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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.

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All differential equations are stochastic differential equations,

but most stochastic differential equations are not differential equations.

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  • $\begingroup$ Nor must they be stochastic. $\endgroup$ – PyRulez Feb 8 '19 at 18:24
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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.
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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.

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

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  • $\begingroup$ the only herrings i've ever seen were blue & white $\endgroup$ – DanielWainfleet Aug 18 '15 at 5:22
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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.

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  • $\begingroup$ The Euler-Cramer paradox is also not a paradox,but it was a good question. $\endgroup$ – DanielWainfleet Aug 18 '15 at 5:24
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For almost any mathematical noun $N$, a nonstandard $N$ by definition can not be a $N$.

That is, for any model $M$, we say that something is a $N$ within the model if satisfies $N$'s definition within the model. For example, a non-standard Turing machine is an element of $M$ that satisfies this definition, but using the models interpretation of it instead of the standard one. This only makes sense if $M$ can interpret $N$'s definition, of course. The main things that could be interpreted differently is what terms set, finite, and transition function mean. For example, a model could interpret finite to mean being "a thing whose size is a hyperinteger" instead of "a thing whose size is an integer". (Of course, we don't want to change too much, or we can not apply meta-mathematical techniques as easily. For example a model that defines finite as "contains a field" would not be a good model. Technically, we would call it a structure, not a model, at that point.)

Anyways, for most theories this something called the "standard model", or the intended interpretation. For example, the standard model of peano arithmetic is $\mathbb N$. Usually if we talk about $N$, without specifying what model we are working in, we assume we are talking about a $N$ in the standard model. There is no standard model of group theory, however, because the axioms of group theory do not have an intended interpretation. Each group has its own interpretation, and none of these are more intended than any other. With set theory, it gets kind of ambiguous whether or not there is a standard model (well, there definitely is not one in the traditional since, since the elements of a model are contained in a set, and there is no set of all sets). Models can also be submodels of other models.

So, what is a nonstandard $N$? For a theory $T$ with a standard model and a model $M$ of $T$ with the standard model as a submodel, a nonstandard $N$ in $M$ is an element of $M$ that satisfies $N$'s definition inside $M$, but is not a $N$ in the standard model. Since talking about $N$ unqualified usually means $N$ in the standard model, we can say that nonstandard $N$ are not $N$, for almost any mathematical noun $N$.


Of course, some of you may be asking "why would you want something like this"? To prevent you from offending model theorists, I'll answer it prematurely, with some examples.

There are nonstandard real numbers that are between $0$ and every positive real number (in certain models). Moreover, this can be done in a model that satisfies the same first order statements as the standard model (in fact, first order formulas even have the same standard solutions). Since theorems are statements, and some are first order, this means we already know a ton of stuff about the nonstandard real numbers. This lets us do Calculus in terms on infinitesimals instead of in terms of limits. This is called nonstandard analysis. Anything true in analysis is true in nonstandard analysis, and anything false is analysis is false in nonstandard analysis, so this is just an extension of regular analysis, and is therefore compatible with it. Although somethings are defined differently, they end up being equivalent. (For example, instead of the epsilon delta definition of a limit, an equivalent one is given in terms of infinitesimals.) There is even two entire text books for introductory calculus courses using nonstandard analysis instead of standard analysis.

You can also use nonstandard models in graph theory. Any nonstandard graph can be turned into a graph, but a finite nonstandard graph might get turned into an infinite graph. In fact, there's a nonstandard model of set theory in which every standard graph (infinite or otherwise) is a subgraph of a finite nonstandard graph. Therefore, you can show that the four color theorem on finite graphs in the nonstandard model implies the four color theorem on all graphs in the standard model.

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