Talk:Mathematics/Archive 10

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Intro

Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.

or

Mathematics is the area of study that deals with concepts such as quantity, structure, space and change.

? —Mets501 (talk) 08:15, 31 July 2006 (UTC)

I prefer discipline. There was a long discussion about the first paragraph above, so I suspect later parts of the article are more amenable to be changed at the moment. The third paragraph is being debated at the moment - see above. Stephen B Streater 08:36, 31 July 2006 (UTC)

I agree with Stephen. Rick Norwood 13:28, 31 July 2006 (UTC)

Me too, I just brought it up because it was changed to "area of study" and I had reverted it to "discipline". —Mets501 (talk) 17:11, 1 August 2006 (UTC)

Third Paragraph

I've been checking back frequently, and haven't seen any objections or revisions yet to my last proposal on the third paragraph (discussion above). Therefore, I'm being bold and changing it in the article. capitalist 02:27, 1 August 2006 (UTC)

Looks good to me. Rick Norwood 13:41, 1 August 2006 (UTC)

Looking into a minor change. Don't feel strongly about the exact change, but there is no comparison as such, more a contrast or recognition that pure and applied complement each other. Stephen B Streater 22:33, 1 August 2006 (UTC)

It was originally "contrast", but Rick Norwood had a concern about that with which I agreed (see the main discussion section above on the first & second paragraphs. Anyway, that's how it ended up as "In comparison". capitalist 02:55, 2 August 2006 (UTC)

Actually, never mind LOL! Paul August just cut the Gordian Knot by removing the phrase entirely. I actually think it sounds much better now. capitalist 03:35, 2 August 2006 (UTC)

Suggested Improvement

In the diagram showing the different types of numbers, the irrationals are not listed. I do not know how to add them to this diagram, but it is very important to do so.

In addition, as for the part of that diagram on the real numbers, only pi, e, and sqrt(2) are listed. This could create some confusion among people who aren't so skilled in math, because the set of all real numbers also contains the rationals.

—Preceding unsigned comment added by 69.231.140.106 (talk • contribs) 01:26, August 3, 2006

I don't think we want to add the irrationals particularly, the types of numbers listed have the nice property that each successive set contains, and is a kind of "completion" of, the previous set (e.g. the integers contain the natural numbers, and "complete" them with respect to subtraction, the rationals contain the integers and "complete" them with respect to division, etc.)

Your second concern has more merit I think, and it applies not just to the real numbers but also to the rationals, and complex numbers. So how about the following?

$1,2,3\,\!$	$0,1,-1,2,-2\,\!$	$1,{\frac {2}{3}},-0.125\,\!$	$3,{\sqrt {2}},\pi ,-e\,\!$	$-2,i,3i+2\,\!$
Natural numbers	Integers	Rational numbers	Real numbers	Complex numbers

I've also removed the elipses, since I thought I needed the space, and I've never really liked them anyway. Comments?

Paul August ☎ 02:54, 3 August 2006 (UTC)

Yes, the only change I would make is to put the examples in the correct order on the number line. The integers for example should be -2, -1, 0, 1, 2 to make it less confusing to someone who may not be as familiar with the concept. capitalist 02:59, 3 August 2006 (UTC)

A significant improvement, Paul. Definitely with you on this one. Soo 10:07, 3 August 2006 (UTC)

It might be useful to leave one complex number in the form $re^{i\theta }$, similar to what is already there. Other than that, I think it looks good. Cheers, darkliight^[πalk] 08:07, 8 August 2006 (UTC)

These improvements are welcome. I think the one thing still lacking is the concept that the earlier sets are included in the later sets (except for 1 in integers). This could either be by including an extra entry in each list, or by saying this explicitely. PS Have we discussed algebraic numbers as a separate entry? Stephen B Streater 09:59, 8 August 2006 (UTC)

I've finally incorporated everyones idea's (I think) and changed the article accordingly. Paul August ☎ 03:56, 24 August 2006 (UTC)

Thanks. Perhaps algebraic numbers was a step too far ;-) Stephen B Streater 20:07, 24 August 2006 (UTC)

You're welcome. I did not add the algebraic numbers because they do not fit into the nice chain of containment, mentioned above (e.g. i is algebraic). — Paul August ☎ 17:03, 25 August 2006 (UTC)

Yes that's a good point. The same would apply to Conway numbers too. Stephen B Streater 17:51, 25 August 2006 (UTC)

I propose to replace the current

$2,i,3i+2,-re^{i\theta }$

by

$2,i,-2+3i,re^{i\theta }$.

