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One thing I discussed with my students here at HCSSiM yesterday is the question of what is a proof.
They’re smart kids, but completely new to proofs, and they often have questions about whether what they’ve written down constitutes a proof. Here’s what I said to them.
A proof is a social construct – it is what we need it to be in order to be convinced something is true. If you write something down and you want it to count as a proof, the only real issue is whether you’re completely convincing.
Having said that, there are plenty of methods of proof that have been standardized and will help you in your arguments. There are things like proof by contradiction, or the pigeon hole principle, or proof by induction, or taking cases.
But in the end you still need to convince me; if you say there are three cases to consider, and I find a fourth, then I’ve blown away your proof, even if your three cases looked solid. If you try to prove something by induction, but your inductive step argument fails going from the case n=16 to n=17, then it’s not a proof.
Ultimately, then, a proof is a description of why you think something is true. The first half of your training is to problem solve (so, come up with a reason something is true) and construct a really convincing argument.
Coming at it from the other side, how can you check that what you’ve got is really a proof if you’ve written down the reason you think it’s true? That’s when the other half of your training comes in, to poke holes in arguments.
To be a really good mathematician you need to be a skeptic and to walk around with a metaphorical gun to shoot holes in other people’s arguments. Every time you hear a reasoned explanation, you look for the cases it doesn’t cover or the assumptions it’s making.
And you do the same thing with your own proofs to help yourself realize your mistakes before looking like a fool. Because putting out a proof of something is tantamount to asking for other people to shoot holes in your argument.
For that reason, every proof that one of these young kids offers up is an act of courage. They don’t know exactly how to explain their thinking, nor do they yet know exactly how to shoot holes in arguments, including their own. It’s an exercise in being wrong and admitting it. They are being trained to get shot down, to admit their mistake, and then immediately get back up again with better reasoning. The goal is to get so good at being wrong that it doesn’t hurt, that it’s not taken personally, and that it’s even fun to be wrong and to improve your argument.
Not every person gets trained in being wrong and admitting it. I’d wager that most people in the world, for most of their professional lives, are trained to do the opposite in the face of being wrong: namely, to wriggle out of it or deflect criticism. Most disciplines spend more time arguing they’re right, or at least not as wrong, or at least they have different mistakes, than other related fields. In math, you can at the most argue that what you’re doing is more interesting or somehow more important than some other field.
转发的“Mathematicians know how to admit they’re wrong”
[I’ve never understood why people would think certain math is more important than other math. It’s almost never on the basis of having applications in the real world, or helping people in some way. It’s just some arbitrary snobbery, or at least that’s how it’s seemed. For my part I can’t explain why I love number theory more than analysis, it’s pure sense of smell.]
Most people never even say something that’s provably wrong in the first place. And that makes it harder to prove they’re wrong, of course, but it doesn’t mean they’re always right. Since they’ve not let themselves get pinned down on a provably wrong thing, they tend to stick with their wrong ideas for way too long.
I’m a huge fan of skepticism, and I think it’s generally undervalued. People who run companies, or universities, or government agencies, typically say they like healthy skepticism but actually want people to drink the kool aid. People who are skeptical are misinterpreted as being negative, but there’s a huge difference: negative means you’re not trying to solve the problem, skeptical means you care enough about the problem to want to solve it for real.
Now that I’ve thought about the training I’ve received as a mathematician, though, and that I’m now giving that training to these new students, I’ll add this to my defense of skepticism: I’m also a huge fan of people being able to admit they’re wrong. It’s the flip side of skepticism, and it’s why things get better instead of stay wrong.
By the way, one caveat: I’m not claiming that mathematicians are any better at admitting they’re wrong outside a strictly logical sphere.
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