[CMP05] The Unexpected Hanging Paradox (Schrödinger’s Cat’s Brother) |Consistent Math & Philosophy 5


This is ln(e). He’s got a famous brother, but to his parents, Lonnie was always number 1, even though he often would get into a lot of mischief. He started his crimes as a cat burglar, but he eventually got into more fur-ocious crimes and was eventually caught flea-ing the crime scene. Today, he is in court being sentenced for his crimes. Hear ye, hear ye. You are hereby sentenced to be executed by means of the ancient black box of mystery. “Black box of mystery? What’s that?” Lonnie asks. Nobody knows the ways of the black box of mystery. But what we know for sure is that the black box does exactly what we ask of it. At exactly noon on one weekday next week, it will emit a poison ending your life. No day shall be more likely than another, but as for what day it happens, nobody will know. It will come as a complete surprise to you. You will not know your execution date until the very moment you witness the poison leave the box. That I promise you. Lonnie’s jaw drops and he says: “Is that guy fur real?” “He’s not kitten me?” “He’s not lion?” ”Everything that has been said is true.” responds the judge. And Lonnie exclaims, “Thank you, thank you! Thank you!” *Surprised by Lonnie’s response, his lawyer asks him “Are you feeling alright?” And Lonnie responds “I’m feline purrfect.” Lonnie goes back to his chamber and explains his generous sentence to his chamber mate. I’ll be alive on Sunday and the black box will deliver a surprise poison on one of the five following days: Monday, Tuesday, Wednesday, Thursday or Friday. The days after I’m poisoned I’ll be dead The days before I’m poisoned I’ll be alive. These are the only five possible cases, right? Well, on Monday, I might be poisoned or I might not. Either outcome is possible so if it happens I’ll be surprised. The same can be said about Tuesday, Wednesday, and Thursday. But if I’m still alive on Thursday night, then there remains only one day to poison me: Friday. So I will know on Thursday night that I will be poisoned on Friday and so I won’t be surprised therefore I can’t be poisoned on Friday. Let’s rule out that option. But now, if I’m still alive on Wednesday night, then there remains only one day to poison me: Thursday. So I will know on Wednesday night that I will be poisoned on Thursday and so I won’t be surprised, therefore I can’t be poisoned on Thursday. Let’s rule out that option. But now, if I’m still alive on Tuesday night, then there remains only one day to poison me: Wednesday. So I will know on Tuesday night that I will be poisoned on Wednesday and so I won’t be surprised, therefore I can’t be poisoned on Wednesday. Let’s rule out that option. But now, if I’m still alive on Monday night, then there remains only one day to poison me: Tuesday. So I will know on Monday night that I will be poisoned on Tuesday and so I won’t be surprised, therefore I can’t be poisoned on Tuesday. Let’s rule out that option. But now, with only one remaining option, I know right now that I will be poisoned on Monday and so I won’t be surprised. So I can’t be poisoned on Monday. There is no case in which I will be surprised therefore there is no day which I can be poisoned. It’s the mice-est sentence I could have received. Well, as it happens, Wednesday comes around, and as Lonnie is napping in his chamber, he awakes to a loud hissing sound. He looks to find a cloud of hydrocyanic acid approach. He screams at the top of his lungs: “Help, help! This isn’t fair!” “Let meow-t right meow!” But they don’t. And with his last breath, he proclaims: “Me-owwwww!” He died that day. And he was surprised. Lonnie’s reasoning was on the right track, but he didn’t go far enough. The moment he became certain that he couldn’t be poisoned, he became vulnerable to a surprise poisoning, so he should have ruled all cases back in. But then following his original logic, he should have then ruled all cases back out, then back in, then back out, then in, then out, and so on to no end. But what should he think in the end? We’d need to take this reasoning to it’s limit and at the limit all cases are simultaneously ruled in and ruled out. At the limit this reasoning is inconsistent and so anything can be concluded. He can be certain of anything and simultaneously certain of nothing. And so when his execution day arrives, he will simultaneously be and not be surprised. So he would simultaneously be and not be executed. He’d simultaneously be dead and alive. But how is that possible? It’s not. If such contradictory states were allowed to actually exist, his universe would explode. These cases are inconsistent and this inconsistency is screaming at us, it’s telling us that we’re missing something. But what? This entire fictional story, Lonnie’s entire fictional universe exists somewhere. It exists in my mind. I’m making it up. And my goal is to allow it to exist from Sunday to Friday without my mind exploding. So stick with me here, as I attempt to construct his universe right now. Okay, so what do I know for sure? Well, on Sunday he is certainly alive and on Friday he is certainly dead. On the days in between, what I know for sure is his potential, the probability for him to be dead each day. On Monday, there is a 1 in 5 chance that he’s dead. On Tuesday, there is a 2 in 5 chance that he’s dead. On Wednesday there is a 3 in 5 chance, and on Thursday a 4 in 5 chance. That’s all I know. So why can’t that be a case? Now, I’m not saying that he’s alive and dead on Monday to Thursday. That’s absurd. I’m saying he’s neither. He has no actual state but instead the potential to be in one of multiple states. He jumps from an actual state of alive on Sunday over a cloud of potential states to an actual state of dead on Friday. And so, he could have been poisoned on any weekday. I don’t know. But this universe exists in my mind, and if I don’t know, nobody could know, including the cats who live in this fictional universe. And so the poison delivered by black box of mystery must come as a surprise as promised. But one might argue that I haven’t completely constructed his universe since in my construction, I haven’t given Lonnie an actual state on Monday, Tuesday, Wednesday, and