Talk:Mathematics

Comments at the bottom, please. Eventually, we will have to refactor (probably just delete a lot of stuff from) this page. Some old talk has been moved to Mathematics/Old_Talk. See also Mathematics/Schemes.

Capitalization in titles

On the question of capitalization in titles, I find a contradiction between the FAQ and somewhere else I can't remember. Should titles of articles be capitalized throughout, or only at the first word? rchase

Contradiction? What contradiction? How could that possibly be, in a collaborative website? :-)

Can I encourage you, mathematicians, to please use lowercase titles? When you use uppercase titles, that forces people to Capitalize Words that Wouldn't Usually be Capitalized in Ordinary Standard English. See naming conventions.

Why isn't there any coverage of K-12 Mathematics? The focus so far is on advanced undergraduate topics.

That's a hell of a good idea. --Juuitchan

I have a suggestion for a slight reorganization of the main page:

"Methods" becomes "Foundations"
"Miscellanea" becomes "History and Miscellanea"
"History of Mathematics" (to be created) goes into "History and Miscellanea"
"Special Functions" moves to "Change"
"Fermat's little theorem" goes on a "Number theory" page (to be created under "Structure")
Links to "Symbolic Logic" and "Set Theory" are placed under "Foundations and History" (Set Theory also remains under Finite Math)
"Finite Math" moves down the list, after "Space"
"Discrete Math" moves under "Finite Math"

What do you guys and gals think? --AxelBoldt

An alternative well thought out classification scheme for math is at http://www.math-atlas.org/index/beginners.html---- As one of the original autors of Math entries here, like Group, Field, Linear algebra, Trigonometric Functions, I think that the new group of articles, are too high level for the average reader, particularly without examples. Many college grads get out of college with a general math course. I think we have to begin to fill in the lower levels. In regard to your placement of Finite Mathematics under Space, I do not agree. We also need people to draw graphics for some of these entries, not to mention a way to do matrices. RoseParks

What should the policy on Greek letters be? I notice that the symbols on the Mathematical constants page appear as blocks in Opera 5.11. Perhaps it would be better just to write them out as pi, gamma, etc., so that they are at least readable.
Zundark, 2001-08-08

I think for now we shouldn't formulate a policy: it should be up to the individual author. If you are interested in widest readability, then spelling out the symbols is best. However, all entities we use (α, ∫, etc., see Howdoesoneeditapage/Quickreference) are valid HTML 4.0 entities and browsers will sooner or later come around to supporting them. I believe the newest versions of Internet Explorer and Mozilla support almost all of them already; I'm not sure about Konqueror. --AxelBoldt

Yes, you're right. I didn't realise it was valid HTML 4.0. So it's Opera's fault for not understanding it. It works in IE4 anyway, and is at least comprehensible in Netscape 4, so I'll carry on doing it.
Zundark

----

Hadn't noticed it, but all the nice math symbols (including Σ, ∫, etc) that work fine w/ NS6.0 on Linux fail utterly w/ Opera 4. Latin-1 works fine with both (of course, latin-1 is the native char set).

So how about things like the inverted Δ (Del), inverted A, reversed E? Being an engineer, not a mathematician, these would provide all the symbols I am likely to need.

--Buz Cory

Found a pretty complete set of math symbols on how does one edit a page/Quick reference. For instance ∇, ∀, ∃.

Also did some more browser checking. StarOffice 5.0 fails utterly on the Greek and Math entities, also. --Buz Cory

Um has any one thought of putting a chaos page on any where? Like the mandelbrot and the like. I don't know much about it but I would love to learn. Michael (Tas)

Apparently nobody has. There's a bit about the Mandelbrot set though, and a stub about fractals. I'll put a

link to "Dynamical systems and chaos theory" on the main mathematics home page under "Change", and hopefully someone will bite. --AxelBoldt

It's a shame the mathematics page still lacks a good general article about mathematics. With so many mathematicians about, you'd think a general discussion and characterization of mathematics would be forthcoming. --LMS

I strongly agree with the above remark. The lack is striking but perhaps people feel this would become too much a subject of dispute?

