Anna Blinstein asks a great question

What's the difference between proof and "really convincing explanation?" #MTBoS #iteachmath

— Anna Blinstein (@Borschtwithanna) August 1, 2017

After a few iterations, this eventually became

v3 -Given a set of shared assumptions, an explanation that's derived deductively and is convincing to a group of skeptical peers is a proof.

— Anna Blinstein (@Borschtwithanna) August 1, 2017

This is tricky because we’re entering into a realm of semantic splicing. A few folks chimed in with an appeal to an axiomatic system and the underlying logical process for what constitutes as a proof in mathematics. Others also gave counterexamples of what is a convincing explanation but not a proof [an interesting circular dilemma since a counterexample is a type of proof!]. Others made an argument about degree vs kind — that is, non-mathematicians are capable of producing sound logical arguments but not to the same degree as a mathematician.

My 140 character version was this

a proof is a type of really convincing explanation. really convincing explanations need not be true. proofs must be true. #mtbos #iteachmath

— M Shah (@shahlock) August 1, 2017

After a few exchanges and a lot more folks, we got to these two questions:

No, I was thinking about opening up the definition of proof to make it more accessible to people of all ages & levels of math sophistication

— Anna Blinstein (@Borschtwithanna) August 1, 2017

So, the way I see it, there are at least three open matters that are intertwined.

- What’s the difference between a proof and a real convincing explanation?
- When something is proven, how can we be 100% confident that it was proven correctly?
- What are the consequences of opening up the “definition of proof” as it pertains to math education?

Here’s an attempt to navigate all this. First, the second question. In general, people make mistakes. We’re still not at a point where we have an infallible proof checking program [though some are working on it.] I do encourage you to read the article in the link. It is at the heart of some of this discussion.

Mathematicians do make mistakes. A team of mathematicians do make mistakes. Thus, to some extent, I can concede that knowing 100% certainty that a proof is a proof [a statement of truth] can’t always be achieved. The more abstract and complex — not just in length, but in content — the more uncertainty can exist. However, those are extreme cases. We are not uncertain about the theorems and their proofs in any of the standard undergraduate or graduate coursework. It’s only when we start reaching the cutting edge research level is there room for doubt. And if anyone does actually believe that standard math content as taught in undergraduate or graduate coursework has an epsilon chance of being fundamentally flawed, then a LOT of mathematics will have to be rewritten. It is more likely that the proofs are correct than the majority of mathematics being flawed. So, to some extent, a “test of time” argument is probably a decent proxy for how confident we are in the absence of really looking into the proof ourselves and being convinced.

This brings me to the first and third questions regarding “convincing”. I maintain that there is a difference in a convincing explanation and a mathematical proof. I think it’s generally harmful to call some things that wouldn’t constitute as a proof a proof for the sake of giving a sense of accomplishment to enterprising learners. I don’t mean in this in a harsh or elitist way, though I understand it sounds that way. What I mean is that we are giving a false sense of accomplishment when we bestow upon those who are very young in their training a sense of elevation that is undue — that is harmful. I am not an artist, though I do draw. My art work is not a work of art. I am not a pianist, though I do play piano. My piano playing is that of a person playing the piano not one of a pianist.

Now, with all that said, one does not have to be an artist to produce a work of art. From time to time my piano playing can sound comparable to that of a pianist. Similarly, one doesn’t have to be a mathematician to be able to produce a proof. But this isn’t about profession, it’s about the act.

Mathematical proofs require sound, logical reasoning. Mathematical proofs *do not* require math symbolism or formal math speak. I say this because the commentary about “degree vs kind” and “opening up the definition of proof” seems to imply that a proof has a style requirement (symbolism, formal speak, for example). And I think this artificially creates a debate.

Here’s what I mean, by example.

Prove that the sum of any even number and two is even.

Here’s what I think is happening in the thought about proof vs real convincing argument. A formal math proof would go something like this.

