Wikipedia:WikiProject Mathematics/Proofs

This is an essay.

It contains the advice or opinions of one or more Wikipedia contributors. This page is not an encyclopedia article, nor is it one of Wikipedia's policies or guidelines, as it has not been thoroughly vetted by the community. Some essays represent widespread norms; others only represent minority viewpoints.

This page (currently under development) is to serve a summary of past discussions on proofs in Wikipedia. It describes the opinions of some of the editors in the WikiProject Mathematics community on our best practices for articles that include, or are about, mathematical proofs. For discussion, see Wikipedia talk:WikiProject Mathematics/Proofs.

In this page, the word "Proof" refers to the garden-variety proof of the working pure mathematician. There are separate issues related to the inclusion of "derivations" and other heuristics.

Proofs within articles

Proofs are often discussed in Wikipedia's mathematics articles, just as axioms, definitions, theorems, and lemmas are. Because much of the published professional literature of mathematics consists of the details of proofs, it would be very difficult to write in any depth about mathematics without including at least some proofs or proof sketches. It does not follow, however, that the bulk of coverage of mathematics on Wikipedia should consist of detailed proofs in the style professional mathematicians use.

Wikipedia articles are not intended to replace the textbooks and advanced monographs that a serious student must use to acquire a detailed understanding of topics. It can be helpful to think of Wikipedia's mathematics articles as akin to survey articles in mathematics journals. Our articles contain a mixture of introductory information, context, definitions, results, pictures, and some discussion of proofs. These articles generally serve their purpose best when they provide a brief, but significant, survey of a topic, along with pointers to the literature. At the same time, our best articles provide a useful reference for readers familiar with a field who wish to look up particular facts. The role of proofs, which may be short but correct arguments or sketches of longer arguments serving more as a map of complete proofs, is to support the "survey" and "reference" ambitions.

Unlike textbooks, Wikipedia does not strive to provide an axiomatic introduction to mathematics. Thus it is not necessary for a Wikipedia article to prove every fact that is mentioned. Our best articles include references to good textbooks that the interested reader can consult for an axiomatic presentation of the field. On the other hand, proofs or proof sketches of a few selected facts can make the article more useful as a reference.

If a proof is not significant enough to place in its own article, but long enough that it may interrupt the flow of the article it occurs in, it may be set off in a collapsed frame, which can be opened if needed. One way to do that is with the following code:

{{Collapse top|title=Proof of ...}}
The proof of ... proceeds as follows:
...
{{Collapse bottom}}

This generates the following:

Proof of ...
The proof of ... proceeds as follows: ...

The use of collapsible frames violates MOS:COLLAPSE. It is likely a better style to instead use explanatory footnotes for short or abbreviated proofs, and to omit long proofs by replacing them with a citation to a good presentation of the proof. For an example of this style of presenting proofs, see Countable set.

Proofs as topics

There are no firm guidelines for when a proof may be given a dedicated article of its own, such as "Proofs of quadratic reciprocity". It is widely accepted, however, that if a proof is made a topic of its own dedicated Wikipedia article, the proof must be significant as a proof, not merely "routine". In particular, WP:NOTTEXTBOOK applies. If the proof can be found verbatim in any standard textbook on a subject, then it is better off transwikied to a project such as Wikibooks, Wikiversity, or ProofWiki. There is also no point in presenting a novel proof not appearing before in print, because it falls afoul of WP:OR. These necessary conditions have not been very controversial. More controversial is what is sufficient for inclusion. WP:GNG is an obvious sufficiency criteria, but as established in various WP:AfD debates, proofs may be significant because the result is "surprising" (Wikipedia:Articles for deletion/Proof that 0.999... equals 1) or prove the property in a non-obvious way (Proof that 22/7 exceeds π).

