Everyone's Got a Proof When They Explode
When do thought experiments stop being experiments?
A specter is haunting social media. (Well, the parts of social media where nerds live). A game theory problem with a lot of pretend lives on the line:
Blue won at 58%. Should it have? What would you pick? If you’re red, are all the blue pushers naive fools who deserve the consequences they face? If you’re blue, are the red pushers a bunch of selfish narcissists who haven’t even considered how tedious it would be to only live with reds? There’s a lot of different ways to frame the problem and they can really make a difference on how it feels. What if it’s a big blender that only suicidal blues would voluntarily step into? What if it’s a red leader who will kill everyone who didn’t vote for him if he wins? Depending on how you think about it, you might dramatically change your mind. But in fact this proliferation of frames points us somewhere else entirely; a long-standing, well-known problem with formal logic.
The problem with logic is that if you do it wrong even once it blows up.1 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.
The problem with logic bombs is that they are quiet. They don’t shake the walls of your house or light your paper on fire or anything. They just tell you that Geodude is your best friend, and if you already kinda wanted that to be true (which is maybe why you were checking), then you might not be that inclined towards skepticism. You might just say, great, let’s check more stuff using logic, the system with no bombs in it! Hmm, let’s see, P OR Geodude has a sick-ass monster truck and he’s going to take me for a ride later. Well, P is true, so…
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.
This is why, while I love thinking and I love experiments, I am generally not a fan of thought experiments. A thought experiment is basically a proof by contradiction but worse. Instead of assuming a starting proposition, it assumes a hypothetical end state. You probably need to make several assumptions to suppose how you’re getting from here to there. The person proposing it usually don’t articulate them, and if they do, it’s with an exhausted “okay, for the rules lawyers…” overtone that implies it’s distracting from the real question. In these vague and unspecified conditions, it’s very easy to smuggle in two assumptions that are individually defensible but will never appear together. I can imagine a world where my insomnia keeps me up all night, and I can imagine a world where I sleep for 12 hours straight tonight and miss my alarm for work, but they are clearly not the same world.
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.
Let’s talk about: buttons! Buttons are little guys you push to signal that you consciously want a thing to happen. You have some in your house. You likely have some for your job that are your sphere of responsibility. It’s easy to push a button. They made it a button and not some other process because you are meant to make an easy, individual decision about that button. If it was something more important, maybe there’s a button buried within the process, but the button is not the master of the process. If you’ve got the button to make the big website go live, it won’t do anything until a lot of other people merge the code, set the configvars correctly, whatever.
Let’s talk about: human death! It’s bad! It’s bad enough that most right-thinking people try to make it hard. Sometimes there are mechanisms that take human life that have fewer process guards than you hope. But these either take the forms of militaries building cocoons of rules around lethal force, or accidental degradation of a risk that would be repaired if it became common knowledge, or some unpreventably dangerous part of life that we make sure to build an oral tradition of restraint around. (How many times have you heard “Never work on garage springs yourself, those things can kill you”?) What does not happen is that everyone makes life and death decisions with easy inputs and no reflection or feedback from their peers.
So you have a button, the thing that encodes small, sufficiently unimportant decisions. You’re using it to make a decision about your own life and death, and potentially the life and death of up to half of the human race. The button is inconsequential and consequential. You’re making this decision individually, as per the dictates of the experiment and the general affordances of buttons. It’s a decision about the life and death of others, which is never made on your own and never disconnected from society-wide norms. P is true and P is false and it’s very very important you make the right decision in this experiment, because it’s not just your life at stake, it’s also your wife Geodude and your five children.
The right answer is just whichever side of the contradiction switchboard you choose to focus on. If you think about the breeziness of a button, clearly no one’s decision on how to press a button should become a death sentence, so your knowledge of buttons makes you want blue. If you think about the massive weight of death, clearly this is one of the massively consequential decisions that everyone has to get right to live every day, so you want to push red for the same reason you don’t work on your own garage springs. If you know the decision was made by lots of people on the fly without warning, then you can be sure there are lots of confused people on both sides, and you know their lives are in your hands. If everyone has had a year to think about their push, then you can be sure society has settled on an equilibrium already and your answer is in relation to that.
The answers can still be a bit meaningful by showing which one of these things first comes to your mind. But these experiments get less interesting the more you think about them, because you just keep rifling through contradictory true statements until you get bored. Saying a thought experiment is highly sensitive to the framing is a tacit admission that you are in an exploded logic. The intellectually correct position is to find the assumption and take away the real lesson: you don’t get buttons that do stuff like this. Less exciting, but with the welcome property of being true. A result that tells us, in the words of Crispy Chicken, “They’re called thought experiments for a reason, assholes.”
If this sentence sounds familiar, it’s because you’ve already read Don’t forget you’re contradicting, dear. Since TIS posts are so short, it seems better to just expand it inline rather than asking you to click. I see the stats on how often people click links in articles and I no longer believe links are appropriate for anything load-bearing.
IRB for thought experiments
I guess the thing that gets me about this viral "logic experiment" is how obvious it is. There's no moral quandary around saving one loved one vs many strangers on a train track. If you pick red nothing bad happens to you. If you pick blue, you might die. Everyone has that same decision in front of them. What is there to think about? What am I missing? And yes, I know that isn't the main focus of your post, I just had to vent after seeing this in too many places.