The problem here is that even if Scenario A and Scenario B are indistinguishable, Amelia's words still constitute Bayesian evidence on which Dumbledore can update his beliefs.
I'm inclined to believe that whatever intelligence is behind capital-T Time is enforcing an intuitive definition of information, in the same way that brooms work off of Aristotelian mechanics.
The problem here is that even if Scenario A and Scenario B are indistinguishable, Amelia's words still constitute Bayesian evidence on which Dumbledore can update his beliefs.
In my formulation, that's "side information." Really, my gedankenexperiment doesn't work unless Amelia Bones happens to be very ditzy concerning time.
I'm inclined to believe that whatever intelligence is behind capital-T Time is enforcing an intuitive definition of information, in the same way that brooms work off of Aristotelian mechanics.
So then, this is a limitation in the "interface" that the Atlantean engine is following. I think my hypothesis is testable.
Given no other information to strictly verify, any supposed time-traveled conversation is indistinguishable from someone not having time-traveled at all and making the information up. The true rule must depend on the actual truth of information acquired, and the actual time such information came from. Otherwise, the rule is inconsistent. It also looks at whether your use of time travel actually involves conveying the information you gained; whether such information is actually transferred to the past, not merely whether it could be. Knowing that Amelia Bones has some information about 4 hours in the future will only restrict your time travel if you would transmit that information to the past - if you would act significantly differently knowing that than you would have otherwise. If you act the same either way, then you are not conveying information.
In short, the rule is that you cannot convey information more than 6 hours into the information's relative past, but that does not necessarily mean that you cannot go to a forbidden part of the past after learning it. It merely means that you cannot change your mind about doing so after learning it. Worth noting: if you plan on going to the past, and then receive some information from 6 hours in the future that changes your mind, you have conveyed information to the past. I'm not sure how that is handled, other than that the laws of the universe are structured as to never allow it to happen.
Your formulation of "indistinguishable" was already invalidated on reddit.com/r/hpmor by a different objection to my hypothesis. When you lie, you leak information. That information just puts the situation into the 6-hour rule. This cuts off the rest of your reasoning below. It also shows how hard the 6-hour rule is to "fool," which in turn explains why it hasn't been figured out yet.
EDIT: Rewrote one sentence to put the normal 6-hour rule back.
EDIT: Basically, if all of the information Dumbledore can receive from Amelia Bones could logically come from her departing anywhere between time X and time Y, then the metadata available to Dumbledore is effectively that, "Amelia Bones came from anywhere between time X and time Y."
In short, the rule is that you cannot convey information more than 6 hours into the information's relative past, but that does not necessarily mean that you cannot go to a forbidden part of the past after learning it. It merely means that you cannot change your mind about doing so after learning it. Worth noting: if you plan on going to the past, and then receive some information from 6 hours in the future that changes your mind, you have conveyed information to the past. I'm not sure how that is handled, other than that the laws of the universe are structured as to never allow it to happen.
I suspect my actual formulation (not your slight misread of it) and yours come out to much the same.
All information is probabilistic, Bayesian. Two scenarios, A & B, may have identical effects, but if the relevant probability ratios are different, then observing this effect can still give you information. If Amelia Bones tells Dumbledore that she's come from six hours in the future, then his objective Bayesian probability (given the information that he possesses) that she'll survive the next six hours goes up, even though it doesn't rise to 100%. And all information is like this; objective Bayesian probability is never quite 100%.
Does Dumbledore use Bayesian reasoning, or is there perhaps something about which he is completely certain?
I think that we can take it for granted that Dumbledore is not a rational Bayesian. (Nobody is perfectly, but Dumbledore is not even close.) Arguments about a perfect Bayesian tell us what information is present, not what any particular person will actually believe.
But if time turners work based on what information enters the actual beliefs of actual people, rather than on what information is theoretically present, then that makes a big difference. If scenarios A & B are indistinguishable, that wouldn't matter; what would matter is which scenario Dumbledore believes is happening.