BozMo (talk)

I doubt that's it. It's just typical of wiki work, that when things are active in the 'specialist' areas, that's where the blood flows.

Charles Matthews 11:44, 6 Apr 2004 (UTC)

Well, I might be tempted to try but churning up a page which has obviously been the object of considerable work would probably just get reversed. It would also be messy for days. Is there a way to get a copy of this page somewhere to work on it? BozMo (talk)

Yes - you can use your own user page, and invite comments: just copy the current text there. By the way, the LMS comment dates back a long way. Charles Matthews 09:45, 14 Apr 2004 (UTC)

I have started a very rough piece here http://simple.wikipedia.org/wiki/Mathematics but I will go back and work on it some more BozMo (talk)

Shouldn't there be an article for Trigonometry? I know that we have the Trig Functions article, but general trig is more broad than that.

Also, shouldn't the number sets be combined under the auspices of one article so that they can have a logical overview and venn diagram to describe their structure?

A page like Numbers could certainly be put under "Quantity" and then it could have an overview and links to all the various number sets and explain their relationships. I would still want to keep links to the reals, complexes etc. right on the main page, so that they are easily accessible. --AxelBoldt

Shouldn't there be a page on transforms? I'm not a mathematician, and can't write it myself. Some transforms I would like to see described are: fourier, cosine, z (used in digital signal processing), laplace, chirp, hilbert, etc. The transforms should be compared regarding their use. --HelgeStenstrom--

I don't think I have ever seen an encyclopedia of mathematics, so I have a question about encyclopedias of mathematics. Would the entry about elementary group theory in such an encyclopedia consist, as it does here, of a system of group theory? Or would it just discuss such a system? Don't get me wrong--I think we should have mathematical systems in Wikipedia. I am asking whether there might be some other information that mathematicians might expect out of an encyclopedia, that we aren't supplying, in most cases, yet. --LMS

I'm only familiar with one Encyclopedia of Mathematics, which is a very large one translated from Russian. The entries usually contain definitions, discuss important results, and give a list of references to the literature. They certainly wouldn't have a article like elementary group theory, since any mathematician should know that anyway. (The article is probably misnamed; proofs of the most basic results in group theory would be more accurate.) Note that we have another article, mathematical group, which discusses group theory. --Zundark

I took the section about Mathematical Books out of the main page. While it would be nice to have a bibliographical listing, right now we don't and it's premature to put it on the main page. --AxelBoldt

Somebody added that topology focuses on the concepts of continuity and direction. I don't see what topology has to do with direction, but I could be convinced. Generally, when I think "direction", I think "tangent space" and hence "differential geometry". How can you talk about direction in topology? --AxelBoldt

perhaps they meant orientable surfaces in algebraic topology. -- Tarquin

"Arithmetic does not count as a "foundation for mathematics"; it is part of elementary algebra"--AxelBoldt

Then why does the elementary algebra article begin: "For this introduction, knowledge of arithmetic (including the use of parentheses) is assumed." --BlackGriffen

No substantive knowledge, except of the English language common to an intelligent high school graduate, should be assumed--unless an exposition of the subject really does require such assumptions. In this case, since we introduce school children to arithmetic all the time, I should think we need a simpler article about arithmetic... In other words, Wikipedia's math articles should, while being maximally useful for mathematicians, also be very useful for non-mathematicians. --LMS

Yes, the elementary algebra article is incomplete and should start out very gently with introducing the order of operations, parentheses, commutativity and associativity etc. (or factor out to an arithmetic article and add a link to elementary algebra if that seems preferable). --AxelBoldt

Should there be a page for arithmetic? There's a page for modular arithmetic, defining it as "a modified system of arithmetic", without any definition of that term. Is it considered to be such a simple concept as to require only a dictionary definition? - Stuart Presnell

See the above discussion. I think we should have a separate article on arithmetic, but I'm not really sure how to write elementary stuff like this, so I haven't attempted it. --Zundark, 2002 Jan 5

What do you think about removing the link to trigonometric functions from the main math page. There's already a link to it under special functions. --Georg Muntingh

