Abuse of notation (nonfiction): Difference between revisions
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In [[Mathematics (nonfiction)|mathematics]], '''abuse of notation''' occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). | In [[Mathematics (nonfiction)|mathematics]], '''abuse of notation''' occurs when an author uses a [[Mathematical notation (nonfiction)|mathematical notation]] in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct [[Logical intuition (nonfiction)|logical intuition]] (while possibly minimizing errors and confusion at the same time). | ||
However, since the concept of formal/syntactical correctness depends on both time and context, many notations in mathematics that are flagged as abuse of notation in some context could be formally correct in other contexts. An instance where abuses of notation is strongly time-dependent occurs when notations are introduced to a theory a long time before the theory is first formalized. In which case, many of such abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration). | However, since the concept of formal/syntactical correctness depends on both time and context, many notations in mathematics that are flagged as abuse of notation in some context could be formally correct in other contexts. An instance where abuses of notation is strongly time-dependent occurs when notations are introduced to a theory a long time before the theory is first formalized. In which case, many of such abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration). | ||
A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but canonically isomorphic. | A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the word ''[[Group representation (nonfiction)|representation]]'' properly designates a [[Group homomorphism (nonfiction)|group homomorphism]] from a [[Group theory (nonfiction)|group]] G to [[General linear group (nonfiction)|GL(V)]], where V is a [[Vector space (nonfiction)|vector space]], it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but [[Isomorphism (nonfiction)|canonically isomorphic]]. | ||
Other examples include identifying a constant function with its value, and identifying a group with a binary operation with the name of its underlying set. | Other examples include identifying a [[Constant function (nonfiction)|constant function]] with its value, and identifying a group with a binary operation with the name of its underlying set. | ||
== Subjectivity == | |||
The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: A → B" for a [[Partial function (nonfiction)|partial function]] from A to B is almost always an abuse of notation, but not in a [[Category (mathematics) (nonfiction)|category theoretic]] context, where f can be seen as a [[Morphism (nonfiction)|morphism]] in the category of sets and partial functions. | |||
== In the News == | == In the News == | ||
Revision as of 03:59, 7 April 2020
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct logical intuition (while possibly minimizing errors and confusion at the same time).
However, since the concept of formal/syntactical correctness depends on both time and context, many notations in mathematics that are flagged as abuse of notation in some context could be formally correct in other contexts. An instance where abuses of notation is strongly time-dependent occurs when notations are introduced to a theory a long time before the theory is first formalized. In which case, many of such abuses of notation may be made formally correct by improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration).
A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but canonically isomorphic.
Other examples include identifying a constant function with its value, and identifying a group with a binary operation with the name of its underlying set.
Subjectivity
The terms "abuse of language" and "abuse of notation" depend on context. Writing "f: A → B" for a partial function from A to B is almost always an abuse of notation, but not in a category theoretic context, where f can be seen as a morphism in the category of sets and partial functions.
In the News
Fiction cross-reference
- Audible (mathematics)
- Crimes against mathematical constants
- Gnomon algorithm
- Gnomon Chronicles
- Mathematician
Nonfiction cross-reference
External links
- Abuse of notation @ Wikipedia