If time turners work at the level of the information available through fundamental physics, then it will be impossible to have more than six hours in any overlapping chain of time turners, even ones on opposite sides of the world whose users never communicate in a human way. But probably, they work on something much more like what people actually believe. So your question is actually a very pertinent one.
Elsewhere I note that it is not likely that time-turners would use information-theoretic meanings of 'information' to prevent paradoxes.
I think it's most likely that time-turners simply cannot create causal chains where the effect precedes the cause by more than six hours. (Is it exactly one-hour increments? Which 'hour' exactly, is it- 1/24th of a mean solar day, or is it 33093474372000 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom at rest and 0K?)
it is not likely that time-turners would use information-theoretic meanings of 'information' to prevent paradoxes
Ultimately, I agree. The characters say that you can't send information back more than six hours, but that's because of the limits of their understanding. Otherwise things are too rigid. But this thread is basically premised on taking the characters' understanding at face value.
it's most likely that time-turners simply cannot create causal chains where the effect precedes the cause by more than six hours
If you're using an information-theoretic definition of ‘causal chain’, then this is no different!
I can't believe that we're using the phrase "information-theoretic definition of ‘causal chain’" in a discussion about magic...
Although !Harry probably would, if he was willing to try further experimentation.
All information is probabilistic, Bayesian.
Is there a rigorous argument for this, or is this just a very powerful way of modeling the world?
In discussions here (ETA: meaning, in the Less Wrong community), I mostly take it for granted that people have adopted the Bayesian perspective promoted in Eliezer's sequences. I think that one can make a pretty good argument (although mathematical rigour is too much to ask for) that receiving information through one's senses can never be enough to justify absolute certainty about anything external. But I'd rather not try to make it here (ETA: meaning, in this discussion thread).
It's more that Bayesian Analysis is a technique you can apply on anything, and under certain conditions is useful.
Consider Scenario C, in which Amelia Bones travels back in time, does not have additional information about it, and tells Dumbledore. This is distinguishable from Scenarios A and B, and this possibility is eliminated. It still conveys some information. All you've shown is that Dumbledore did not have all the information. There are still two possible scenarios. This would be true if he was told about everything except the path of a single neutrino. If that's all that's necessary to keep time turners from breaking, how could they have possibly figured out they break?
I don't think the path of a single neutrino could do it. Answer this, from the informational POV of Dumbledore's location in space-time, is path P of that neutrino any less consistent with Scenario A or Scenario B?
I don't think you quite get what I'm saying.
You have given two scenarios that Dumbledore cannot distinguish. This proves that he has incomplete information. It does not prove that he has no infomation. That would require that he be unable to distinguish any two scenarios.
Imagine that Amelia Bones tells Dumbledore everything that happened up until her departure from the future, except the path of some neutrino. Furthermore, thanks to Dubledore's legillimancy, he knows she's telling the truth. Can he distinguish between Scenario C, what actually happened, and Scenario C', which is just like C, except that the neutrino went left instead of right?
Is the inability to distinguish C and C' enough for Dumbledore to be able to go back another six hours? If not, how is distinguishing C and C' different from distinguishing A and B?
No, because if she was able to provide that much information as a conscious communication, she will have provided enough information to have affixed her departure at a specific time.
In any case, there's probably some reason that would make it impossible for her to convey that much information inside 6 hours, anyhow.
How about this:
Scenario A: Amelia Bones comes back from six hours in the future and provides large amounts of evidence of this fact, and of what happens.
Scenario B: Due to quantum randomness, a large number of particles happen to jump into the spot to create a clone of Amelia Bones who believes she is from the future and carries evidence of this.
It is, again, impossible for Dumbledore to tell which of these situations happens. Yet the time turner does not work.
It's very possible for to distinguish the two situations. The same probabilistic mechanism that determines the arrow of time precludes scenario B. Also, it's not really that Dumbledore is actually doing the distinction. It's more if he could do it.
