Examples of the Mathematical Red Herring principle

Question

I read the Mathematical Red Herring principle the other day on SE and wondered what some other good examples of this are? Also anyone know who came up with this term?

The mathematical red herring principle is the principle that in mathematics, a “red herring” need not, in general, be either red or a herring.

Frequently, in fact, it is conversely true that all herrings are red herrings. This often leads to mathematicians speaking of “non-red herrings,” and sometimes even to a redefinition of “herring” to include both the red and non-red versions.

The only one I could think of a manifold with boundary which is not a manifold in the usual definition.

Answer 1

All differential equations are stochastic differential equations,

but most stochastic differential equations are not differential equations.

Answer 2

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.

Answer 3

The Division Algorithm is not an algorithm, it's a theorem.

Answer 4

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.

Answer 5

Russell's Paradox and the Banach-Tarski Paradox are not paradoxes,they are theorems. Russell showed that the assumption of the existence of a set with certain properties leads to a contradiction, hence no such set exists. Banach-Tarski is a highly counter-intuitive property of 3-D Cartesian space, which may seem to contradict Lebesgue measure theory, but it uses non-measurable sets. Anyone have some more "paradoxes"?..... Russell's Paradox : If X is a widget which dapples every widget that does not dapple itself, and does not dapple any widget that dapples itself, then X dapples itself if and only if it doesn't.