The Most Controversial Problem in Philosophy

Don't hit the like button! Or the dislike button, at least not yet. I want you to consider a

problem

that has been one of the

most

controversial

in mathematics and

philosophy

for the last 20 years. There is no agreed upon answer. So I want you to listen to the

problem

and then vote for the answer you prefer using the Like and Dislike buttons. Well, here's the setup: Sleeping Beauty volunteers to be the subject of an experiment. And before starting she is informed of the procedure. On Sunday night they will put her to sleep. And then a fair coin will be tossed.

If that coin comes up heads, they will wake her up on Monday and then put her back to sleep. If the coin comes up tails, they will also wake her up on Monday and put her back to sleep. But then they will also wake her up on Tuesday and then put her back to sleep. Now, every time they put her back to sleep she will forget that they ever woke her up. In the brief period in which she is awake she will not be told any information. But she will be asked a question: What do you think is the probability that the coin will come up heads?

So how should she respond? Feel free to pause the video and answer the question yourself right now. This was my reaction after hearing the problem for the first time. I mean, the intuitive answer that comes to mind is clearly one in three. It could be Monday when it comes up heads. Or it could be Monday when Tails showed up. Or it could be Tuesday when Tails showed up. But what's really interesting is that you just answered that the probability of a coin coming up heads is one-third. I think a lot of this comes down to the specific question being asked.

What is the probability that tossing a fair coin will come up heads? That's 50 percent. What is the probability that the coin comes up heads? I would say the answer is a third from your perspective. Yes, it's surprisingly the same question. The simple reason Sleeping Beauty should say that the probability of her coming up heads is half is because she knows the coin is fair. Nothing changes between the moment the coin is tossed and the moment she wakes up and she knew for a fact that she would be woken up and she receives no new information when that happens.

Imagine that instead of tossing the coin after she is asleep, the experimenters toss it first and immediately ask her, "What is the probability that the coin comes up heads?" Well, she would certainly say half of it, so why should anything change after she goes to sleep and wakes up? This is known as the Halfer position. But there is another way to look at it. Others would say that something changes when she wakes up. I mean, she doesn't seem to receive any new information. There are no calendars, no one tells her anything and she knew they would wake her up.

But she actually learns something important. She learns that she has gone from existing to a reality where there are two possible states. The currency came into existence in a reality where there are three possible states: Monday heads, Monday tails or Tuesday tails. And therefore she should assign the same probability to each of these three outcomes where she only came up heads in one. So the probability of the coin coming up heads is one-third. This is known as the third position. I know it seems wrong to suggest that a fair coin should have a one-third chance of coming up heads, but that's because the question being asked is subtly different.

The implicit question is: "If you are awake, what is the probability that the coin comes up heads?" And that's a third. Now the midfielders would respond that just because there are three possible outcomes does not mean that each of them is equally probable. In the Monty Hall problem, for example, the contestant ultimately has to choose between two doors. But it would be a mistake to assign them 50/50 odds. In reality, the prize is twice as likely to be behind one door as the other. In the Sleeping Beauty problem we know that a heads and tails outcome are equally probable.

So the probability of waking up on Monday with a head is 50 percent and the probability of waking up on Monday or Tuesday with a tail should be 50 percent. Therefore, Tails' probability is divided into two days at 25 percent each. but if you repeat the experiment over and over again, which you can try for yourself by tossing a coin repeatedly, you will find that she wakes up one third of the time on Monday heads, one third of the time on Monday tails, and one third of the time on Monday tails. Tuesdays. tails, not 50-25-25 as the above analysis would suggest.

So if you were Sleeping Beauty and you were woken up and asked "What is the probability that the coin comes up heads?" What would you say? If you say a third, hit the like button. If you say half, press the Dislike button. The answer may seem obvious to you, but you should know that to other people the other answer seems equally obvious. And that is why over the last 22 years hundreds and hundreds of

philosophy

articles have been published on this problem. There have been many variations of this problem, such as what if instead of waking up twice if the coin lands tails, you woke up a million times?

If the coin comes up heads, it will wake up only once. Doesn't it seem absurd in this case, when Sleeping Beauty wakes up, to say that a coin was as likely to come up heads as tails? When we know that there are a million more awakenings in the case of tails than in the case of faces. I mean, if you reach into a bag containing one white marble and a million black marbles, what are the chances that you will pull out that white marble? This convinced me quite a bit and I considered myself a third party, but this same argument is used to convince people that we live in a simulation.

The idea is that our computing technology has improved so dramatically, even in the last 40 years, that we can imagine a time in the not-too-distant future when we can create a completely realistic simulation of our world. And once that happens, it should be trivial to make unlimited copies of that simulation. And then if you asked someone if they were living in a simulation, they would have to admit that they probably are because there are many more instances of that existence than there is the one true external reality. But how do we know that this has not already happened and that we live inside a simulation?

I mean, if it can happen, then it probably has happened and we're living in a simulation. This seems to be the logical conclusion of the third world view. Now, I personally don't think I'm living in a simulation and I think

most

people don't. But maybe that's just illogical prejudice. But there is another thought experiment that makes me seriously reconsider the third position. Let's say there is a soccer match between a great team like Brazil and a less dominant world team like Canada. So the odds are 80:20 in Brazil's favor now that an investigator will put you to sleep before the game starts.

And if Brazil wins, they will wake you up once. But if Canada wins, you'll be woken up 30 times in a row and, like Sleeping Beauty, you won't remember if you've been woken up before. Okay, so the game is about to start, you fall asleep... and now you're awake. Who do you think won the game? The third would say Canada, but I would almost certainly say Brazil. I mean, why should I make a big deal about what the researcher would have done if Canada had one when I'm pretty sure it won't? To expand on this, let's say Brazil plays Canada five times and we do this experiment each time.

Okay, so if you say Brazil every time you wake up, you'll probably get four out of five games right. but if you said Canada every time, you'd be wrong about those four games, but you'd be right 30 times in a row when asked repeatedly about Canada's only win. If you can win a bet by answering the question correctly, then you should bet in Canada. But if you want to correctly pick the winner of more games, then you should say Brazil. And this is what is at the heart of the dispute between Halves and Thirds in the Sleeping Beauty problem.

If you want to be right about the outcome of the coin toss, you should say that the probability of it coming up heads is one-half, but if you want to answer more questions correctly, then you should say one-third. I want to leave you with one last thought experiment. Imagine that you know with certainty that before our universe began, a coin was tossed in the air and if it came up heads, only one Universe would be created. But if Tails arose, a nearly infinite Multiverse would be created and in each of those Multiverse universes you would find every possible variation of the Earth and the people on it.

In some versions there would be no Earth. Now becoming conscious is like Sleeping Beauty waking up. There is no way to know if you are in that one universe or one of the universes in the Multiverse, but you know that there are many more. So, you'd think you're safe in the Multiverse? Or are the chances 50/50? The best way to develop intuition about probability is to work on scenarios or run simulations like we did for Sleeping Beauty. Brilliant, the sponsor of this video, offers probability courses that will guide you through many situational probability questions. Do you want to learn how to create your own simulations?

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