It doesn't preclude scenario B. It just makes it unlikely. The same could be said about the original scenario A. It's possible that Amelia Bones was mistaken about when she came back, but it's unlikely. The probability is more extreme, but the information is still there.
It doesn't preclude scenario B. It just makes it unlikely.
I have a "Many Worlds/QM" style interpretation of time turner mechanics. Basically, all of the possible interpretations of the information+metainformation you have transmitted via time turner "exists" or is in a kind of superposition, until receiving information precludes them. Making Scenario B overwhelmingly unlikely is precluding it.
I am going to have to accuse you of making a grave Mind Projection.
Physics, to be believable should never talk about "information." I know from study that most of modern physics have this property, and I know EY knows from the QM sequence and the Epistemology 101 sequence.
Special and General Relativity might talk about "observers" and "observables," but these are very distinct from the "information" discussed in Bayesian Stats.
Bayesian stats (i.e. non-omniscient Agents) is the only place you are ever allowed to talk about "information," (Thermodynamics is applying agents to physics).
An observable in relativity is usually taken to mean something you can slap a unit on and call it a day. Mass of bodies, relative velocities, energy, etc.
I am going to have to accuse you of making a grave Mind Projection
Apparently Black Holes preserve information. There are other connections to physics and information theory, Such as the theoretical computers that can use ever smaller quantities of energy, so long as all of their operations are reversible. Given that, it doesn't seem unreasonable that there would be an information theoretic component to the rules of magic. My formulation doesn't require a human mind. If I talk about minds or arbiters, or use language suggesting that then that's just lazy writing on my part.
The most obvious instance of Information Theory/Bayesian Statisics overlapping physics is Thermodynamics and Statistical Mechanics, which both deal with the notion of Entropy (all physical entropy is Shannon Entropy). The Information Problem of black holes is a question of their entropic behaviour, and really, black holes are sort of a grey area of our map of reality.
Your formulation exactly hinges on information, not observables. I read it twice. It is much more likely that the 6 hours is a conceptual limitation and that the HPMOR-verse is consistent if not causal, either by recomputation or being the solution to an equation, and just has time travel built in.
Conceptual limitations are apparent in other branches of magic, and I would hazard a guess that timetravel requires energy correlating to the amount of time jumped back. This would put a practical limit on it too.
Magic clearly does care about information, and it even uses a different definition than Bayesian stats. Luckily, magic doesn't care about physics.
Magic is clearly agent-like. Magic is clearly embedded in reality. Reality is by definition physics.
I don't know if EY has planned to pull something along the lines of Sam Hughes' Ra, but it would highly surprise me if he was about to throw Reductionism out the window.
There is literally no other definition of Information than the one being the moniker in Information Theory. Bayesian statistics heavily overlaps with Information theory. Everything axiomatized differently doesn't lend itself to Agenthood and Statistics. Anything else than Bayesian Stats doesn't lend itself to Agenthood either.
Also, I have spent inordinate amounts of thinking on solid academic grounds and in conversation with several sharp rationalists. Most of these counterpoints I have already considered.
Magic tells agents, reality, and physics to take a hike.
Unless you think that the Grand Unified Theory has a "WIngardum Leviosa" term.
I thin you might have missed that Harry at one point figures out this almost verbatim, that there must be some sort of very powerful system out in the territory that makes "Wingardium Leviosa" into a force application of some kind.
Also, I am trying to make a point, not trying to fight-argue with you.
It also seems that you do not know what Agent, Reality and Physics really mean.
Reality is whatever you can point at using Causality or Axiomatisations. Causality is just about any time where you have a verb and a noun, not anything special "sciency" like thing. "Dumbledore levitated a book" is causality. This definition also allows for the Everett Interprentation and for ships passing the Cosmological Horizon. Axiomatizations is the thing mathematicians are employed to work with, proofs, theorems, deduction.
Physics is whatever we find in the end. I believe we are done when there is no question left to ask of elementary physics when we can derive everything mathematically. Constants, interactions, gravity, the whole shebang.
