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There are a couple of terminologies that I find mildly confusing:

  • Using the definition of graph on wikipedia, a graph is defined as having a finite number of vertices and edges. It states that there is the concept of an infinite graph, but the term graph does not include them by default.
  • Probably an even clearer is a non-deterministic Turing machine. Any time a Turing machine is defined, the definition is that of a deterministic Turing machine. A non-deterministic Turing machine is not actually a Turing machine under this definition.
  • A Non-associative ring is not necessarily a ring or even non-associative.
  • Traditionally, a plane is defined as being Euclidean. That is, if someone says in a traditional geometry textbook "let P be a plane", it is implied that it is Euclidean. When talking about non-Euclidean geometry, however, it can refer to either a euclidean or a hyperbolic plane, or even some other things, depending on context.
  • The field with one element. It is neither a field nor does have one element. It is not even a set, necessarily.
  • The most extreme example, at least to me, is the term non-standard as used in model theory. Though typically used when describing models, it can also be used to describe the elements of that model. In fact, for almost any mathematical concept, you can create a non-standard version of it by just applying that concept's definition within the model. For example, you can have nonstandard graphs, Turing machines, rings, planes, and fields. You can even have nonstandard versions of unique objects, like the real numbers or the free group with two generators. They will in general, however, not actual be an example of what term they are based on. A non-standard turing machine, for example, need not be a turing machine. Non-standard objects are perfectly well-defined objects, of course; they just aren't actually what their name seems to say they are.

Now, I am not arguing against such definitions. I know that once you get used to them, they can be quite convenient, and that they usually do not cause problems. After all, it would not make sense to say that something needs to fail associativity; we just sometimes do not mandate it. My question is what such a definition is called?

Christopher King
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    Here is a reverse example: an which is not . An orthogonal projection matrix, which is a projection matrix but (except for trivial examples) is not orthogonal. – Misha Lavrov Feb 05 '19 at 04:40
  • @MishaLavrov Yeah, those definitions pop up sometimes as well. – Christopher King Feb 05 '19 at 04:41
  • A one-parameter subgroup isn't a subgroup, or even a group. https://en.wikipedia.org/wiki/One-parameter_group – Travis Willse Feb 05 '19 at 04:42
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    These might be considered "abuses of terminology". – Blue Feb 05 '19 at 04:47
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    It's not like these things only occur in mathematics. Consider "forged Picasso", "imitation crab", "fake fur", "honorary citizen", ... –  Feb 05 '19 at 04:49
  • @Rahul Hmm, that's true. And wonder what the general term for the phenomenon is, then? (I do think with mathematics it is a little different, since (1) mathematicians are usually much more literal with their language than in any other discipline and (2) sometimes in math either the noun, adjective, or both can be incorrect.) – Christopher King Feb 05 '19 at 04:51
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    A "generalized what-have-you" is an expansion of the definition of a what-have-you. One could see things like "infinite graph" as including an implicit "generalized". – Blue Feb 05 '19 at 04:54
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    @Blue I guess you could say the adjective "generalized" is the "mother" of all these adjectives. Its also another example of where the adjective is not literal, since even though rado's graph could be said to be a generalized graph, it would not be correct to say that rado's graph is "generalized", whatever that would mean. – Christopher King Feb 05 '19 at 04:57
  • I don't think the graph example fits. Nobody would claim that an infinite graph is not a graph just because they prefer to only work with finite graphs (and to drop the qualifier "finite" when it is understood). – Misha Lavrov Feb 05 '19 at 04:59
  • @MishaLavrov Wikipedia would, at least. Well, I guess no one would say "an infinite graph is not a graph" explicitly, but they may implicitly assume it during a proof, which they are justified in doing if they defined graphs that way. For example, they may say "all graphs have property $P$", when in actuality only finite graphs have $P$. – Christopher King Feb 05 '19 at 05:09
  • @MishaLavrov In fact, the graph example is what sparked this question. See this comment. – Christopher King Feb 05 '19 at 05:10
  • A free abelian group is usually not a free group. Different thing, but kind of similar. – Zeno Rogue Feb 05 '19 at 14:01
  • @ZenoRogue yeah, the idea that free has different meanings in different contexts – Christopher King Feb 05 '19 at 18:14
  • In a different field of science, a dwarf planet is not a planet. – Michel Fioc Oct 17 '24 at 07:58

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I don't know if there is a more standard term for this phenomenon, but these are examples of what the nCatLab calls the red herring principle. Quoting from the article:

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.

See also this Math SE question for additional examples.

Community
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pregunton
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    I had a professor (in my French-speaking university) who called it the "pink elephant" principle, with the same reasoning. – Arnaud D. Feb 08 '19 at 13:16