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Intrism comments on Harry Potter and the Methods of Rationality discussion thread, part 19, chapter 88-89 - Less Wrong
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Hmm. You're right, that particular oddity ''did'' work. Has anyone here noticed conclusive rules about when a paradox will and will not form?
... Anyhow, though, if a Time-Turner could be used without paradox, it wouldn't be here. When Hermione was missing, it would have been very, very easy to make her missing somewhere safe, without changing anyone's observations. The fact that we got to the death scene without that happening, or any other visible Time-Turner intervention, tells me that it probably can't happen.
It's not clear what happens when there are multiple possible stable time loops. For example, why he ended up with a paper that says "DON'T MESS WITH TIME TRAVEL" instead of the semiprime factored, even though either would work. However, it can't do anything that results in a paradox. If Harry's going to go back and try to save Hermione whether or not he sees his future self save her, there is conceivable some loophole in which she is not saved, but it seems much more likely she'd just get saved.
This isn't like trying to factor a semiprime. This is like trying to write a specific phrase and send it back. You can mess it up, and if you don't get anything back you know you did, but it doesn't seem very plausible.
Because if he'd gotten back a number, he would have kept pushing until he did something that broke time, thereby causing a paradox. The only way to head off paradox was to cut his line of inquiry off then and there.
One of the established rules was that information can't be sent back more than six hours. Would he have broken time within six hours?
Also, there is always a way to head off a paradox. Consider the multiple universes model. What came out of the time machine in one universe is a function of what came out in the last universe. Furthermore, it's a continuous function. Theoretically, you could do something like send back what came out plus a mote of dust, but I think it can be assumed that there's a limit to whatever the time turner can bring back, meaning the domain of the function is bounded. Since every continuous function mapping a bounded set into itself has at least one fixed point, there must be at least one stable time loop.
Knowing HJPEV, yes, he would have broken time within six hours. He wouldn't have even had much trouble doing it.
And yes, there's a stable time loop. It's the one where he writes "DO NOT MESS WITH TIME".
He already went back in time once. He knows he's not supposed to go back more than two hours a day, and while he breaks that rule quite often, he wouldn't do it for something as minor as not wanting to wait until tomorrow.
If he was going to use the time turner again, he'd have just only gone a half-hour back from the beginning.
Addendum: Why would it be a continuous function? Human decisions can be binary. Consider, as a trivial example, "If I see 1 on the paper, I'll write 0 and send it back in time. If I see anything else on the paper, I'll write 1 and send it back in time".
Humans are made of atoms. Atoms behave continuously. Therefore, human decisions are continuous. A human can no more follow the algorithm you gave than Buridan's ass can follow the algorithm of "Eat the bigger bale of hay first, or the one on the left if you can't tell which is bigger".
There is the whole quanta thing with quantum physics, but when you get to that point there are bigger problems. Which future does the Time Turner come from? How can you have single universe time travel when the physics you're using already established more than one universe?
I think this is the compositional fallacy: Humans are made of atoms. Atoms are smaller than pennies. Therefore, humans are smaller than pennies.
I don't mean that your conclusion is false, just that your argument is (as read) invalid.
Let me rephrase that.
Humans are systems of atoms. Systems of atoms behave continuously. Therefore, human decisions are continuous.
It's the same principle that lets me conclude that any machine you make obeys the law of conservation of energy when all I know is that it's made of atoms.
Lots of things are discreet to within measurement error. Flip a coin. Its heads or its tails or you flip again. There is some (not exactly zero) probability the coin along with many other atoms spontaneously reconfigure into a Velociraptor and eat you. But in practice its binary.
What measurement error? If there was some sort of god that was not quite omniscient that tried to find a self-consistent timeline, then it would have a hard time. To my knowledge, there's no way that god could generally even be sure any given point is anywhere near a stable time loop, no matter how precise its measurement. I don't get the impression there is such a god. It's just that one of the self-consistent timelines happen. The universe has a set of laws of physics and a set of initial conditions, and some timeline where all that applies happens.
What does "behave continuously" mean?
It means that a sufficiently small change in the initial value will cause an arbitrarily small change in the final value. This is true for real life systems, but if you had a truly binary system, it wouldn't be. For example, if you have a switch that conducts electricity if it's flipped more than half way, and you flip it exactly half way, then if you flip it any more, no matter how little you move it, it will instantly go from being an insulator to being a conductor.