Having stated that there is a "significant proof" criterion that allows inclusion under conditions that WP:GNG does not, it is a little hard to explain precisely what a "significant proof" is. One measurement might be the interest a general Wikipedia reader might have in such an proof. We can imagine several facets that may be brought up in an AfD debate:

The result (it may be celebrated, or surprising)
Features of the proof that are external (remarkably long or short; or machine-assisted)
Internal or technical features of the proof that have implications (elementary in the technical sense, constructive or non-constructive where there is some point to knowing that)
Mathematical or conceptual features (unobvious in the sense of neat or elegant or unexpected, or contains an idea interesting in itself)
Social or historical features (a proof that has generated controversy, or had influence).

This is not an exhaustive list, but it should be clear that, given sufficient evidence for a proof being significant, it is natural that Wikipedia should contain information about that proof.

However, even if the proof is significant, there may be little to say about the proof. The proof most likely qualifies as a sub-topic of a relevant article on the theorem or subject of the theorem. Therefore, under Wikipedia:Summary style, it is only if the article is quite long that it is worth splitting out the proof on a separate page. Specifically the criteria in WP:SIZESPLIT applies.

Article names

A common suggestion is that articles with a proof as a topic have names ending in '/Proof' or '/Proofs'. This practice has been determined to be a violation of WP:SP, specifically the prohibition on using subpages for permanent encyclopedia content. Currently, many articles of this type begin with the words 'Proof of ...', 'Proof that ...', or 'Proofs involving ...' as in Proof of Bertrand's postulate. When more specificity is desired, a name connected with the proof may be added as in Furstenberg's proof of the infinitude of primes.

Style

Proofs in Wikipedia should conform as much as possible to the style guidelines given in Wikipedia:Manual of Style (mathematics). In particular, the proof should be presented in conversational prose, as if being given in a lecture at a blackboard. Unnecessary use of symbols (such as ∀ and ∃) and jargon should be avoided. Avoid phrases such as "Clearly..." and "Obviously...", though it may be helpful to replace lengthy calculations with a summary. Avoid putting Q.E.D. at the end of a proof since many readers will need to look up the meaning. Instead, the end of the proof should be marked by a section heading or the end of the article itself.

Use of the imperative

Proofs normally use the imperative heavily, for example "Let ABC be a triangle…." This is standard practice in mathematical writing and should not be confused with instruction manual material.

Formalized proofs

As a matter of practice, Wikipedia articles do not typically include fully formalized proofs as would be generated by Metamath, apart from articles that specifically deal with formal proofs and formal deduction systems. The same is true in the mathematics literature; few journals publish fully formalized proofs.

Examples

Many articles containing proofs are in Category:Articles containing proofs.

The following articles demonstrate widely accepted ways of including and writing about proofs.

Isosceles triangle theorem (pons asinorum) is a result in geometry made notable by the difficulty generations of students had in mastering the proof.
Difference of two squares uses a walk-through of a proof to explain the scope of a mathematical identity.
Proofs of quadratic reciprocity is a short survey addressing the several hundred known proofs.
Kleene's recursion theorem is an article about a particular theorem, whose proof is short but requires a technical trick. The article includes a description of this proof, as a reference for the reader.

The following versions of articles include or included proofs in ways that are less widely accepted.

Parabola/Proofs contains a lengthy derivation of an elementary fact. Moreover, the proof is presented in a "two-column" style instead of prose.
Linear/Proof includes a lengthy proof of relatively elementary fact: If a function f on a vector space over the rationals is additive (meaning f(a+b) = f(a) + f(b) for all a, b in the space) then f is homogeneous (that is, f(qa) = qf(a) for all a in the space and every rational q).

Wikibooks and other projects

Wikibooks is a project that lets editors collaborate on writing freely-licensed textbooks. Like Wikipedia, Wikibooks is run by the Wikimedia Foundation. Wikibooks is one option for editors who would like to work on textbook-style treatments of mathematical topics. In addition to many books on specific subjects, the book Famous Theorems of Mathematics has a collection of theorems in many different areas. Occasionally, it is suggested that proofs that are removed from Wikipedia articles might be good source material for Wikibooks, although it is not obvious how to accomplish this easily.