Yup, that makes sense. What we really need is a Trigonometry article though. --AxelBoldt

Okay. How can I be sure most people agree? Shall I remove it? (It looks very logical to me.) --Georg Muntingh

Just do it - you'll soon find out if someone doesn't agree. :-) --Zundark, 2002 Jan 12

A question for you all: is is the inequality

||u ± v|| ≥ | ||u|| - ||v|| |

(quoted from Normed vector space) known as the Cauchy-Schwartz Inequality, or am I thinking of something else? --Tarquin

I've found it in my notes, I'll answer my own question :-) -- The Cauchy-Schwartz Inequality is:

|<u , v>| ≤ ||u|| . ||v||

I just added the page mathematical symbols. I would like to hear your opinion on the idea of including a link to this page in each article that uses math. symbols and can potentially be made more readable for a mathematical beginner by inserting the link. --Rade

I like the symbols page; I think we don't need links from *all* articles that use math symbols, but certainly from those which are mainly directed at or will be read by beginners. Some articles are completely incomprehensible without some math background, and adding a link to the symbol page wouldn't make them any less so. AxelBoldt 18:14 Aug 20, 2002 (PDT)

Agreed. --Rade

I removed the text "2. -thematos" from the etymology part, because it's unclear: what is the two about, what is the hyphen, and because I couldn't verify it in the Oxford English Dictionary or in Merriam Webster. AxelBoldt 22:59 Sep 29, 2002 (UTC)

I noticed that the formating for the mathematical topics was changed. I liked the old style because it was more concise and it actually fits on one page. The new style is very cluttered on the screen. Am I the only one who thinks this? -- Ram-Man

I also liked the old style better. AxelBoldt 03:13 Oct 24, 2002 (UTC)

On the mathematical branch listing: Under "Finite Mathematics", "Basic Set Theory" is coined, but the article itself is called "Naive set theory", shouldn't one of these be changed? (213.8.129.12)

Going to revert "logically" to "naturally" as "naturally" has a specific meaning in mathematics - PML.

I have coloured the statements of the theorems in Pythagoras' theorem and Fermat's last theorem. The coloring could make the statements standing out in the article. If this practice is acceptable, I will do it for other theorems and conjectures (with different colour, maybe). -- User:Wshun

This seems like the kind of thing that could be very useful for many mathematical articles - does anyone else have suggestions or ideas about what format would be best? One with a white background may be necessary in order to accommodate our TeX markup; it also ensures a maximum amount of contrast for readability. I think a nice colored border would work well. For others who are interested, I have contributed an alternative formatting for theorem/formula highlighting. See Wikipedia talk:TeX markup for another example. (Actually, considering it now, maybe blue is not the best color - too similar to the link color - maybe green or orange or something?) -- Wapcaplet 02:50 17 Jun 2003 (UTC)

I've gone ahead and formatted several of the theorem article with dotted-outline purple boxes. See, for example, Pythagorean theorem, Fundamental theorem of calculus, and Fermat's last theorem. Sticking to traditions I'm familiar with in math textbooks, I've only done this with formal statements of theory (or sometimes hypothesis), and not just any old "suppose such-and-such..." Someone better versed in mathematics should probably take a look at them, since formality is not my strong suit :) I think these look quite a bit better, though. Before, there was a hodge-podge of styles for highlighting important theorems and other statements (bold, italic, indentation, etc.) Comments welcome! -- Wapcaplet 18:56 18 Jun 2003 (UTC)

The "outline" proposal seems to be an acceptable idea. Maybe we should submit a proposal on "styles of mathematics articles" before we proceed...-- Wshun

Something that would be useful to us non-math inclined people would be some sort of pyramid that goes from the building blocks of mathematics to the more advanced concepts. Give people an idea of where to start.--

I made my own version of a 'distributed roadmap' for mathematics on the page fractional calculus and some of it's subpages (the ones that aren't stubs). Perhaps something like it can be adopted as a conventional practice?

