Wikipedia:WikiProject Mathematics

First, an important note for everyone to remember:

A few Wikipedians have gotten together to make some suggestions about how we might organize data in articles about mathematics. These are only suggestions, things to give you focus and to get you going, and you shouldn't feel obligated in the least to follow them. But if you don't know what to write or where to begin, following the below guidelines may be helpful. Mainly, we just want you to write articles!

Title

WikiProject Mathematics

Scope

This WikiProject aims primarily to organize articles in the area of mathematics; in its broadest terms, this may include overlap into the areas of physics, computer science, and other areas.

The goals of this WikiProject are:

• provide a standard "bare bones" format for mathematical articles
• provide useful links for article writers
• provide a location to discuss issues relating to this section of Wikipedia
• provide standards for mathematical notation using wikified HTML.

Some issues to think about

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 general statements as the article proceeds. The structure describe below is one way of achieving this.

As you write your article, remember that Wikipedia is constantly growing; and so it may make your job easier to link to definitions of terms used in your article rather than to attempt to define / prove them in-line. If you use a term that you have good reason to think will be used again in another article, by all means create a good stub; if you list it on the list of mathematical topics (see below), the odds are good that someone will expand on it.

Since some terminology varies from author to author in the literature, you can check the Wikipedia article on an ambiguous term (if one exists) to see what usage is established here (or to see if you want to try to change that).

Only use the linking feature on the first occurence of a term; it is more distracting to read "the set of all sets having sets as members" than it is to read "the set of all sets having sets as members".

It's worth a bit of time to just peruse what's already in the 'pedia; this will give you a feel for what type of information is already available, and how much detail you need to provide.

Proofs

This is an encyclopedia, not a collection of math texts; but we often want to include proofs, as a way of really exposing the meaning of some theorem, definition, etc. A downside of including proofs is that they may interrupt the flow of the article, whose goal is usually expository. Use your judgement; as a rule of thumb, include proofs when they are part of an explanation; don't include them when they are a justification whose conclusion is merely "... therefore, P is true".

Since many readers will want to skip proofs, it is a good idea to set them apart in some way, for instance by giving them a separate section.

Suggested structure of a mathematics article

Mathematical articles typically rely highly on an exact definition of the article title; but in general a definition only begins the process of explaining the idea under consideration.

A general format that seems to be working well is as follows:

• An introductory paragraph (or two), including the article title in bold, which describes the subject in general terms, and giving the mathematical context in which the term appears; for 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. Continuous functions are the raison d'être of topology itself.
• An exact definition, in mathematical terms; often proceeded by a subheading "==Definition(s)=="; 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.
• Some examples (often proceeded by a header ==Examples==), which serve to both expand on the definition, as well as provide some context as to why one might want to use the defined entity. You might also want to list non-examples -- things which come close to satisfying the definition but do not -- in order to refine the reader's intution more precisely.
• Often, you will need to introduce some notation (again, often in its own subheading). Remember that not every one understands that, for example, x^n = x**n = xⁿ; try to use the standard notation (listed below) if you can. If you need to use non-standard notations, or if you introduce new notations, define them in your article.
• Often you will want to then add subheadings for applications or motivations which help illuminate the use of the mathematical idea and its connections to other areas of mathematics.
• A section about the history of the concept is often useful and can provide additional insight into the motivation.
• Finally, most mathematical ideas are amenable to some form of generalization under the subheading ==Generalizations==; for example, multiplication of the rationals can be generalized to other fields, and so on. Given the amount of pretty abstract stuff already on the 'pedia, this is a good place to link out from.

Standard notation

Life would be wonderful if TeX markup or some equivalent could be easily used in Wikipedia articles; but at this point in time, these features are not supported for a number of reasons (see m:Math Markup for comments and thoughts on how this could be accomplished in the future on Wikipedia).

In spite of this lack, some standardizations and the use of a little bit of cleverness allow us to present almost any mathematical description.

Italicization and bolding

To start with, we generally use italic text for variables (but never for numbers or symbols). Most editors prefer to use emphasised text (with '', i.e., apostrophe-apostrophe) rather than italic text (with the <i> tag, resembling HTML), since the former is easier to type and read in the edit box, even though the latter is more correct from the point of view of the HTML that it produces. Thus we write "''x'' = (''y'' + 2)" for "x = (y + 2)". Note that the parentheses, equals and plus signs are not italicised.

