Examples of the Mathematical Red Herring principle
Solution 1:
All differential equations are stochastic differential equations,
but most stochastic differential equations are not differential equations.
Solution 2:
My understanding of this principle is that sometimes, adjectives widen the scope of nouns (or modify their scope in other, more complicated ways) and this can be confusing. Examples:
- partial functions aren't necessarily functions
- non-unital rings aren't necessarily rings
- non-associative algebras aren't necessarily algebras (under my preferred definition)
Another funny one is:
- a partially ordered set isn't necessarily an ordered set.
In this case, an adverb (partially) is widening the scope of an adjective (ordered).
There's a related phenomenon whereby we give a black-box meaning to phrases of the form [adjective]-[noun], and that meaning isn't a compound of the meanings of these two words individually. E.g.
- Topological spaces aren't "spaces" because the term "space" lacks a technical meaning
- Lawvere theories aren't "theories" because the term "theory" lacks a technical meaning
- etc.
Solution 3:
A "set of measure zero" is often defined without saying what measure is used, or what value it takes on the set. Thus, neither the "measure" nor the "zero" are defined/true on their own.
Solution 4:
The Division Algorithm is not an algorithm, it's a theorem.