This post arose when I was pondering the Bible and how easy it is to justify. In the process of writing it, I think I’ve answered the question for myself. Here it is anyway, for the sake of discussion.
Suppose that there’s a world very much like this one, except that it doesn’t have the religions we know. Instead, there’s a book, titled The Omega-Delta Project, that has been around in its current form for hundreds of years. This is known because a hundreds-of-years-old copy of it happens to exist; it has been carefully and precisely compared to other copies of the book, and they’re all identical. It would be unreasonable, given the evidence, to suspect that it had been changed recently. This book is notable because it happens to be very well-written and interesting, and scholars agree it’s much better than anything Shakespeare ever wrote.
This book also happens to contain 2,000 prophecies. 500 of them are very precise predictions of things that will happen in the year 2011; none of these prophecies could possibly be self-fulfilling, because they’re all things that the human race could not bring about voluntarily (e.g. the discovery of a particular artifact, or the birth of a child under very specific circumstances). All of these 500 prophecies are relatively mundane, everyday sorts of things. The remaining 1,500 prophecies are predictions of things that will happen in the year 2021; unlike the first 500, these prophecies predict Book-of-Revelations-esque, magical things that could never happen in the world as we know it, essentially consisting of some sort of supreme being revealing that the world is actually entirely different from how we thought it was.
The year 2011 comes, and every single one of the 500 prophecies comes true. What is the probability that every single one of the remaining 1,500 prophecies will also come true?
Let P(2,000) be the probability that all 2,000 prophecies come true, and P(500) be the probability that the initial 500 all come true. Suppose P(2,000) = 2^(-2000) and P(500) = 2^(-500). We know that P(500|2,000) = 1, so P(2,000|500) = P(2,000)*P(500|2,000)/P(500) = 2^(-2000)*1/2^(-500) = 2^(-1500). A probability of 2^(-1500) is not pretty darned high, so either P(2,000) is much greater than we supposed, or P(500) is much lower than we supposed. The latter is counterintuitive; one wouldn’t expect the Believable Bible’s existence to be strong evidence against the first 500 prophecies.
And this doesn’t depend on prophecies in particular. Any claims made by the religion will do. For example, the same sort of argument would show that according to our subjective probabilities, all the various claims of a religion should be tightly intertwined. Suppose (admittedly an extremely difficult supposition) we discovered it to be a fact that 75 million years ago, an alien named Xeno brought billions of his fellow aliens to earth and killed them with hydrogen bombs. Our subjective probability that Scientology is a true religion would immediately jump (relatively) high. So one’s prior for the truth of Scientology can’t be anywhere near as low as one would think if one simply assigned an exponentially low probability based on the complexity of the religion. Likewise, for very similar reasons, komponisto’s claim elsewhere that Christianity is less likely to be true than that a statue would move its hand by quantum mechanical chance events, is simply ridiculous.
So one’s prior for the truth of Scientology can’t be anywhere near as low as one would think if one simply assigned an exponentially low probability based on the complexity of the religion.
If nobody had ever proposed Scientology, though, learning Xenu existed wouldn’t increase our probabilities for most other claims that happen to be Scientological. So it seems to me that our prior can be that low (to the extent that Scientological claims are naturally independent of each other), but our posterior conditioning on Scientology having been proposed can’t.
In proportion to the complexity of the claim given that humans exist, which is much lower (=> higher prior) than its complexity in a simple encoding, since Scientology is the sort of thing that a human would be likely to propose.
The prior for “Scientology is proposed” is higher than the simple complexity prior of the claim, to the (considerable) extent that Scientology is the sort of thing a human would make up.
You’ve got it a little backward, I think. The fact that someone makes a particular set of prophesies does not make those things more likely to occur. In fact, the chances of the whole thing happening… the events prophesied and the prophesies themselves is much lower than one or the other happening by themselves. This means that if some of the prophesies start coming true the probability the other prophesies come true goes up pretty fast. But predicted magic is even less likely than magic.
Where A = “events occur” and B = “events are predicted”, you’re saying P(A and B) < P(A). Warrigal is saying it would be counterintuitive if P(A|B) < P(B).
