The problem with logic is that if you do it wrong even once it blows up. The principle of explosion states that, if your logical system has a single contradiction in it, you can prove anything in that system. When you have a statement P that is both true and false, combine it with whatever you want: “P OR Geodude from Pokemon is real and my best friend.” Well, P is true, so the whole statement is true, since a true statement OR anything is still true. But P is false, so the only way that whole statement is true is if Geodude from Pokemon really is real and your best friend. You use the contradiction as a sneaky little u-turn to drive wherever you want to go.
Yet despite the hair-raising risk of explosion mathematicians will still edge themselves into almost exploding, all the time, for fun. This is called a proof by contradiction. You start by making up exactly one thing and asserting it is true without evidence. Then you do math using your new toy. If you reach a contradictory conclusion — that is, if you find a single bomb, which has enough destructive power to blow up all of math — then you say “Okay this is a bomb, but math has not been exploded, which means math has zero bombs. So the thing I made up must not be true, because it led to a bomb.” And then you come away with a bit of useful knowledge: the thing you made up is not true. The benefit was in the turning away, not from blowing yourself up for no reason.
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If you are going to assume something without evidence, it is really, really important to a.) remember the thing that you assumed and b.) recognize the first contradiction you see as a signal that your assumption is wrong, not an excuse to walk up “P is true” and then walk down “P is false” and keep proving stuff. Because if you take that path just once, that’s it. Everything you find afterwards is meaningless, but will seem locally true, and the further you go the more it seems like you’re on to something because you keep proving stuff. This is the wretched loop behind most screeds about new physics, new philosophy, etc.
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The whole point of the proof by contradiction is to be hyper-vigilant for when you’ve hit the bomb so you can stop trying to prove anything and declare a victory of discretion. The whole point of a thought experiment is to try to prove things that follow from your experiment; that is, to assume you haven’t hit a bomb. And if you hit the bomb and don’t notice, you probably can come up with a great proof of the outcome you’re arguing for, because you can come up with a great proof for anything.
This is why, while I love thinking and I love experiments, I am generally not a fan of thought experiments. [...]
From the article:
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