# “Proof” meaning in maths and society

When we ask students to prove a particular result in a math class, students often reply with examples. For example, if I state: if a number is even its square will be even, and ask the students to prove it, they will reply with an example (such as, "The square of two is four, and both are even").

I believe that the problem lies in the wording (which we cannot change). Proof in a social context means supporting your arguments with evidence, which is different from the mathematical definition of the word.

How can we handle such problems in class?

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• This sounds less to me like an English question, and more like a math pedagogy question: how do you communicate to your students that "proof" in math class has a different evidentiary standard than "proof" in social contexts. As it happens, SE has the helpful matheducators.SE, which might be a better venue for this question. – Codeswitcher Jul 26 '14 at 5:19
• See also Is a proof also "evidence"? on Mathematics SE. – J W Jun 20 '16 at 5:40

The mention of "evidence" in addition addition to "proof" is a good way to start.

One can explain that the fact that the square of 2 is also even is evidence for the assertion/conjecture/hypothesis/claim made. The fact that the square of 6 is also even is further evidence for it. And so on.

But no matter how much evidence of this form one generates there might always rest some possibility, some doubt, that the general claim could be false.

Then, one could recall that in court one common standard is to say that for something to be considered proved there needs to be evidence beyond a reasonable doubt; but for other situations one might only need to show a "Preponderance of the evidence" or still something else. See the Wikipedia site Legal burden of proof for various concepts.

Finally, one could say in mathematics the standards of proof are very strict and there needs to be evidence beyond any doubt whatsoever for something to be considered as proved.

To sum it up: also in everyday usage there is (or at least there should be) a distinction between "evidence" and "proof," and this is a good opportunity to recall this. Which quantity of supporting evidence is considered as a proof differs depending on context, and in mathematics is especially high.

• AFAIU, in the US the standard of "beyond a reasonable doubt" is only applicable in certain cases (criminal cases?), in others it is "preponderance of the evidence" only. E.g. OJ got a "not guitly" veredict in criminal court, but was sentenced to pay damages for wrongful death in civil court. This distinction could be used to clarify... – vonbrand Jul 26 '14 at 16:17
• @PurpleVermont there is a compilation of "Eventual counterexamples" on MathOverflow. Not few examples there seem however too advanced. One thing that could work are Fermat numbers being "always" primes or perhaps also Mersenne Primes or possibly still better that $n^2 + n + 41$ is prime for "all" nonnegative integers (true until 39). – quid Jul 26 '14 at 18:03
• @quid the example of $n^2+n+41$ (not) always being prime seems like a perfect example. I may play with that with my middle school students. – PurpleVermont Jul 26 '14 at 18:06
• A natural example given to me by a staff member from the Learning Centre at RMIT (Melbourne, Australia): If you made the statement "I am the best tennis player in the world" and were challenged to prove it, the only way to know for sure is to play every other person in the world. Just playing everyone in your class wouldn't prove it. – DavidButlerUofA Jul 26 '14 at 23:33
• I feel like commenting that I have heard stories of some US prosecutors explain the difference between "beyond reasonable doubt" and "beyond a shadow of a doubt" to a group of jurors. I think that in math we want that "beyond a shadow of doubt" level of confidence before we are satisfied. The difference may or may not explain the (thankfully rare) occurences of wrongful (death) sentences. But, it is probably best to leave that discussion to a different forum. – Jyrki Lahtonen Jul 28 '14 at 10:17

I think it would be helpful to simply reiterate the differences between a "proof", in the real-world sense of the word, and a mathematical proof. Indeed, it would be best to keep adjoining the "mathematical" part to "proof" whenever you ask your students to write a proof. This will continually remind them that they are working on a specific kind of argument style with rigorous standards.

Another idea is to provide students with written arguments (they could either created by you, or come from past students' assignments) and have them critique them. Have some that are correct proofs of correct results, some that are false proofs of correct results, and some that are false proofs of false results. Have the students read them and discuss them, for both correctness and clarity. By getting them to think and talk about different arguments, they will get a better understanding of what constitutes a mathematical argument, as opposed to a regular ol' argument.

• Still, one might reasonably claim that "mathematical proof" differs from "colloquial proof" only in a matter of degree, nevermind that the mathematics community's mythology pretends a genuine absolute, etc. – paul garrett Sep 13 '14 at 22:26

Here's a few things that may help.

Borrow a concept from science -- the notion of falsifiability.

There repeated verification of a theory is just that -- verification but not proof.

In science we cannot prove something correct.

You may find help in the phrase "formal proof" or in the word "derivation" Introduce them to conjectures too such as Goldbachs. By using a different label, you can help separate the concepts in their head.

Have a discussion about 'what is truth?' Show that in everyday life we require different levels of evidence and different kinds of proof. I am more likely to believe stuff I see in the Washington Post rather than in the National Enquirer. NBC news has a better rep than Fox. When a friend tells you that Susan is sleeping with Mike, you require less evidence than you would if that same friend tells you that Challenger has exploded. (Happened to me. Took him 10 minutes to convince me that he wasn't kidding.) At this point Mathematical Proof becomes yet another type of proof.

This is material that should be covered in a course on logic, but alas, we don't generally teach logic anymore.