Agents are the object studied in Decision Theory. Unless you think magic somehow does away with brains, world models, sensory input, motorical output, utility functions, cost-benefit analyses, choices and a few other useful concepts, then you have misunderstood something,
Is there a "Magic" term is the True Causality Truth? Is there a "Magic" factor in the math from which Decision Theory is created?
Broomsticks, for gods' sakes! Broomsticks break relativistic physics by having a maximum groundspeed (airspeed? I don't think it's explicitly stated) rather than thrust and drag. But there's a method to ground broomsticks that aren't protected against the grounding charm; it's possible that physics always was the way that imperfect thinking beings imagined it should be, but it's more likely that imperfect thinking beings hacked physics at some point to work the way they thought it ought to- and they thought that flying broomsticks ought to go where they were pointed, and ought to have a fixed maximum speed relative to the ground or air.
Likewise, someone thought that faux-Latin and wand movements should be required to lift stuff with magic, so they mostly are (obviously there was a dissenting opinion somewhere). Someone decided that Seeker was an interesting game rule, and someone decided that Hogwarts need not have a consistent internal layout, so all of those things are true enough.
Okay. It occurs to me we have some inferential gaps that I don't see my duty to fill. Go read HPMOR again, go read some more sequences. This is beginner level stuff, you are not presenting strong-evidence-based arguments, you are asking rhetorical questions, making non-deep analogies and taking things literally.
Is there a "Magic" term is the True Causality Truth? Is there a "Magic" factor in the math from which Decision Theory is created?
There is not a term for "magic" anywhere because first of all, we are arguing about a fanfic, so let us get that mind projection out of the way.
I know Eliezer's core beliefs and messages: He will not allow his work to portray a universe where anything is inherently "magic" because that is bad reductionism. He will maybe not introduce a new kind of quantum mechanics, because it is very much not his style to techno-babble.
Broomsticks, for gods' sakes!
Broomsticks do not conclusively break anything. The speeds are too low for the gamma factor to be measurable with equipment that can function around magic. Broomsticks probably have maximum velocity only in the apparent. Remember, there has not in canon been any precise measurement of the nature of broomsticks.
That the physics is the way imperfectly thinking beings thing is not a credulous hypothesis, because QM and GR are experimentally right, yet counterintuitive, and because the program that describes Aristotelian Physics is more complicated than the QFT lagrangian, for any representation of rational numbers. I know the math, evidently better than you.
Likewise, someone thought that faux-Latin and wand movements should be required to lift stuff with magic, so they mostly are (obviously there was a dissenting opinion somewhere). Someone decided that Seeker was an interesting game rule, and someone decided that Hogwarts need not have a consistent internal layout, so all of those things are true enough.
Have you completely missed the chapter where Harry thinks that the "heart of magic" is a thing the Atlanteans built? In the most recent chapters he thinks about, and talks to Quirrel about how one goes about making a new spell!
Seekers existing does not have anything to do with magic, it has to do with the culture surrounding games. You can see it in many existing games and sports today. Wizards are still human, they have human habits.
Why are we even arguing? I agree to almost everything you say which is not a blatant argument soldier!
I don't accept that "strong evidence-based arguements" is a meaningful phrase in the presence of a counterfactual premise. If there was introduced some kind of unexplained techno-babble that wasn't consistent both internally and with the entire story (including the exchange rate of magical currency), it would simply be pushing 'magic' up or down the meta-scale.
The (reletivivistic) speed of a broomstick is roughly equal to the speed of the Earth, which almost has measured relativistic effects (GPS satellites in LEO observe relativistic effects from their orbital velocity, but their orbital velocity is inherently greater than any sustained orbital velocity on the surface).
And the math doesn't have to be done in-universe. Can you estimate the mathematical complexity of turning into a cat, getting your tail stepped on, and then returning to human form with a bruised coccyx? In a world where magic acts this way, it makes sense for physics to be a cultural phenomenon as well.
I have conferred with a few of my most trusted rationalist friends, It seems we agree that you conceive of a "magic" being separate from "physics" in a way which we have collectively edited you of our cognition.