Rationale: (1) Putting the imaginary part before the real part as in $3i+2$, although not wrong, is unconventional. (2) The "polar" formula is the only expression in the display that is a general form, and then its minus sign, which does not add to the generality, is a bit curious. (3) But we want a minus sign somewhere, so I moved it to the real part of the truly complex number. (4) This is a minor matter of taste, but I find it pleasing that the first three concrete numbers happen to be ordered by increasing value of their arguments in the range [0, 2π).  --Lambiam^Talk 00:35, 26 August 2006 (UTC)

Yes, fine with me. Paul August ☎ 03:39, 26 August 2006 (UTC)

Same here. And I bet some of our readers will actually appreciate these improvements :-) Stephen B Streater 07:09, 26 August 2006 (UTC)

Agreed; this piece is vastly improved now. What's next? capitalist 02:27, 27 August 2006 (UTC)

And so replaced. --Lambiam^Talk 06:46, 27 August 2006 (UTC)

I don't agree that $re^{i\theta }$ is a complex number. It is just a function of two variables. r and θ could be quaternians or matrices for all we know. --McKay 06:42, 29 August 2006 (UTC)

Do you have a concrete proposal, such as removing that form, or adding a domain for r and θ? --Lambiam^Talk 08:08, 29 August 2006 (UTC)

Maybe something like $e^{i{\frac {\pi }{4}}}$ or $2e^{i{\frac {\pi }{4}}}$? JPD (talk) 11:02, 29 August 2006 (UTC)

Yeah I think just a concrete example of a complex number in polar form might be helpful, and your second second example looks as good as any to me. darkliight^[πalk] 11:57, 29 August 2006 (UTC)

Yes - I would have used $2e^{i{\frac {4\pi }{3}}}$ myself - $2e^{i{\frac {\pi }{4}}}$ is slightly unfortunate if we are making the numbers more complex as we go along the table, and we have been moving round 0 up until now. Stephen B Streater 13:38, 2 September 2006 (UTC)

I've put in $2e^{i{\frac {4\pi }{3}}}$. Even mathematically illiterate readers will appreciate that this is indeed a complex number. :) --Lambiam^Talk 16:00, 2 September 2006 (UTC)

Hehehe, yep! darkliight^[πalk] 17:10, 2 September 2006 (UTC)

Math and Maths in first sentence.

I propose that 'Maths' and 'Math' should be included in bold in the first sentence, something like:

"Mathematics, often abbreviated to Maths or Math, is the discipline..."

Wherever a subject has more than one commonly used name or short-form of a name they should be included in the introduction. Look at the article on the United States for example. Noting these abbreviations further down the page is not good enough. You might not believe it, but there are people out there who have only ever called it Math or Maths and have no idea what it is short for, not just stupid people, I mean people with English as a 2nd language for example, we need to accomodate for all users. Abc30 03:34, 4 September 2006 (UTC)

When I reverted your change I hadn't realized how far out a recent edit had moved the sentence (and to a place where it does not properly belong to boot). If mentioned in the first sentence, and including the (not irrelevant) information about the distribution of the two forms, we get something rather ugly:

Mathematics, often abbreviated to math in the U.S. and Canada and maths in Britain, Ireland, Australia and many Commonwealth countries, is the discipline that deals with concepts such as quantity, structure, space and change.

So what should we do? Mention this twice, once in the first sentence without the distribution information and once again with? Or just forget about explaining who use "math" and who say "maths"? In any case, there is no good reason to capitalize the abbreviations. --Lambiam^Talk 07:43, 4 September 2006 (UTC)

Is Math used anywhere outside of the US and Canada? If not that could make the sentence alot shorter. Likewise for commonwealth countries if it makes it easier. darkliight^[πalk] 08:23, 4 September 2006 (UTC)

Given how much we've packed into the first paragraph already, shorter is better. We could have: Mathematics (also maths or math) ... with more detail later on. Stephen B Streater 08:27, 4 September 2006 (UTC)

Yes, that's exactly how it should be done. I had considered making that change myself but I know that the intro is fiercely defended by those who wrote it and, given the level of junk that is added to this page daily, that's probably a good thing. Soo 09:45, 4 September 2006 (UTC)

Yes I think the abbreviations should just be mentioned in the first sentence (as in my example above), and then the usage of each abbreviation should be explained later in the article. Abc30 12:41, 4 September 2006 (UTC)

So can I make the change or not? The few comments received so far seem to reach a consensus that the current wording should remain, but with something like (also maths or math) included after the first word. If you oppose this please say so soon otherwise I'll make the change and it will be unfair of you to revert it. Abc30 21:50, 4 September 2006 (UTC)