Thursday. One might say that my construction is just an imperfect shadow of the true construction, and that if I had access to all of the information, including the “hidden variables”, I would be able to complete the construction and give Lonnie an actual state every day. NO. That’s impossible. That’s just case 4 and we’ve already determined that case 4 is inconsistent. These decoder glasses explode. Let’s stop and ask ourselves, why must everything be actualized? Why do we cling to this notion of completeness? What is more important to you: Completeness or consistency? To me, consistency is paramount. So I believe that my construction is the real deal. There is no truer construction. My incomplete construction describes everything about Lonnie’s universe and it’s consistent. But still that nagging question probably remains in your mind: is he actually alive or dead on Tuesday? My answer is that until I decide his state on Tuesday, until I make that measurement in my mind, he has no actual state on Tuesday. What he has is the potential to be in one of multiple states. He has a potential state. But if I want to give him an actual state on Tuesday, I would give him a 2 in 5 chance of being dead that day, so I pull the handle and test my luck… well, his luck. And in this case, I measure that he’s dead on Tuesday. The remaining potential states need to then be updated to reflect this new information. There’s now a 100% chance he’ll be dead on the days that follow, and a 50% chance he’ll be dead the day before. And this is another valid case. It’s valid because a potential state lies between the actual states of alive and dead, meaning that uncertainty of his execution date remains, and the element of surprise is preserved. If you’re into quantum mechanics, I’m sure that all of this all seems remarkably familiar and you understand why Lonnie’s brother is Schrödinger’s Cat. If not, check out the Minute Physics video on Schrödinger’s Cat, linked in the description, to see for yourself. The actual and the potential are opposites in the sense that they cannot both be simultaneously real. If Lonnie has an actual state, then there is no potential for his state to be otherwise. But if he has a potential to be in one of multiple states, then he must be in no actual state. And in Lonnie’s case, a potential state must lie in between distinct actual states for his story to remain consistent. Or in other words, the actual and the potential are both needed for Lonnie’s story to evolve. The actual and the potential are complementary. This is a demonstration of the Principle of Complementarity formulated by Neils Bohr in the context of quantum mechanics. For details on how this applies to quantum mechanics, check out the video by PBS Space Time linked in the description. But let’s go back to Lonnie’s story and have another attempt at jumping from Sunday to Tuesday. But this time, I’ll measure that he’s alive on Tuesday, so as before, I need to update the remaining potential states. And suddenly, with one measurement on Tuesday I now know with certainty that he must have also been alive on Monday. My measurement on Tuesday appears to have changed history. But the measurement hasn’t actually changed history, it’s made history. I had jumped straight from Sunday to Tuesday so there never were any actual moments in between. But when I update his state on Tuesday, I have additional information that allows me to construct a story of the past as if it actually did happen. In quantum mechanics, this is called delayed-choice, first proposed by Gilbert N. Lewis and later expanded upon by John Wheeler. But there’s more. I construct his universe, from start to finish, based on the information stored in my memory now, but if somehow I completely forgot about my last measurement, then it can no longer be considered in my construction of his universe and it will be as if that measurement never occurred. In quantum mechanics, this is called a quantum eraser. If you’re interested in learning about my favourite science experiment, the delayed-choice quantum eraser experiment, check out the videos by PBS Space Time and Eugene Khutoryansky linked in the description. In 1999, Yoon-Ho Kim and collaborators performed an experiment that showed that delayed-choice quantum erasers are real. And that’s INCREDIBLE. Truth is quantum. Oh ancient black box of mystery, show me the creator of this universe. Oh goodness, the great one is more beautiful than I imagined. Ohh, an itch has he? Perhaps a booger. It appears to be an immovable booger but it meets an unstoppable finger. What shall be the fate of that immovable booger? Oh, I just threw up a little. In the next video we look at the Liar’s Paradox. This is one of the oldest and most famous paradoxes of all time. I’ve mentioned it in previous videos but in this video I propose a solution to it. Check it out. And if you like this video, show your support. Click that like button, click that subscribe button because, honestly, every click counts. Thank you.

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3 Responses

  1. The Weird Math Guy says:

    RYAN’S NOTES

    Please keep in mind that this series is composed of short and raw videos containing many original ideas. The videos are not textbooks. Let’s use these videos as a starting point for a conversation. Whether you agree or disagree with the ideas in the videos, leave your thoughts in the comments and we’ll talk it through.

    I will be posting comments, corrections, and FAQs here once I get some feedback on this video.

  2. Tyler W says:

    cool video it was really good

  3. Mikko Pulkkinen says:

    This argument cannot be correct. In your construction you claim there is a 1/5 chance of being alive after noon on Thursday. That contradicts the premise which is that the execution will come as a surprise, because in those 1/5 of cases the execution on Friday would not be a surprise. That is the heart of the issue and cannot be sidestepped by thinking probabilistically or drawing quantum mechanical analogies.

    The same mistake occurs even earlier when you are talking about "ruling in" the cases again. The case of living on Thursday afternoon cannot ever be ruled in.

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