What is a proof?
I recently described (here) a proof to be a convincing argument of why you think something is true. I’ll stick to that definition in spite of a few commenters who want there to be axioms or postulates, because I really don’t think that’s what happens in real life (which is a good thing! It would be an incredibly boring life!). Since I’m a utilitarian, I only care about and only want to discuss what actually happens.
The above definition immediately begs the question, convincing to whom? Can a proof to someone be a non-proof to someone else? Absolutely, proofs are entirely context-driven. If I’m trying to prove something to you and you remain unconvinced, then it is no proof, even if I’ve used the same argument before successfully.
This brings me to my first main point, which is that it the responsibility of the person proving something to convince his or her audience that it’s true. Likewise, it is the responsibility of the audience to remain skeptical (but attentive) and be open to being convinced or to finding a flaw in the argument.
Things get trickier when it’s not a live interaction, but when things are written down, like in published articles. On the one hand, written proofs give the audience more time to understand the reasoning and to come up with problems, but on the other hand there’s no opportunity to say “I just don’t get what you’re talking about,” which is the feeling one typically has at least 85% of the time.
In an ideal world, those who write proofs understand the goal to be that the reader should be able to understand the argument, and thus make the arguments coherent and understandable to their “typical reader.” Who is this typical reader? Someone who is probably relatively fluent in the basic objects of the field, say, but hasn’t recently thought about this problem.
Now that I’ve described the ideal situation, I’ll rant for a bit about how people game this system. There are two things that creep into the system that give rise to its gaming, and those two things are status and credit. People like to be high status (and like to signal high status even more), and of course people like to take credit.
First, status. It turns out that people often really want to explain their reasoning no to the typical audience, but to the expert audience. So they don’t give sufficient context, and they are lazy reasoners, because the experts can be expected to understand how to fill in the details.
It’s not only insecure young mathematicians that are guilty of this – there are plenty of experts who themselves fall prey to this habit (thus the signaling). I think it’s driven by a combination of feeling kind of smug and smart when people who are trying to follow your conversation leave because they’re exhausted and confused (and possibly ashamed), and the echo chamber that remains after people who don’t get it (or who admit to not getting it) leave. Whatever the reason, there are plenty of experts who get less and less understandable over time, in person and in print.
The other side of this status play is those experts get away with it. The papers written by these people are often accepted in spite of the fact that they are nearly unreadable to all but the 5 people in their field for whom they have been written, since after all, these guys are experts.
But does this approach constitute a proof? I claim it doesn’t, not if I have to be one of 5 people to read and understand it. The writer has choked, bigtime, on his or her responsibility to convince the reader.
Second, the credit thing. People want to get credit for proving things, because that’s how they get high status. But they don’t always want to prove everything they claim, because it’s hard work. So sometimes you see people proving something and then claiming an even more general thing is true, and giving a “sketch of a proof” for that more general thing (this is one example where “sketches” come up, but actually there are plenty of them).
Let’s examine that concept for a moment, the “sketch of a proof.” Usually this implies that the basic outline is there, but many details of how to rely on so-and-so’s theorem or what’s-his-name’s method are left out. It’s a proof lying in the shadows, and we’ve only seen it highlighted every few feet or so to wend our way through it.
Is a sketch a proof? No, it’s not. Best case scenario, it would take a typical reader a few minutes, maybe up to two hours, say, to turn that sketch into a proof.
But what if the typical reader can’t do it in two hours?
The problem with the concept of a sketch of a proof is that it’s too difficult to refute. If I am a reader and I say, “this is a false sketch” then I could just be opening myself up to people who tell me I didn’t spend my two hours wisely, or that I’m not good enough to complain about it. They may even expect me to prove that that method cannot be used to prove that result.
But that’s bullshit! As far as I’m concerned, if you claim to have sketched a proof, and if I’ve tried to prove it using your notes and I’ve failed, then that’s your fault, not mine. It’s your responsibility to prove it to me, and you haven’t.
Conclusion: let’s all remember when you claim a result, you are claiming credit, and it’s your responsibility to convince the audience it’s true – not just 5 experts. And second, if you aren’t willing to actually prove something, don’t claim it as a result. Instead, say something like, “this may generalize using so-and-so’s theorem or what’s-his-name’s method….”. Consider it a gift to the next person who reads your paper and wants to prove something new.