Functions are also frequently italicized; but this depends more on the application. For example, the standard sin and cos functions are rarely italicised, but we italicise f when we define the function f by f(x) = sin(x) cos(x)

Sets are usually written in upper case, and italicised; for example, "A = {x : x > 0}" (in wikified text, that would be "''A'' = {''x'' : ''x'' > 0}").

Greek letters should not be italicised; for example, as in "λ + ''y'' = π''r''²", for the expression "λ + y = πr²".

In a differential equation, the leading "d" indicating the derivative should not be italicised; for example, we write

   ∫ ''u'' d''v'' = ''uv'' - ∫ ''v'' d''u''

to get

∫ u dv = uv - ∫ v du

Commonly used sets of numbers are typeset in boldface, as in the set of real numbers R; see Blackboard bold for the types in use. Again, typically we use three apostrophes (''') vs the <b> tag for bolding.

Using special symbols

You may want to have a look at the table of mathematical symbols. Not all of these symbols are displayed correctly on all browsers; it is generally better to be rather conservative in the use of HTML character entities in order to reach a larger audience, for example by writing "x in Y" rather than "x ∈ Y".

As noted above, we are shooting for clear expositions; choose your variable names and your wording to maximize the ability of the reader to follow your drift. Often, if you can explain an idea by using a series of formulas, both you and the reader will benefit from the exercise of explaining it in words.

Indented vs. non-indented

Important equations are indented by starting the line with one or more colons (:), as in

:''x''<sub>''i''</sub><sup>2</sup> + ''y''<sub>5</sub><sup>''n''</sup> = 1

to get

x_i² + y₅ⁿ = 1

Don't indent using spaces; although this may make layout easier, any line which starts with a space is rendered in an ugly mono spaced font like this:

    xi2 + y5n = 1

or this (without the italics)

    xi2 + y5n = 1

The latter style is more appropriate to articles in the area of computer programming or algorithms, where the formatting structure (indents, etc.) is important to understanding the content.

This is probably the single biggest drawback to "wikified" math notation; one often wishes to write, for example

        π2
   x = ---
        4

but has to live with instead

x = π²/4.

The use of spaces to indent sometimes cannot be avoided (for example, for setting up a table); use your best judgement. The preference is to stay with the nice "serifed" fonts, and avoid the mono-spaced fonts.

Another reason to avoid formatting with spaces is that this will make the formula essentially impossible to read with a nonvisual web browser (such as a voice browser for the blind).

Common constructions

Some examples of how to get desired effects:

Inverse: x · x^-1 = 1

Wikified text: ''x'' · ''x''^-1 = 1

Function composition: (f o g)(x) = f(g(x))

Wikified text: (''f'' <small>o</small> ''g'')(''x'') = ''f''(''g''(''x''))

Integration: ∫_-a^∞ (x sin(x))dx

Wikified text: ∫_-''a''^∞ (''x'' sin(''x''))d''x''

Summation: ∑_i=1ⁿ (i! + i)

Wikified text: ∑_''i''=1^''n'' (''i''! + ''i'')

Indexing an operation over a set: ∩_{i in A} (f(φ_i))

Wikified text: <big>∩</big>_{''i'' in ''A''} (''f''(φ_''i''))

(This example also shows how to get a large operator that has no character entity of its own.)

Quantification without symbols: X = {x in Y : there exists a in A such that, for all b in B, b^-1xb = a}

Wikified text: ''X'' = {''x'' in ''Y'' : there exists ''a'' in ''A'' such that, for all ''b'' in ''B'', ''b''^-1''xb'' = ''a''}

Some useful links and resources

Diagrams are often a great help in explaining mathematical concepts; User:Chas_zzz_brown (amongst others) would be happy to create them (given time, ability, etc.).

The article List of mathematical topics is used by contributors to keep track of changes to the entire content of mathematics in Wikipedia, in a fashion similar to the more general "Recent Changes" link. If you add new articles which are remotely related to mathematics (including biographies of mathematicians, and so on), please add them to that list, so that everyone can review / add to / mercilessly savage your contributions.

The list of topics is also a useful place to check to see what other material on Wikipedia already exists that you can use to link with your material. This helps reduce the effort of defining terms and proving statements; and can help reduce the duplication of definitions and proofs.

Other lists of topics for subdisciplines:

• List of group theory topics
• List of mathematicians

Participants in this WikiProject

• Chas_zzz_brown. My knowledge of topics outside of group theory is a monotonically decreasing function of their relationship to abstract algebra.
• AxelBoldt
• Toby Bartels