Precise definitions are vital in high-level mathematics. We need precision so that we can all agree on what we are talking about. Nonetheless, owing to personal preferences, definitions may vary slightly from mathematician to mathematician so it pays to be vigilant.

The reason for giving a mathematical object a particular name is often lost in the mists of time. Frequently names come from ordinary English words but the mathematical meaning may be different to that in everyday speech and give no clue to the definition. For example, the major objects of interest in algebra are groups, fields and rings! Names for objects can be humorous, such as greedoids, or can be derived from a person, such as Gorenstein rings. In the latter case there is no guarantee that the person had anything to do with it – Daniel Gorenstein claimed that he did not know the definition of a Gorenstein ring, despite it being named after him.

What is a definition?

A mathematical definition gives the meaning of a word (or phrase) in a specific way. The word (or phrase) is generally defined in terms of properties.