Where A = “events occur” and B= “events are prophesied” and C = “the events prophesied come true” I am saying that when the events in A= the events in B, P(A|B) < P(B) or P(A) because A ^ B entails C.
Could be simple time travel, though. AFAICT time travel isn’t per se incompatible with the way we think the world works. Not to the degree sufficiently fantastic prophecies might be at least.
If someone just observed events in 2011 and planted a book describing them in 1200, the 2011 resulting from the history where the book existed would be different from the 2011 he observed.
I think the important bit here is that even if you could just “play time backwards” and watch again, there’s no reason to think you’d end up in the same Everett branch the next time around.
Makes perfect sense to me if you assume a single time-line. (This might be a big assumption, but probably less big than the truth of sufficiently strange prophecies.) You can think of this time line as having stabilized after a very long sequence of attempts at backward time travel under slightly different conditions. Any attempt at backward time travel that changes its initial conditions means a different or no attempt at time travel happens instead. Eventually you end with a time-line where all attempts at backward time travel exactly reproduce their initial conditions.
We know that we live in that stabilized time-line because we exist (though the details of this timeline depend on how people who don’t exist, but would have thought they exist for the same reasons we think we exist, would have acted, had they existed).
By the way, that sort of time-travel gives rise to Newcomb-like problems:
Suppose you have access to a time-machine and want to cheat on a really important exam (or make a fortune on the stock marked or save the world or whatever. The cheating example is the simplest). You decide to send yourself at a particular time a list with the questions after taking the exam. If you don’t find the list at the time you decided you know that somehow your attempt at sending the list failed (you changed your mind, the machine exploded in a spectacular fashion, you were caught attempting to send the list …).
But if you now change your mind and don’t try to send the list there never was any possibility of receiving the list in the first place! The only way to get the list if for you to try to send the list even if you already know you will fail, so that’s what you have to do if you really want to cheat. And if you really would do that, and only then, you will probably get the list at the specified time and never have to do it without knowing you succeed, but only if your pre-commitment is strong enough to even do it in the face of failure.
And if you would send yourself back useful information at other times even without having either received the information yourself or pre-commited to sending that particular information you will probably receive that sort of information.
Why was this post voted back down to 0 after having been at 2?
Newcomb-like problems are on-topic for this site and I would think having examples of such problems in a scenario not specifically constructed for them is a good thing? If it was because time travel is off topic wouldn’t the more prudent thing have been voting down the parent? The same if the time travel mechanics are considered incoherent (though I’d be really interested in learning why?) . If you think this post doesn’t actually describe anything Newcomb-like I would like to know why. Maybe I misunderstood the point of earlier examples here, or maybe I didn’t explain things sufficiently? Or is it just that the post was written badly? I’m not really happy with it, but I don’t see how I could have made it much clearer.
It’s an interesting point. It actually came up in the most recent Artemis Fowl novel, when he managed to ‘precommit’ himself out of a locked trunk in a car. :)
Anyone who can travel through time can mount a pretty impressive apocalypse and announce whatever it is about the nature of reality he cares to. He might even be telling the truth.
For the two examples of the mundane prophecies that you gave it seems possible some on-going conspiracy could have made them true… but it sounds like you’re trying to rule that out.
I did mean those to be positive examples. There’s no way we can guarantee that we’ll discover an ancient Greek goblet that says “I love this goblet!” on March 22, 2011. There’s also no way we can guarantee that a woman born on October 15, 1985 at 5 in the morning in room 203 of a certain hospital will have a baby weighing 8 pounds and 6 ounces on January 8, 2011 at 6 in the afternoon in room 117 of a certain other hospital.
The Believable Bible
This post arose when I was pondering the Bible and how easy it is to justify. In the process of writing it, I think I’ve answered the question for myself. Here it is anyway, for the sake of discussion.
Suppose that there’s a world very much like this one, except that it doesn’t have the religions we know. Instead, there’s a book, titled The Omega-Delta Project, that has been around in its current form for hundreds of years. This is known because a hundreds-of-years-old copy of it happens to exist; it has been carefully and precisely compared to other copies of the book, and they’re all identical. It would be unreasonable, given the evidence, to suspect that it had been changed recently. This book is notable because it happens to be very well-written and interesting, and scholars agree it’s much better than anything Shakespeare ever wrote.