Also, where is this so-called "styles of mathematics articles". I haven't been able to find it. It should be made easier to find. - Kevin Baas

Well, time is becoming too expensive, I've become proactive and decided that there 'is' a Styles of Mathematics Articles page, for the discussion of just that, so that the pages can be more readable and organized. Anyone who wants to discuss the format of mathematics articles, style, presentation, ect., please discuss it there. Thanks. Kevin Baas

We need more on the history of mathematics. Here is a public domain encycloepdia entry from the Jewish Encyclopedia on this subject. Maybe we can adapt some of this material into an article on the history of math. RK 15:54, 30 Aug 2003 (UTC)

History of Mathematics, public domain article

I put your suggestion on the History of mathematics in the links section. -- Ap 19:16, 30 Aug 2003 (UTC)

Definitions For Math Aren't Dictionary Definitions

There is a fundamental problem in trying to arrange mathematical knowledge in a dictionary or encyclopedia and it is made worse by hypertext links. This is that mathematical definitions are not definitions of words or short phrases. For example, there is no mathematical definition of words such as "limit" or "point".

A mathematical definition may contain the word "limit" but it involves explaining a relation between complete statements, not trying to equate a single word with some statements. We don't define "limit". We give a statement, such as "The limit of the function f(x) as x approaches a is equal to L" Then we say this is equivalent (or "means") another statement such as "for each epsilon greater than 0 there exists a delta greater than 0 such that ...". And all this must take place in a context where we state or imply by notation conventions that f is a real valued function and epsilon and delta are real numbers.

This seems tedious and even abstract mathematical textbooks make concessions to the ordinary use of language. For example, after a long definition is given for "f is a measure on the sigma algebra S of subsets of the set Omega", the text may say "A probability measure" is "a measure such that f(Omega) = 1". This would appear to be a definition that says one phrase is another phrase rather than saying one statement means another statement. The books assumes the reader will recognize that it amounts to saying that the statement "f is a probability measure on the ..." is equivalent to the statement "f is a measure on the ... and f(Omega) = 1".

In the case of "point", this word is usually taken as an undefined term. A fact that is disturbing to non-mathematicians is that undefined terms must be a central feature of any mathematical system. A layman's idea of a definition is an explanation that gives him an intuitive understanding. The trouble with such a definition in mathematics is that an intuitive idea can tempt us to introduce properties that mathematics does not assume. For example, if the teacher says "Let f(x) be a function such that f(3) = 7", a student may object: "That can't happen. The point 3 is infinitely small, so f couldn't do anything there because there is no time or room. And f couldn't find it exactly because it's too small to see." We regard this as a confusion originating from intutive ideas. The student misinterprets a function as a matching-up process that must have some physical realization. His idea of a point is something that has no temporal or spatial extent.

In the case of words that mathematics takes as undefined, I suppose an encyclopedia is obligated to give the reader intuitive guidance. In such a case it would be wise to mention the fact that what is being offered is an intuitive idea that explains the motivation for creating certain mathematical models. And that some intutive ideas about the term may not be reflected in the mathematical assumptions used in the model.

In the case of words like "limit" we have a term that is embedded in various mathematical definitions. (For example the definition for "f(x) is the uniform limit of the sequence of functions f[i](x) as i approaches infinity" is different than the definition given above.) I would prefer to see mathematical definitions gven in the form of one statement being defined as equivalent to another statement. This doesn't preclude giving the intuitive and cultural background for the isolated word. But a clear distinction should be made between the formal and informal discussions.

I have a friend who gets upset at the imprecise use of words in cultural and political discussions. He asks, with exasperation, "When will people learn that words have meanings?". My reply is that "Words don't have meanings. Only statements have meanings." This summarizes the cultural conflict between the dictionary and the mathematics textbook.

Stephen Tashiro

I can't agree with all of that. There seem to be different cases. With some things, like the integral sign, you do have an 'incomplete symbol', and the Frege-style analysis of the meaning as recovered from complete propositions is a good match to what is going on. In other cases such as 'limit', there may be difference usages, implied in different contexts, but there is a perfectly good definition (although you can also treat 'lim' as an incomplete symbol, you don't really have to). And in the case of 'point', it is effectively undefined in standard pure mathematical usage, but reflects the way we say or understand: point is to space as element is to set.