I usually wait for some 24 hours, so that I know for sure users from all time zones may have had a chance to see it (if they connect daily). --Lambiam^Talk 22:24, 4 September 2006 (UTC)

Yeah don't worry that's what I intend to do. Anyway I'm guessing there will be some objecions to this change, people seem incredibly protective of the article... Abc30 23:15, 4 September 2006 (UTC)

I also agree with mentioning the three terms briefly in the first sentence and explaining later. The consensus still holds! :0) capitalist 02:14, 5 September 2006 (UTC)

Abc30 says:

You might not believe it, but there are people out there who have only ever called it Math or Maths and have no idea what it is short for, not just stupid people, I mean people with English as a 2nd language for example....

You're right, I don't believe it. Oh, I'm sure there are some, but not many, and ESL speakers least of all; people who've studied English in class will have been taught to say "mathematics", not "math" or "maths". --Trovatore 06:19, 5 September 2006 (UTC)

I've moved the existing sentence about the abbreviations back from the section Etymology (where they don't belong) to the lead section. This is not meant to pre-empt the issue of whether "math" and "maths" should be mentioned parenthetically in the first sentence. --Lambiam^Talk 07:23, 5 September 2006 (UTC)

If that sentence stays, I don't think there is any reason to repeat it in the first sentence. I'd rather see this placed in the first sentence instead of the lone paragraph. Could we have something like

Mathematics (often abbreviated math in the United States and Canada or maths elsewhere) is the discipline that deals with concepts such as quantity, structure, space and change.

Ofcourse, that's only helpful if it's actually true :) Is it actually abreviated in other predominatly english speaking countries? Is math confined to just the US and Canada? darkliight^[πalk] 07:33, 5 September 2006 (UTC)

The intro sentence should be kept as concise as possible. The details of where and when the various other names are used should be placed elsewhere in the article. Currently we don't have anywhere to put them, but that's a separate issue that needs to be resolved. Soo 10:23, 5 September 2006 (UTC)

darkliight: Yes it is almost always abbreviated (to maths) in english speaking countries. I think the only time anyone ever says "mathematics" where I live in the UK, is if they are talking about it in some kind of technical sense, and even then you would seem highly pretentious. Trovatore: You are naive to think everyone out there is so well educated.

I don't think keeping the sentence concise is a good enough reason to leave it out. There are thousands of articles on here which could have much shorter and easier to read opening sentences, but instead they include the short-forms and alternative names because the editors of these articles recognise that it is important to ensure that the reader knows exactly what the article is about. Abc30 12:05, 5 September 2006 (UTC)

I'm agreeing with you, mate, I don't think you understood my comment. We should mention math and maths in the opening sentence. The discussion of when and where each of those terms is used should be further into the article, since it's not of any great significance to anyone who doesn't know already. Soo 12:36, 5 September 2006 (UTC)

I'm fine with just the abreviations being used in the introduction. I don't know where we should mention their usage though. darkliight^[πalk] 13:20, 5 September 2006 (UTC)

Hmmm. There is an obvious advantage to having the abbreviations in the first sentence, but the explanation of their usage doesn't really fit in the article apart from the introduction. I suppose it is possible to mention the abbreviations twice in the intro. JPD (talk) 13:25, 5 September 2006 (UTC)

About the shortest I can come up with having the abbreviations in the first sentence and covering the usage is:

Mathematics (abbreviated to math in North America and maths elsewhere) is the discipline that deals with concepts such as quantity, structure, space and change.

While this does not explicitly state that the abbreviation "math" applies to North American English, the "elsewhere" part does not limit "maths" to extra-American English use either anyway. (Coincidentally, les maths is understood in France.) I don't think the word "often" is needed.

How do we proceed from here. Do we need to have a poll? --Lambiam^Talk 15:29, 5 September 2006 (UTC)

Yes, I hope it's not too bold of me to say so, but I am the one who started all this, so I am going to say we should have a poll. If you agree to the wording proposed by Lambiam above please put Support below. If you disagree please put Oppose. Abc30 16:35, 5 September 2006 (UTC)

Maybe you should peruse WP:STRAW first. --Lambiam^Talk 02:31, 6 September 2006 (UTC)

Support Abc30 16:35, 5 September 2006 (UTC)

We haven't needed a poll for any of the above discussions, so I don't see why we need one now. Would Lambiam's wording be made too lengthy if "North America" were replaced with "the United States and Canada"? JPD (talk) 16:55, 5 September 2006 (UTC)