Let \(n\) by any even integer. By definition, \(\exists\ k \in \mathbb{Z}\) such that \(n = 2k\). Let \(s = n + 2\). Then \(s = 2k + 2 = 2(k+1)\). Since \(k \in \mathbb{Z}\), then so is \(k + 1\). Thus, \(2(k+1)\) is even by definition. QED

Here’s what I think people mean by “real convincing explanation in degree but not in kind” that should be considered as a proof.

I know that numbers go in a pattern of even, odd, even, odd, … So if I have an even number and I add two, then I would skip one odd number and then end up at an even number.

This *is* a valid proof and there’s no need to classify it as anything less. What it lacks is an element of formalism, but that’s not required for a proof. That’s more for presentation. We might not submit this to a math journal (and to be honest proving that \(n + 2\) is even if \(n\) is even is so trivial that one wouldn’t even write a proof for it in a journal paper). But even if we did, nothing would be rejected about the proof itself. If anything, we’d get commentary for a bit more formalism (or to omit it because it’s so obvios). That’s the world of math professionalism. Part of it is about polish.

So if we can agree that a proof only requires an adherence to the generally accepted logical reasoning process, then there really is no debate between the two examples. We should call both of them proofs. If we want to take it a step further and groom the student in the ways of math formalism, then that’s a different matter.

In a bleak view, mathematics is “just” a tautological extension of definitions and theorems. Define \(X\), then if \(A(X)\) then \(B\). Now suppose we define \(Y\) then if \(C(X,Y) \mbox{ then } D\) and so on.

On the other hand, consider this argument, more typical for someone new to proving statements:

Well, \(2\) is even and if I add two, I get \(4\) which is even. If I add two to \(4\), then I get \(6\) and that’s also even. So, yeah, an even number plus two is even.

This is *not* a proof. This is headed in the direction of a proof. This is beginning to sound like a convincing explanation by way of a few examples. But this isn’t a proof. It’s a few examples showing the case we wanted, but this method requires a case by case check of every even number. It’s missing a generalization step. Either an induction argument or at minimum a number line argument like in the second example.

If it is this example that we want to open up to be called a proof, then I would advise against it. What we have in this third example, is one of many ways that mathematicians begin to think about proving statements. It’s acceptable to work out some test cases to convince oneself. From there, a next step is to find ways to express these examples more generally. We should try to encourage and develop this next step of generalization / abstraction. We can tell our students, “Hey that sounds convincing.” and we should add to it, “But how can we prove it?” This helps to distinguish between convincing explanations and proofs.

In fact, if we are so inclined, we can go down a route of meta-proofs to show that the idea of extrapolating from a handful of examples isn’t a guaranteed way to prove things. A famous example is of a prime generating quadratic: \(n^{2} – n + 41\). Remarkably it produces prime numbers for \(n\) from 1 to 40. But when \(n = 41\) the formula breaks down since \(41^{2}\) is not prime. And this is a proof by counter example that we can’t use a handful of cases to make a general statement.

Finally, there is another layer of semantics. A proof can be confusing. A proof can be roundabout. A proof can be unnecessarily long. A proof can lack elegance. A proof can be uncheckable [a whole ‘nother can of worms]. However, within a mathematical framework, if it is a proof then it is a convincing explanation. It’s just not convincing to humans.

Finally, finally, mathematics doesn’t care about age, sex, gender, race, color, creed, etc. A kid can produce a proof. A non-mathematician can produce a proof. An old, tired mathematician who hasn’t proved anything in decades can produce a proof. The name of the game is logic and you have to know how to play it.

What’re your thoughts?

[As an aside, if you want to discuss over Twitter, then just know that if too many people are in the tweet thread, I’ll abandon the thread. Nothing good comes of a free for all “conversation” on Twitter. I prefer you write a blog response or leave a comment below.]

Erik PostmaI find myself most in agreement with the “sliding scale” view, where a few positive examples < a convincing informal argument < a proof as we learn to write in undergrad < such a proof with one or more iterations of independent review < a proof written and verified with proof checkers such as HOL light or Coq. I always wonder how popular proof checkers will end up being in the limit as time goes to infinity…