Charles Matthews 08:35, 11 Sep 2003 (UTC)

Another tension is the one between the dictionary and the encylopedia. Encylopedia articles are usually self contained while technical dictonaries often leave the reader with the impression that he is being told that one term, which he does not understand, is a special case of another term, which he does not understand. Tracking down what something means becomes something like a pinball game.

In mathematics, I think that the dictionary approach would be useful for a limited audience. It would help mathematicians who already understand most of the various technical terms mentioned and just need a little hint about how they are related. Perhaps all the highly technical terms will go the way of the dictionary and the simpler terms will find a good expositor.

Stephen Tashiro

Perhaps the "Definitions for Math .." thread should be moved to the styles for mathematical articles page. But there should be a link to the styles page on the mathematics page. Stephen Tashiro

Perhaps the "Definitions for Math .." thread should be moved to the discussion page for styles for mathematical articles. But there should be a link to the styles page on the main mathematics page. Stephen Tashiro

How come math education is completely absent from the page? It should be listed in one place. I know that math education is a controversial subject, but it should at least be mentioned.

I searched for Eric (of treasure troves/mathworld fame) in this page and the old one, and I couldn't find any references to him. There is a link to his pages at the bottom of the article; shouldn't there be a warning there? Or does everyone know about the events surrounding his website already?

MrJones 19:10, 18 Oct 2003 (UTC)

I had a question. In law of sines and law of cosines, are the angle measures capital letters (i.e. A, B, C), or greek letters (i.e. α, β, γ)? Because the formula:

${\sin A \over a}={\sin B \over b}={\sin C \over c}$

looks kinda tacky. I always learned it as:

${\sin \alpha \over a}={\sin \beta \over b}={\sin \gamma \over c}$

What should I use in the formulas on the mentioned pages?

I don't see anything tacky about using A, B and C, and it's a common notation for this sort of thing. A problem with the Greek letters is that they don't work in some browsers (unless you do them as images, which does look tacky if used everywhere). --Zundark 10:14, 15 Nov 2003 (UTC)

That makes sense... and I have weird senses of things that I think are tacky.

Common definition of

I (a simple user of the fruits) read with much confusion the stated definition of Mathematics. Space? Change? Structure?! And here I always thought that the art had more to do with understanding the rules for working with (operating on) abstract concepts/objects (and the rules for working with those rules...)

Oh, this aticle looks for me as written from formalistic point of view. Hell, I think mathematics is not 'investigation of axiomatically defined abstract structures'. There were some problems with foundations of mathematics and then many people thought about axioms, sets, structures, definition numbers, etc. But these problems were solved so modern math concentrtes on real work.

I've deleted the word 'modern' before 'formalistic'. Sorry, but this is 50 (Bourbaki) or 100 (Hilbert) years old. Ilya

I removed the like to "Axiom of Countability" from the list of famous theorems and conjectures. Whatever axioms of countability are, they are not famous theorems or conjectures. Now nothing links to "axiom of countability", which is a really odd little article anyway.

Mathematics is Not...

In the article, what is the purpose of the single member list (which BTW is not in standard Wikipedia form for a trivially populated list) captioned "Mathematics is Not..." ? Is there supposed to be any difference between this presentation of information and just a simple sentence that says the same thing? Bevo 18:32, 23 Feb 2004 (UTC)

Formula pages/tables

I was just thinking, isn't it a nice idea to have a link on the main mathematics page, to a page which is an index for the various formula pages? --Georg Muntingh 13:59, 10 Apr 2004 (UTC)

North America is not a country

"Mathematics is often abbreviated to math in North America and maths in other English-speaking countries." seems to suggest that North America is a country, which it isn't. Or maybe I'm being nitpicky. Elektron 09:15, 1 May 2004 (UTC)[reply]

Wikipedia:Be bold, pitch in and change it if you feel strongly about it :) Dysprosia 09:17, 1 May 2004 (UTC)[reply]

I assume nobody has objections to changing it to "math (American English) and maths (British English)". Elektron 16:36, 1 May 2004 (UTC)[reply]