Article count?

Is there any way to obtain a count of how many articles are in the Mathematics category or any of the categories beneath it, that is, articles that are in the scope of this project? What about other science projects such as Physics, Chemistry, etc.? CMummert 16:43, 26 October 2006 (UTC)

User:Jitse's bot says its 14953 which is updated daily. Its dificult to give an exact answer as it all depends by what you mean by a mathematics article. --Salix alba (talk) 17:36, 26 October 2006 (UTC)

Thanks. The goal I has was to get a relative sense of the sizes of the various projects. Obviously article counts don't tell the whole story, but they do give interesting numbers to compare the different projects. Is it true the Jitse's bot runs with regular user permissions (no SQL queries or anything like that)? If so, I might someday try writing a script to count the articles. CMummert 17:48, 26 October 2006 (UTC)

What you can do is download a database dump, parse that to find the links table and play around with that to your hearts content without bothering the servers (avoid importing into MySQL as it take forever). I guess there are about 1000 maths categories so querying that does impose some load. Theres lots of other ways to do queries meta:toolserver and http://en.wikipedia.org/w/query.php are both options. --Salix alba (talk) 19:49, 26 October 2006 (UTC)

It's kind of easier to figure that mathematics is 1% of enWP and then you count using the Main Page. (The proportion has been dropping, but slowly ...) Charles Matthews 21:27, 26 October 2006 (UTC)

Unfortunately, the category system here is sometimes surprising. For instance, Lute is in Category:Musical instruments is in Category:Music is in Category:Sound is in Category:Waves is in Category:Differential equations is in Category:Differential calculus is in Category:Calculus is in Category:Mathematical analysis is in Category:Mathematics. For this reason, User:Oleg Alexandrov maintains list of mathematics categories, which lists the categories that are considered mathematics.

There is also a bot called User:Pearle which does something similar to Wikipedia:WikiProject Mathematics/Current activity, but I don't quite know what it does or how it works. -- Jitse Niesen (talk) 02:54, 27 October 2006 (UTC)

And the line between math and nonmath can be blurry indeed. A few days ago my bot added the article Robert Byrd about the US senator to the list of mathematics articles because the guy has been put in the Category:Mathematics education reform. Gosh. Oleg Alexandrov (talk) 03:19, 27 October 2006 (UTC)

There is a shorter path for Lute. Category:Differential equations is in Category:Equations is in Category:Mathematics. JRSpriggs 09:15, 27 October 2006 (UTC)