# Right English wording for "counterexamples to a theorem"

This question is about the right English wording.

I give here what I call "counterexamples to Banach fixed-point theorem". What I do, is that I look to what happen if some hypothesis of the theorem are not fulfilled.

I call this "counterexamples to Banach fixed-point theorem". However, I was told that this is not a proper English wording as a theorem cannot have a counterexample! I'm a French native speaker and it seems that a straight translation of "contre-exemples au théorème du point fixe de Banach" is not good.

What would be the appropriate wording to this situation?

• I don't think this an English vs. French issue, I think it's more to do with how rigorous you are in your wording and whether the audience understands what you mean. Perhaps it is clearer to say that you're giving counterexamples to a "strengthened version of HB". Aug 31 '15 at 12:01
• While it may not be strictly correct, "Counterexamples to the Banach fixed-point theorem" is certainly catchy! Specially if you're giving a talk.
– lhf
Aug 31 '15 at 12:01
• It is not an English vs French issue. A counterexample to a statement (logical sentence) is a an example that shows the statement is not true. A theorem is known to be true. A better wording would be "examples that the hypothesis of Banach fixed-point theorem can not be dropped". Aug 31 '15 at 12:21
• I do not believe that the French expression says what you claim, but the English one needs an article or a genitive. Aug 31 '15 at 17:12
• @Ramiro: Mathematics, like other languages, often has idioms which are widely understood as having a meaning different from their literal meaning, but which might not appear in a dictionary. Aug 31 '15 at 19:11

How about simply "counterexamples related to the Banach fixed point theorem"?

Or if you want to be more precise, "counterexamples to possible strengthenings of the Banach fixed point theorem".

• Wouldn't "regarding" be more concise than "related to"? Aug 31 '15 at 18:18
• @guest: Same number of letters :-) And I actually think "related" fits better here, though it's hard to explain why. Aug 31 '15 at 19:09
• I thank you all for your constructive comments and proposals. It helped me to: (1) better understand the meaning of counterexample in English, (2) have proposals with various points of view and (3) highlight what can be catchy (I discovered the word) which is one of the objective of a website post. Aug 31 '15 at 20:13
• +1 The "precise" statement is good but verbose. Seems to me that an agreed formal term for this would be useful. Seems similar to "edge cases" and "boundary cases". Sep 1 '15 at 4:20

I am also not a native English speaker, so not sure it this is a good suggestion, but I would use something like Necessity proofs or Necessity demonstrations or Necessity examples (meaning: examples that prove that every assumption of the theorem is necessary).

• Nice. A slightly different perspective, but much less misleading than "counter-examples". Aug 31 '15 at 19:41

A possible English wording is "what happens if we drop the hypothesis of the Banach fixed-point theorem?"

• Or "hypotheses", if there's more than one of them :) Aug 31 '15 at 17:49

I have looked at your website before, and I remember being a bit dissatisfied with the terminology "counterexamples"- from a technical point of view. For me, a counter-example is some constructible object that demonstrates the falsehood of some statement. Equivalently, a counter example demonstrates the truth of the negation of the original statement. So for (counter-)example, take the statement "All functions are invertible". That is, for every $f$ there is some $f^{-1}$ such that for all $x$ in the domain of $f$, it is true that $f^{-1}(f(x)) = x$. A counter example to this is the function $f(x) = x^2$. Because, assuming an inverse function $f^{-1}$ we have

$$1 = f^{-1}(f(1)) = f^{-1}(1) = f^{-1}(f(-1)) = -1$$ which is clearly a contradiction. So this is a counter example to the claim.

On the other hand, as one commenter points out, from a "marketing" point of view, "counter-example" sure sounds catchy. But ultimately I find it misleading.

Edit: I realise I didn't really answer your question! In your case, I would say "Counter-examples to variations on the theorem statement" or something along those lines. Or go with the answer above. There are really a lot of ways to say it, but I would certainly avoid "counter examples to the ____ theorem".

• I like "Counter-examples to variations on the theorem statement"; it suggests very nicely that the theorem itself is correct, but making slight changes produces something incorrect. Aug 31 '15 at 19:17
• Thanks, you like it more than me in that case! Technically it's OK but I think verbally we could do with something a bit more succinct. Aug 31 '15 at 19:24

This is much more a mathematical question than an English language question. (Note that I didn't even look at the maths)

On your website you say "We look here at counterexamples to the Banach fixed-point theorem when some hypothesis are not fulfilled."

Basically this theorem, like most theorems, takes the form "if conditions A, B and C are fulfilled, then we get the result X". And you are giving results that demonstrate that if A is not fulfilled, or B is not fulfilled, then we don't get that result X. You call A, B, and C "hypothesis". That's wrong. They are not hypotheses. They are preconditions of the theorem.

You then demonstrate that you don't get the result X. That is not a counterexample to the theorem, since the theorem never claimed you would get the result X without all the preconditions.

I'd suggest "Examples for the necessity of the preconditions of the Banach fixed-point theorem". There are probably many possible ways to put this.

All that said, when you were told "a theorem cannot have counterexamples", that is not quite right. "Theorem" is not used for a mathematical statement that is true, but usually for one that is widely believed to be proven. Sometimes not even that is necessary; Fermat's Last Theorem was called Fermat's Last Theorem for many, many years when no proof was known, and when it was not even know if a proof existed. So it is entirely possible that something called a "Theorem" is actually wrong and that there are counterexamples. If you managed to give an actual counterexample and that counterexample has been widely accepted, the name would probably get changed.

• I have a few issues with the above, all just notation though, nothing serious. 1) I have seen A, B, C called hypothesis before. I worked with the Coq theorem prover and in that tool, your "preconditions" or "arguments" or "parameters" to a theorem are exactly called "hypothesis" 2) I'm not sure I entirely agree on your classification of "Theorem" as "believed to be true". But it's an interesting standpoint, and I can't fault you on it. Personally I think it should have been "Fermat's last conjecture" though. Usually something like that is called a conjecture e.g. Goldbach. Aug 31 '15 at 19:31

I would go for non-inclusive examples or exclusive examples, as in essence any violation to the hypotheses/preconditions of the theorem are supposed to produce results that are excluded from when every condition is fulfilled.

For instance, on the level of definitions, when defining the term 'rectangle', you can state that a square and a triangle are inclusive and exclusive examples of a rectangle respectively.

Great question, great discussion.

Perhaps "special case" or "classic example" is appropriate. For instance when discussing the implications between continuity and differentiability the special case or classic example is the Weierstrass function continuous everywhere, differentiable nowhere. This demonstrates that continuity does not imply differentiability.

• Please avoid making trivial edits to very-low-quality posts since this removes them from the very-low-quality queue, so reduces the chance that they will be properly handled (that "answer" should either be converted to a comment or deleted). If you can re-flag it as VLQ that would be appreciated. Mar 7 '19 at 22:58
• Same here and here. Mar 7 '19 at 23:12
• @Bill Dubuque: You make a good point on VLQ. The other two posts came up in the First Posts queue. As a courtesy, the text was converted to MathJax after posting the "Welcome, Please use MathJax" comment. There is some queue malady on MSE VLQ and FP seem to blend. And thanks for being so specific. Mar 7 '19 at 23:32
• Ah, I see. I rarely use them so I don't know their quirks. I though it would be worth pointing out the VLQ issue in case you didn't know that the edit terminates the review. Mar 7 '19 at 23:34