Ok... Well I think 'North America' is fine. Most people speaking English in Mexico learn American English so presumably say 'math'. There may be some caribbean islands with more British influence that say 'maths', but I can't see anyone point in being so picky. 'North America' makes the point. Abc30 17:16, 5 September 2006 (UTC)

If we must (I'm not really convinced this is necessary or good) mention "math" and "maths" in the first sentence then I definitely would favor the short form: "Mathematics (also maths or math) …" Paul August ☎ 02:43, 6 September 2006 (UTC)

I've already said that I favor the short form as well. Putting the geographic stuff in the first sentence tends to diffuse its point and gives the reader the impression that a bunch of editors debated the wording for days...ahem! Anyway, why can't the discussion about who uses "maths" vs. "math" go into the Etymology section? I know etymology is supposed to be about the origins of the word, but it seems the most likely spot for this stuff. capitalist 04:04, 6 September 2006 (UTC)

I also prefer the short form "Mathematics (also maths or math) …" with more information later. Anything more complex will expand over time as people make it more accurate, which will detract from the main thrust if the sentence. Stephen B Streater 08:26, 6 September 2006 (UTC)

Yes, exactly. Many featured articles are dogged by over-elaborate sentences which are clearly the result of "compromise" between lots of different editors. Let's just keep it simple. Further discussion can go elsewhere - we can rename the Etymology section if necessary. Soo 09:24, 6 September 2006 (UTC)

Is anyone who suggested a longer version in the first sentence happy with the more concise wording? Stephen B Streater 18:50, 6 September 2006 (UTC)

Is anyone who suggested a longer version in the first sentence unhappy with the more concise wording? Lambiam^Talk 21:14, 6 September 2006 (UTC)

Well I don't know if my opinion actually matters, but my only concern is that the two words are included, in whatever way. So if it seems like a consensus is reached, then somebody make the change (to the concise version). Abc30 23:41, 6 September 2006 (UTC)

I think everyone's opinion counts .... anyway, I'd rather see the word abbreviated instead of also used, but other than that, it's fine with me. darkliight^[πalk] 01:43, 7 September 2006 (UTC)

So... was all this discussion for nothing? Abc30 22:45, 10 September 2006 (UTC)

Sometimes the best course of action is to do nothing. If it ain't broke, don't fix it. That's how I read the consensus in this case. capitalist 03:04, 11 September 2006 (UTC)

After all this work, I propose one last suggestion: Mathematics (abbr. maths or math). This allows the alternatives near the front as convention requires, is short, and includes abbreviation, although in its more concise form. Stephen B Streater 08:12, 14 September 2006 (UTC)

I'm happy with that. JPD (talk) 09:08, 14 September 2006 (UTC)

I've added it in. I think it meets consensus, and also FA requirements (we are moving towards featured article status here). Stephen B Streater 09:17, 14 September 2006 (UTC)

I'm happy with this choice (I think the inclusion of "abbr." is better than "also"). Paul August ☎ 13:40, 14 September 2006 (UTC)

I'm in...looks fine. capitalist 03:32, 16 September 2006 (UTC)

Unrestricted Theorem Proving in "What Mathematics Is Not".

As some branches of logic and theoretical CS do consider unrestricted theorem proving or the automation of mathematics, should this really be here? 64.59.43.150 13:38, 12 September 2006 (UTC)Aaron Kaufman

As I've said before, I'd be happy nuking the whole "is not" section; like "misconceptions" sections in general, I think it causes more problems than it solves. Still, I think you might have missed the point a little. It's not that proof of arbitrary theorems is not a valid object of study of mathematics; it clearly is. The point is that it isn't mathematics itself. That is, while mathematicians may be interested in how random-theorem-provers behave, they are not themselves random-theorem-provers. --Trovatore 18:09, 12 September 2006 (UTC)

It invites confusion then that the phrase "theorem proving" is wikilinked to automated theorem proving. While it may be a revelation to some that mathematicians are not themselves random-theorem-provers, we may take it for granted that it is well known that they are not themselves automated theorem-provers. --Lambiam^Talk 20:13, 12 September 2006 (UTC)

I'd like to see it gone as well. The whole idea is wrong-headed. Perhaps we should list all known mammals in there, since mathematics is not them either. Soo 16:07, 13 September 2006 (UTC)

