# Help:Multiple Definitions

## Motivation

Different sources may give different Definitions for the same thing.

Because $\mathsf{Pr} \infty \mathsf{fWiki}$ aims to be accessible to a large public, equivalent definitions are treated equally.

## Introduction

Instead of choosing only one definition as the "main" definition and taking the other definitions as theorems, on $\mathsf{Pr} \infty \mathsf{fWiki}$ equivalent definitions are bundled on the same definition page.

Their equivalence is proved on a separate page.

This has multiple advantages:

- Every student feels at home at $\mathsf{Pr} \infty \mathsf{fWiki}$, regardless of the definition their textbook uses.
- Seeing different definitions in one glance, increases the chance for a visitor to understand the concept being defined.
- Different viewpoints on the same concept makes $\mathsf{Pr} \infty \mathsf{fWiki}$ more coherent, without making the pages appear too encyclopaedic.
- The easy-to-verify versus usefulness trade-off: The ideal definition is easy to verify in practice, yet strong enough to deduce many interesting properties.

- These two are often incompatible, and multiple definitions allow to profit from both advantages.

## Page Structure

The structure of page with multiple equivalent definitions is roughly as follows:

== Definition == Let ... ... Let ... === [[Definition:Definiend/Definition 1|Definition 1]] === {{:Definition:Definiend/Definition 1}} === [[Definition:Definiend/Definition 2|Definition 2]] === {{:Definition:Definiend/Definition 2}} ... == Also see == * [[Equivalence of Definitions of Definiend]] {{Link-to-category|Definiends|definiends}} [[Category:Definitions/Definiends]] etc.

As with multiple proofs, the introductory part is copied identically on the transcluded subpages, but not inside the `<onlyinclude> ... </onlyinclude>`

tags.

A link to a proof showing their equivalence (see below) is placed in the Also see section, preferably as the first item in the list, so as to make it easier to find the link.

Pages that severly deviate from this structure have to be refactored accordingly.

### Sources

A source backing up a definition is only placed at the corresponding transcluded definition, not on the main definition page.

## Equivalence Proofs

The equivalence of the definitions has to be proved on a separate page.

More instructions at Help:Definition Equivalences.

## When and when not to

### Equivalent Characterizations

One should not overdo the use of multiple definitions.

Especially if there are many equivalent characterizations which are otherwise not likely to contribute to the comprehensibility of the concept, it is better to set up a page that bundles all equivalent characterizations.

You may also want to enhance the corresponding Also see section or Also defined as section.

As a general rule, when a definition can be found in a textbook, it is considered appropriate to add it as an alternative definition rather than a theorem.

If the equivalence proof requires some theory to be built on one of the definitions, it is often a better idea to treat them as characterizations, not equivalent definitions.

Consider that, strictly speaking, one cannot prove theorems about something before the definition is settled.

But there is no general rule, and such difficult cases have to be considered case by case.

### Related definitions

While unrelated definitions with the same name have to be **disambiguated**, very closely related definitions bearing a same name can be treated in the same way as generalizations.

This article is incomplete.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Stub}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### Generalizations

It is not uncommon that a definition has one or more generalizations.

Naturally, this does not mean that only the most general definition should be given:

- $\mathsf{Pr} \infty \mathsf{fWiki}$ aims to be accessible to people with different background
- Nobody knows if an even more general definition will be invented some day.
- The way in which definitions generalize may be not totally ordered.

If one definition applies to a significantly more general setting, it should not be treated as just another equivalent definition.

Instead, the less and more general definitions are placed on separate pages.

Depending on the situation, there are roughly two ways to do this:

- $(1): \quad$
**Transclude**the pages on a common page.- They can either be transcluded as subpages, which is appropriate in most cases, or as independent pages.

- $(2): \quad$ Set up a
**disambiguation page**.- This becomes especially more convenient if there are several related definitions, which would otherwise compromise the conciseness of the page on which everything is transcluded.

### Additional Constraints

This article is incomplete.In particular: explain the policy about definitions that are only valid when a minor additional condition is addedYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Stub}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

### Nonequivalent definitions

*Also see Help:Also defined as*

This article, or a section of it, needs explaining.In particular: explain what to do with definitions that are equivalent in a more abstract sense, such as Tangent SpaceYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Explain}}` from the code. |