What is a proof?

High-school math teachers and those who know them need to be concerned with what a proof is. That certainly isn’t a new claim. But for most students a proof is something you learn to construct in high-school geometry class and then never see again.

So why do we think it’s so important? And, stepping back a little, we have to ask what a proof really is anyway.

Three articles on the subject have recently come to my attention. Two are from last month; one is actually from ten years ago, a piece by Keith Devlin in Discover Magazine. Let’s start with that one, as it’s the oldest. I have to say that I like and admire Devlin a great deal. Usually I agree with what he writes. But not in this case. In the first place, his strange distinction between “left-wing” and “right-wing” conceptions of proof is irritating at best:

The right-wing (the “right or wrong” or “rule of law”) definition, is that a proof is a logically correct argument that establishes the truth of a given statement. The other answer, the left-wing definition (fuzzy, democratic, and human centered), is that a proof is an argument that convinces a typical mathematician.

This distinction immediately rubbed me the wrong way, but I couldn’t quite articulate (to myself or to others) just why it did that. I suppose it was partly because it left the student unable to tell whether a given argument was really a proof or not, since s/he is unlikely to know “a typical mathematician.” It reminded me of victim-defined crimes, where the alleged perpetrator has no way of knowing whether a crime has actually been committed or not. But my unease was then put into words by one of my colleagues who has a degree in philosophy:

As a philosopher I feel I must comment. I believe the article contains a philosophical error which itself dates back to ancient Greece: conflating metaphysics (the study of what is) with epistemology (the study of how we come to know what is). The “right-wing” definition tells us what it means for a proof to be valid, while the left-wing definition tells us how we come by the knowledge that a proof is valid. In order to be a proof, it must meet the standard in the right-wing sense; however, we may sometimes accept arguments as proof though we cannot say with certainty that they are proofs, whether that acceptance is derived from our own perusal of the paper, the democratic agreement of experts, hearsay, etc.

Yes, exactly…even though I couldn’t have expressed it that way myself. But read it again. He’s right.

And what’s the student’s point of view? Well, there are three types of geometry students. (It’s always three, or course. Except when it’s four.) Let’s pick three students who illustrate the three groups:

  • Chris looks at a proof that the teacher considers perfect and declares it valid, more out of faith or indifference than out of deep thinking. The proof may actually be flawed in many ways, but Chris is credulous and believes it: of course we’ve proved that the sum of the angles of a triangle is 180°.
  • Sam looks at the same proof and finds flaws in it: lots of hidden assumptions, skipped steps, and invalid reasoning from the diagram. Like a real mathematician, Sam is skeptical: we only think we’ve proved that the sum of the angles of a triangle is 180°. Sam knows better.
  • Casey looks at the same proof and rejects the entire endeavor. This whole proof business is just a game required by the teacher; it doesn’t and couldn’t actually prove anything. Casey is cynical: we can go through the motions in order to get a good grade, but it’s meaningless.

Almost every class contains a mixture of Chrisses, Sams, and Caseys, though the proportions can vary wildly. A teacher who imposes the same set of requirements on everyone in the class is going to convince only 23% of the students. (A completely fabricated statistic, of course, but you know what I mean.) I suspect that part of the issue in high-school geometry comes from our insistence on two-column proofs. For more on this issue, see some two-column proofs that two-column proofs are terrible.






Two-column proofs are a tradition in American high schools dating back only to 1893. Real mathematicians don’t use them. Read the link for a thorough discussion of this particular point.

For a very different point of view about proof, read the always-worth-reading Kate Nowak. Be sure not to skip Stuart Jeckel’s video embedded in the post that that link points to. It may give you a different perspective on what a proof is.


Categories: Math, Teaching & Learning