Whoaaaaa everyone...slow down. There are actually some very valid points in these sections. If we are to remove the entire "Misconceptions" section or even just the "What mathematics is not" subsection, I strongly favor finding a new home for the information somewhere else in the article. The distinctions between mathematics and numerology, accountancy and the physical sciences are important ones. They are important not for those who are well versed in mathematics, but for the greater number of those who are NOT. Eliminating popular misconceptions should be (if it's not already) part of Wikipedia's mission. capitalist 03:17, 14 September 2006 (UTC)

No, I disagree. WP is a reference work, a place where you look things up. It has no pretensions to teach, much less to correct misconceptions. I say to /dev/null with the whole section. --Trovatore 03:58, 14 September 2006 (UTC)

Your general point about that is fine. However, pressing general remarks to the extent of making articles less useful is not so fine. This is a central article, and one can argue (I do) that delineating mathematics on a few sides is part of making it a comprehensive introduction. Charles Matthews 09:31, 14 September 2006 (UTC)

I'm not convinced that the section is really all that useful. Also, in addition to my point above about the nature of WP, sections like this are serious POV bait. In particular the current assertion that

empirical testing of mathematical predictions is not a mathematical undertaking

is completely unacceptable. I'll give some thought as to what I think should be done about that particular one. But I really think the cleanest solution is just to dump the whole section. --Trovatore 00:38, 15 September 2006 (UTC)

After reviewing the "Verifiability, not truth" section of Wikipedia's policy, I would have to now agree with Trovatore that it's not Wikipedia's place to eliminate popular misconceptions. However, it is acceptable to include statements such as "mathematics is not numerology" in an article as long as the article cites reputable sources for such statements. I think the solution is to continue to "delineate mathematics on a few sides" as Charles said, but to do it by citing sources for those delineation statements.

I can't find the statement that, "empirical testing of mathematical predictions is not a mathematical undertaking" in the article. I was trying to find out the context of it, but it's not there now. Looks like that problem is solved for now? capitalist 03:58, 16 September 2006 (UTC)

Never mind; I found your recent edit. I don't really have a dog in the "empiricist vs. rationalist" debate since I subscribe to the Randian/Objectivist view that eliminates the false dichotomy and integrates the two. I'll happily watch that battle (if it comes) from the sidelines. capitalist 04:08, 16 September 2006 (UTC)

Capitalist, as I'm studying philosophy of math right now, I would absolutely love to hear your point of view in some other medium (email or user discussion). It is a philosophy I am unfamiliar with. As to the topic of this current debate, I would agree with ditching the whole section for the same reasons as have already been stated. Archmagusrm 07:25, 16 September 2006 (UTC)

I left a message on your talk page. :o) capitalist 03:49, 17 September 2006 (UTC)

I'm not against replacing material by better material. As far as delineation goes, mathematics has boundaries with: physics/theoretical physics/mathematical physics, engineering, computer science (now probably distinct from engineering), logic (where mathematics ingested formal logic, but philosophical logic arose), economics, the information theory/cryptography area, statistics. These are all significant boundaries, with neighbouring but distinct areas. The boundaries with accounting, astrology, numerology, pseudomathematical work are tenuous/non-existent, though in the past they have played some part. I suppose navigation can serve as an example of an area of applied mathematics that has been in effect completely solved. Charles Matthews 15:12, 17 September 2006 (UTC)

Classification of math

The article uses "structure" to mean discrete structure, but "space" is the study, essentially, of continuous structure. I think both divisions of math should lie under an uber-division called structure.

ALSO -- re the question of applied math / pure math: These are extremely different endeavors. Pure math is not technically related to the real world (despite taking much inspiration therefrom); applied math's whole point is to apply what is known in pure math to learn things about the real world.

Of course, applied mathematicians sometimes find they need to develop new mathematical insights (theorems) in order to better get the results they seek about the real world; when this occurs, the applied mathematician is doing pure math.

CONCLUSION: Pure math and applied math each deserve their own article; in each article, the connection to the other kind of math should be discussed at least briefly. —Preceding unsigned comment added by Daqu (talk • contribs)

Fear of Math

Why hasn't anyone mentioned Math-phobia? its the number one reason why students fail at Mathmathics and it is a very real problem that everyone must know about. Defeating this, in my opinion, will solve the top problem with Students who do not understand Mathmatics.Magnum Serpentine 18:49, 25 September 2006 (UTC)

That may be a good candidate for its own article, as long as the information is verifiable and it does not constitute original research. capitalist 03:50, 26 September 2006 (UTC)

There is a section on Innumeracy in the article numeracy and also an article on something called Dyscalculia. These cover the topic to some extent, though more could be said--agr 14:25, 26 September 2006 (UTC)