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Typical usage is with math, or philosophy, proofs. Also typically the simple example disproves the theory, but is of a arbitrarily contrived nature and not something that would naturally arise.

Is that clear enough?

A mathy example would be, in differential equations. If Brent said all functions are differentiable. The Professor X would say what about f(x)=x^2 for all x not equal to 3, when f(3) = -100. Then f(3) is a jump discontinuity and is not differentiable. This is an arbitrary, simple example disproving Brent's statement.

The example is ______

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  • 1
    Are you looking for the word counterexample?
    – Peter Shor
    Commented Jul 10, 2015 at 15:11
  • 1
    Trivial is close, but trivial is more passive than the word I can't think of. The elusive word connotes more intention in the example.
    – Brent
    Commented Jul 10, 2015 at 15:26
  • Oops I misspoke, No the word is not counterexample.
    – Brent
    Commented Jul 10, 2015 at 15:30
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    Have you ever heard of this en.wikipedia.org/wiki/Spherical_cow
    – maxwell
    Commented Jul 10, 2015 at 15:40
  • No hadn't heard of that! Funny and sadly true...
    – Brent
    Commented Jul 10, 2015 at 16:58

3 Answers 3

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The phrase you're thinking of is pathological example.

This is used in maths for an unnaturally contrived situation, often devised just to disprove some theory which usually does hold true.

For instance, pretty much any function that's continuous everywhere is also differentiable somewhere (and indeed almost everywhere). But that's not true as a theorem, without the "pretty much", because of functions like the blancmange function or Weierstrass's function which are continuous everywhere and differentiable nowhere.

Source: I'm a mathematician :-) Also, Wikipedia link.

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  • This is the closest to what I was thinking about. And admittedly it may be exactly what I was thinking and I may be mis-remembering the what I thought the word sounded like(yes my memory of the last situation I heard the word, in a Real Analysis class, had the word not sounding like "pathological"). And your notion of my meaning is spot on.
    – Brent
    Commented Jan 5, 2016 at 15:59
  • Out of curiosity and irrelevant to the question: what precise meaning could be given to "pretty much any" in this case, referring to continuous function which are differentiable somewhere? My naive expectation is that in fact, almost none of the continuous functions are differentiable, not anywhere, unless we limit the scope to "continuous functions which a human might come up with".
    – Jonathan Y.
    Commented Feb 1 at 14:00
  • @JonathanY. Yes, in a mathematical sense "almost all" continuous functions are not differentiable, but in practical examples, most continuous functions that we actually meet are differentiable. On the other hand, there are also mathematical senses in which "almost all" continuous functions are polynomials - see the Weierstrass approximation theorem. It all depends on how you define that "pretty much any" / "almost all" and what kind of mathematical structure you build around it.
    – Rand al'Thor
    Commented Feb 1 at 15:58
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I think you're talking about multiple possible things here.

I've heard the word 'toy example' or 'toy problem' (see Wikipedia and Google) to denote simple (or simplified) problems demonstrating the application of a theory or technique in the natural sciences.

Since proofs are supposed to cover all cases, not just simple ones, I'm not sure they're so relevant to proofs. I suppose in the very specific case of proofs by mathematical induction (https://en.wikipedia.org/wiki/Mathematical_induction), the 'base case' is something resembling what you originally describe.

A counterexample that decisively disproves something may be a refutation (e.g., 'That example serves as refutation of the theory.')

Simple examples are relevant to arguments, as opposed to merely proofs, are often referred to as 'thought experiments' in philosophy. See https://en.wikipedia.org/wiki/Trolley_problem for a popular example. Or, there's always the ever popular Schrodinger's cat.

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  • A toy problem is definitely the right idea, but there is a word that is used or at least was used by my professors at university.
    – Brent
    Commented Jul 10, 2015 at 19:47
  • A none mathy example could be, John and Henry are walking down the street and they see the "The Elephant Man". John says to Henry, "You don't see that every day", to which Henry quips "Unless your his wife!" ("unless your him" works to but its slightly less quippy). Henry's statement is a simple "counter" example that is manufactured in a way to be very relevant, its ...(the word I can't think of).
    – Brent
    Commented Jul 10, 2015 at 19:53
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Degenerate. Base case. Special case. Interesting because it shows how a general model does or does not encompass specific models. -tim

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  • +1 for "base case"; "special case" is more general (!) than what the OP is looking for.
    – Rand al'Thor
    Commented Sep 9, 2015 at 16:23

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