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!

Table of contents 1 Title 2 Scope 3 Participants 4 Some issues to think about 5 Proofs 5.1 Examples 6 Suggested structure of a mathematics article 7 Typesetting of mathematical formulas 7.1 Italicization and bolding 7.2 Using special symbols 8 Other special formatting 9 Some useful links and resources

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, operations research, 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/TeX.

Participants

• 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
• Ram-Man - If I can ever find the time to put my education to use!
• Jeff - My background: dynamical systems, real analysis, and complex systems. I love to edit.
• Mark Dominus
• Pierre Abbat
• LittleDan I know up through geometry, and a fair amount of group theory and vector spaces[?]. I can usually pick things up from wikipedia articles, if not from mathworld[?], then I can edit wiki articles for clarity.
• User:Charles Matthews I've added a couple of dozen pages recently, so I suppose I should try to get to grips with the issues mentioned here.

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 described below is one way of achieving this.

When you need to describe a concept in terms of some other concept (for example, explaining rational numbers in terms of integers), be sure to:

• Add a (prominent) link to the relevant article (in this case, integer). As for other Wikipedia articles, avoid duplicate links.
• If it makes sense, add a very quick (and naive) explanation/description (in this case, "positive or negative whole numbers" might work).

If the relevant article has not been written yet, then create a good stub, and 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).

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.

Examples

As with proofs, examples should be included where appropriate. Most of the time, they should be put on a separate section or separate page.

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:

• Optionally, for articles which beginners might be expected to find difficult, a header line, stating something on the lines of this:

  If you are having difficulty understanding this article, you might want to learn more about functions and naive set theory first

  followed by a horizontal rule to separate it from the rest of the article.
• An introductory paragraph (or two), including the field(s) of mathematics this concept belongs to, the article title in bold, which describes the subject in general terms, and giving the historical motivation and mathematical context in which the term appears, giving names and dates; for example, like the following:

  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.

• 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.
• Optionally, an informal introduction to the topic, without rigour, suitable for a school student or first-year undergraduate, as appropriate. This should 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 your readers may be non-mathematical scientists.
• 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.
• 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.
• 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.

Typesetting of mathematical formulas

Wikipedia allows to typeset mathematical formulas in (a subset of) LaTeX markup; the formula is normally translated into a PNG image which is included into the text. For the mechanics of this, see Wikipedia:TeX markup.

The LaTex formulas work inline (like this: <math>\mathbf{x}\in\mathbb{R}^2</math>) as well as displayed on their own line:

<math>\int_0^\infty e^{-x^2}\,dx</math>

The former inline method is generally discouraged, for several reasons:

• the font size is somewhat larger than normal, making text containing inline formulas hard to read
• the download speed of a page is negatively affected if it contains many images
• HTML (as described below) is adequate for most simple inline formulas and better for text-only browsers.

When displaying formulas on their own line, we indent the line with one or more colons (:); the above was typeset as

:<math>\int_0^\infty e^{-x^2}\,dx</math>

If you find an article which indents lines with spaces in order to achieve some formula layout effect, you should convert the formula to LaTeX markup.

If you plan on editing LaTeX formulas, it is helpful if you leave your preference settings (link in the upper right corner of this page, underneath your user name) in the "rendering math" section at the default "HTML if very simple or else PNG"; that way, you'll see the page like most users will see it.

The following sections cover the way of presenting simple inline formulas in HTML:

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. Some prefer using the HTML "variable" tag, <var>, since it provides semantic meaning to the text contained within. Which method you choose is entirely up to you, but in order to keep with convention, we recommend the double-apostrophe emphasis method. Thus we write:

''x'' = (''y''<sup>2</sup> + 2)

which results in:

x = (y² + 2)

Note that the parentheses, equals and plus signs are not emphasized; try to keep them outside of the double-quoted sections.

Fixed names for functions, such as sin and cos, are not emphasized, but we emphasize f when we define the function by f(x) = sin(x) cos(x).

Sets are usually written in upper case, and emphasized; for example:

A = {x : x > 0}

would be written:

''A'' = {''x'' : ''x'' > 0}

Greek letters should not be emphasized; for example, as in &lambda; + ''y'' = &pi;''r''<sup>2</sup>, for the expression "λ + y = πr²".

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 (''') rather than 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".

Other special formatting

In many math textbooks, formalized statements, axioms, or theorems are often set apart in a box, perhaps with a colored background. Some Wikipedia editors have experimented with similar formatting in mathematical articles. For example, the Continuum hypothesis, when stated formally, could be formatted using the following Wikipedia code:

<blockquote style="padding: 1em; border: 2px dotted purple;">
There is no set whose size is strictly between that of the integers and that of the real numbers.
</blockquote>

The above is rendered like this in your browser:

There is no set whose size is strictly between that of the integers and that of the real numbers.

Graphical browsers which support Cascading style sheets should render the above in an indented box, surrounded by a purple dotted line. This formatting technique helps set apart the most important statements in mathematical articles. Again, whether you do this is entirely up to you; it's just something a few of Wikipedia's editors have found to be helpful in making math articles clearer.

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

See also Wikipedia talk:Naming conventions (theorems)