Wikipedia:WikiProject Mathematics/Proofs

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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[edit]

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:

Proofs as topics[edit]

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". It is a little harder to explain what that means, but the criteria for inclusion seem to involve some necessary conditions that are not very controversial. Sufficient conditions are less easy to formulate, but clearly enough they describe what interest a reader might find in such an article.

Work is needed in this area to get a better understanding

  1. of what the necessary conditions should be (e.g. exclusion of "original research" in the sense of offerings of novel proof ideas here);
  2. how the sufficient conditions relate to content policy in general, and how they should be formulated in such a way as to encourage authors of articles interesting to the general Wikipedia reader.

Under (2) there are several facets. What we can say about a proof may relate:

  1. to the result (it may be celebrated, or surprising);
  2. to features of the proof that are external (remarkably long or short; or machine-assisted);
  3. to 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);
  4. to mathematical or conceptual features (unobvious in the sense of neat or elegant or unexpected, or contains an idea interesting in itself);
  5. to social or historical features (a proof that has generated controversy, or had influence).

This is not an exhaustive classification, however.

Under (i) it is natural to say that Wikipedia's article on the result should contain information about the proof, but it is not automatic (under Wikipedia:Summary style) that the proof deserves its own article.

Article names[edit]

Some of the existing 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#Articles do not have sub-pages (main namespace). 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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.

Relevant policies and guidelines[edit]

See also[edit]