nLab red herring principle

Idea

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

Given an adjective “foo”, and a term “bar”, the term “foo bar” is an example of the red herring principle if and only if at least one object referred to by the term “foo bar” fails to be both foo and a bar.

Notes

“Red herring”, as used here, is to be interpreted neutrally: it refers to a name of a concept which might throw the reader off-track, by accident as it were. This should not to be conflated with its usual usage, which typically refers to a deliberate attempt to divert or throw one off track (for instance, as a rhetorical tactic). One should also avoid taking it as perjorative (indicating, for example, ineptitude or lack-of-care in naming). It may be that some term “bar” once meant one thing, but – over time – had its meaning changed. In that case, there may be terms of the form “foo bar” that seem strange according to bar’s modern definition, but made much better sense under the older (but mostly forgotten) meaning of “bar”.

When examples of the red herring principle appear, it is usually due to a deficiency in existing terminology. Most commonly, one finds that “herring” is defined in such a way that all herrings are red herrings. If the associated near-herrings (objects that would be herrings, if only they were red) are interesting in their own right, this can lead to mathematicians speaking about the “non-red herrings”, despite the fact that – strictly speaking – no such things exist. When this happens, it is usually an indication that the definition of “herring” should be revised to include both red and non-red herrings.

Examples

• A manifold with boundary is not a manifold. This leads to the use of “manifold without boundary”.

• In linear algebra, an algebra is usually defined to be associative, so a nonassociative algebra is not an algebra in this sense. This leads to the technically redundant use of “associative algebra”. Furthermore, a “nonassociative algebra” might happen to be associative; it just doesn't have to be!

• Similarly, noncommutative geometry is really about not necessarily commutative or possibly non-commutative geometry. In fact, many tools developed in “non-commutative geometry” are also useful tools in ordinary commutative geometry.

• A *-autonomous category is not an autonomous category, but the reverse is almost true: a symmetric autonomous (a.k.a. compact closed) category is a fairly special case of a $*$-autonomous one. This red herring is apparently an accident of history: at the time $*$-autonomous categories were invented, “autonomous category” was sometimes used to mean a closed monoidal category, of which $*$-autonomous categories are indeed a special case, but nowadays that usage has mostly disappeared.

• A linearly distributive category has essentially nothing to do with a distributive category.

• If a localizer refers to a class of maps in a presheaf category, as it sometimes does, then a basic localizer, being a class of maps in Cat, is not a localizer.

• A multivalued function is not a function, but a function is a special case of a multivalued function.

• A nondeterministic automaton may be deterministic; it merely happens to lack nondeterministic transitions. In terms of the previous example, a nondeterministic automaton has a multivalued transition function, while a deterministic automaton has a transition function.

• A planar ternary ring? is not a ring. A general ring is not a planar ternary ring either; their “intersection” is the class of division rings.

Some adjectives are almost universally used as “red herring adjectives,” i.e. placing that adjective in front of something makes it more general in some way. Some red herring adjectives almost always have the same meaning, such as “pseudo” and “lax”, but others, such as “weak”, have different meanings in different contexts.

• A weak factorization system is not a factorization system, as originally defined, but rather the reverse. This has led some people to the use of orthogonal factorization system in place of the classical term “factorization system”.

• A weak limit is not a limit in a similar way: it satisfies only existence but not uniqueness.

• Originally, n-category referred only to the strict version, so that a “weak $n$-category” was not an $n$-category. Nowadays some people (including many authors of the nLab) are trying to reverse this, so that $n$-category means the weak version, while the strict version needs an adjective added. Likewise for related concepts such as 2-limit and strict 2-limit.

Sometimes a red herring is only a red herring in degenerate cases, perhaps only one case:

• Fixing a natural number $n$, a homogeneous polynomial of degree $n$ is almost always a polynomial of degree $n$, but there is one exception: the zero polynomial has no degree (or is degree $-1$ or degree $-\infty$, depending on conventions), yet it is homogeneous of any degree.

Non-examples

• The term “red herring principle” is a non-example of the red herring principle. Following the disclaimer in this page's notes section, the adjective “red herring”, when used to describe this principle, will be misleading if taken in the usual sense. Of course, this makes the usage of “red herring” in “red herring principle” a red herring. The red herring principle is thus both a red herring and a principle, and so the term “red herring principle”, used to describe the red herring principle, is not an example of the red herring principle.

Some uses of terminology are similar in some ways, but don’t quite fall under the same category. For instance, in a number of cases mathematicians working in a particular field tend to omit niceness adjectives, e.g.:

• The generally accepted definition of a ring does not include commutativity, but in algebraic geometry and commutative algebra it is an almost universal convention that all rings are taken to be commutative. However, almost every book concerning the subject announces its conventions early on.

• Algebraic topologists usually work with a convenient category of topological spaces rather than the category of all topological spaces, but usually they simply say “space” to mean an object of their chosen convenient category. Some algebraic topologists (particularly of the MIT school) take this even further and say “space” to mean simplicial set. In the Chicago school, this is occasionally stretched even further, with “space” meaning spectrum (although this is arguably more of a joke than a common usage).

These terminological uses can create situations that appear similar to actual red herrings, such as the use of “noncommutative ring” by people who are familiar with using “ring” to mean “commutative ring”. However, since it is generally accepted that the use of such abbreviations does not change the actual definitions of terms like “ring” and “topological space”, these are not true red herrings.

Semi-non-examples

Sometimes, adding an adjective does not destroy the property that the noun has, but the noun now has its property in a different way, e.g.:

• A contravariant functor from $C$ to $D$ is still a functor, however not from $C$ to $D$, but from $C^{op}$ to $D$ (or equivalently from $C$ to $D^{op}$).

References

Early discussion making the terminological issue explicit:

• Mark Dominus: Nonstandard adjectives in mathematics, blog entry (Jan 2008)