This book also happens to contain 2,000 prophecies. 500 of them are very precise predictions of things that will happen in the year 2011; none of these prophecies could possibly be self-fulfilling, because they’re all things that the human race could not bring about voluntarily (e.g. the discovery of a particular artifact, or the birth of a child under very specific circumstances). All of these 500 prophecies are relatively mundane, everyday sorts of things. The remaining 1,500 prophecies are predictions of things that will happen in the year 2021; unlike the first 500, these prophecies predict Book-of-Revelations-esque, magical things that could never happen in the world as we know it, essentially consisting of some sort of supreme being revealing that the world is actually entirely different from how we thought it was.
The year 2011 comes, and every single one of the 500 prophecies comes true. What is the probability that every single one of the remaining 1,500 prophecies will also come true?
Pretty darned high, because at this point we already know that the world doesn’t work the way we think it did.
So it sounds like even though there are 2,000 separate prophecies, the probability of every prophecy coming true is much greater than 2^(-2000).
Maybe you just need to explain more but I don’t see that.
Let P(2,000) be the probability that all 2,000 prophecies come true, and P(500) be the probability that the initial 500 all come true. Suppose P(2,000) = 2^(-2000) and P(500) = 2^(-500). We know that P(500|2,000) = 1, so P(2,000|500) = P(2,000)*P(500|2,000)/P(500) = 2^(-2000)*1/2^(-500) = 2^(-1500). A probability of 2^(-1500) is not pretty darned high, so either P(2,000) is much greater than we supposed, or P(500) is much lower than we supposed. The latter is counterintuitive; one wouldn’t expect the Believable Bible’s existence to be strong evidence against the first 500 prophecies.
And this doesn’t depend on prophecies in particular. Any claims made by the religion will do. For example, the same sort of argument would show that according to our subjective probabilities, all the various claims of a religion should be tightly intertwined. Suppose (admittedly an extremely difficult supposition) we discovered it to be a fact that 75 million years ago, an alien named Xeno brought billions of his fellow aliens to earth and killed them with hydrogen bombs. Our subjective probability that Scientology is a true religion would immediately jump (relatively) high. So one’s prior for the truth of Scientology can’t be anywhere near as low as one would think if one simply assigned an exponentially low probability based on the complexity of the religion. Likewise, for very similar reasons, komponisto’s claim elsewhere that Christianity is less likely to be true than that a statue would move its hand by quantum mechanical chance events, is simply ridiculous.
If nobody had ever proposed Scientology, though, learning Xenu existed wouldn’t increase our probabilities for most other claims that happen to be Scientological. So it seems to me that our prior can be that low (to the extent that Scientological claims are naturally independent of each other), but our posterior conditioning on Scientology having been proposed can’t.
Right, because that “Scientology is proposed” has itself an extremely low prior, namely in proportion to the complexity of the claim.
In proportion to the complexity of the claim given that humans exist, which is much lower (=> higher prior) than its complexity in a simple encoding, since Scientology is the sort of thing that a human would be likely to propose.
The prior for “Scientology is proposed” is higher than the simple complexity prior of the claim, to the (considerable) extent that Scientology is the sort of thing a human would make up.
You’ve got it a little backward, I think. The fact that someone makes a particular set of prophesies does not make those things more likely to occur. In fact, the chances of the whole thing happening… the events prophesied and the prophesies themselves is much lower than one or the other happening by themselves. This means that if some of the prophesies start coming true the probability the other prophesies come true goes up pretty fast. But predicted magic is even less likely than magic.
Use \* to get stars * instead of italics.
Oops! It seems I assumed everything would come out right instead of checking after I posted.
Edit: Yeah, I was being dumb.
Where A = “events occur” and B = “events are predicted”, you’re saying P(A and B) < P(A). Warrigal is saying it would be counterintuitive if P(A|B) < P(B).
Where A = “events occur” and B= “events are prophesied” and C = “the events prophesied come true” I am saying that when the events in A= the events in B, P(A|B) < P(B) or P(A) because A ^ B entails C.
You’re talking about P(A and B). Warrigal is talking about P(A|B).
But not necessarily over .99, since the prophecies could have been altered by another author sometime before the beginning of modern records.
Could be simple time travel, though. AFAICT time travel isn’t per se incompatible with the way we think the world works. Not to the degree sufficiently fantastic prophecies might be at least.
If someone just observed events in 2011 and planted a book describing them in 1200, the 2011 resulting from the history where the book existed would be different from the 2011 he observed.
Depends if it’s type one time travel. Fictional examples: Twelve Monkeys, The Hundred Light-Year Diary.
I think the important bit here is that even if you could just “play time backwards” and watch again, there’s no reason to think you’d end up in the same Everett branch the next time around.
Insofar as I understand that page, that would mean that the world worked even less the way we thought it did.
Makes perfect sense to me if you assume a single time-line. (This might be a big assumption, but probably less big than the truth of sufficiently strange prophecies.) You can think of this time line as having stabilized after a very long sequence of attempts at backward time travel under slightly different conditions. Any attempt at backward time travel that changes its initial conditions means a different or no attempt at time travel happens instead. Eventually you end with a time-line where all attempts at backward time travel exactly reproduce their initial conditions. We know that we live in that stabilized time-line because we exist (though the details of this timeline depend on how people who don’t exist, but would have thought they exist for the same reasons we think we exist, would have acted, had they existed).
By the way, that sort of time-travel gives rise to Newcomb-like problems:
Suppose you have access to a time-machine and want to cheat on a really important exam (or make a fortune on the stock marked or save the world or whatever. The cheating example is the simplest). You decide to send yourself at a particular time a list with the questions after taking the exam. If you don’t find the list at the time you decided you know that somehow your attempt at sending the list failed (you changed your mind, the machine exploded in a spectacular fashion, you were caught attempting to send the list …). But if you now change your mind and don’t try to send the list there never was any possibility of receiving the list in the first place! The only way to get the list if for you to try to send the list even if you already know you will fail, so that’s what you have to do if you really want to cheat. And if you really would do that, and only then, you will probably get the list at the specified time and never have to do it without knowing you succeed, but only if your pre-commitment is strong enough to even do it in the face of failure.
And if you would send yourself back useful information at other times even without having either received the information yourself or pre-commited to sending that particular information you will probably receive that sort of information.
Why was this post voted back down to 0 after having been at 2? Newcomb-like problems are on-topic for this site and I would think having examples of such problems in a scenario not specifically constructed for them is a good thing? If it was because time travel is off topic wouldn’t the more prudent thing have been voting down the parent? The same if the time travel mechanics are considered incoherent (though I’d be really interested in learning why?) . If you think this post doesn’t actually describe anything Newcomb-like I would like to know why. Maybe I misunderstood the point of earlier examples here, or maybe I didn’t explain things sufficiently? Or is it just that the post was written badly? I’m not really happy with it, but I don’t see how I could have made it much clearer.
It’s an interesting point. It actually came up in the most recent Artemis Fowl novel, when he managed to ‘precommit’ himself out of a locked trunk in a car. :)
Anyone who can travel through time can mount a pretty impressive apocalypse and announce whatever it is about the nature of reality he cares to. He might even be telling the truth.
For the two examples of the mundane prophecies that you gave it seems possible some on-going conspiracy could have made them true… but it sounds like you’re trying to rule that out.
I understood those to be negative examples, in that the actual prophecies don’t share that characteristic with those examples.
I did mean those to be positive examples. There’s no way we can guarantee that we’ll discover an ancient Greek goblet that says “I love this goblet!” on March 22, 2011. There’s also no way we can guarantee that a woman born on October 15, 1985 at 5 in the morning in room 203 of a certain hospital will have a baby weighing 8 pounds and 6 ounces on January 8, 2011 at 6 in the afternoon in room 117 of a certain other hospital.
That’s not clear to me, but I acknowledge that it doesn’t affect the original question.