I really must emplore you to read the sequence Highly Advanced Epistemology 101 for Beginners. It argues strongly for why Mathematics and Physics/Causality is the only way of talking "meaningfully" about the universe.
Also, you are veering dangerously close to my trolling pattern matching algorithm's threshold.
Given the object-level information I have about the universe of HPMOR, I assign a significant probability to the possibility that the universe there does not follow Mathematics as we can conceive of in this universe. With the meta-level knowledge I have, I an certain that it will not be confirmed to be the case, and that magic will have the properties needed for the plot without having any epicycles which are not foreshadowed far enough in advance. It is currently too late to foreshadow anything new.
Also, you are veering dangerously close to my trolling pattern matching algorithm's threshold.
Your algorithm has the risk of false positives and false negatives; you can reduce the false results by improving the algorithm, or you can shift the threshold to reduce one and increase the other.
The thing to note here, is that, from the informational point of view of Dumbledore, provided he doesn't have some additional side-channel information, Scenario A and Scenario B are indistinguishable.
For Dumbledore, sure. Are the scenarios really indistinguishable for the whatever-it-is that keeps histories consistent and, presumably, enforces the six-hour rule? If not, Dumbledore has an obvious means of distinguishing between the scenarios: Attempt to travel six hours into the past, in steps of one hour, precommitting to telling someone what Amelia told him. In scenario B, the third hop will fail; in A, only the sixth.
This is precisely what I meant when I mentioned the empirical side information detector. The "informational point of view of Dumbledore" is "whatever-it-is that keeps histories consistent," and the indistinguishability only has to come into play in the local context of whenever Dumbledore uses the time turner. In the way I've envisioned it to work, Dumbledore can only use your algorithm to detect leaked information or side-information that was available to him which he might not be aware of.
That only really works if Bones is perfectly unreliable, though - if she's more likely to come back under certain circumstances, there's information.
At a guess, I would say data below a certain (low) threshold is allowed, and time travellers don't try to deduce such things in case they pass said threshold.
For that matter, Amelia could be wild-guessing when she tells you about something that will happen six hours from now. It doesn't make sense that if she came back six hours and tells you something it would be any different than if she came back one hour and told you the same thing, or said the same thing without any time travel involved.
Some people have been asking the question about the 6-hour limit on time turners in Harry Potter and the Methods of Rationality. Apparently, if a character, let's say it's Amelia Bones, goes back in time N hours and relays a piece of information about the future to another character, say Dumbledore, then Dumbledore cannot go back in time more than (6-N) hours.
The 6 hour limit is a useful rule to keep the HPMOR universe from becoming over-complicated with time travel, however several people have brought up the following objection. The claim is that Amelia Bones, by traveling back in time and saying that she has information she hasn't yet revealed, has already revealed information about the future in the form of Metadata: That the future still exists N hours in the future and that Amelia Bones was in it, etc. I will show, however, that this is not the case.
Imagine that Amelia Bones has traveled back in time from 1 hour in the future, but she is confused about the time after having apparated across time zones and mistakenly tells Dumbledore that she has information for him from 4 hours in the future. Call this Scenario A. Now imagine that the same scenario happens, but that Amelia is not mistaken about the time. Call that Scenario B.
The thing to note here, is that, from the informational point of view of Dumbledore, provided he doesn't have some additional side-channel information, Scenario A and Scenario B are indistinguishable. (In the same sense that being in an accelerating room is indistinguishable from being in a gravity field in General Relativity.) This is what I mean by "time turner meta-informational relativity." Provided that the act of arriving at some time and place with a time turner doesn't itself leak information about how far in the future you arrived from, meta-information about the future is not the same as information. The time-space coordinate meta-information conveyed when Amelia Bones tells Dumbledore, "I used a Time Turner and I have information about the future," is smeared out over the possible 6 hours. This tells us that meta-information cannot be the same as particular information about the future.
Additional consequence: From this, we can hypothesize that *any* time-indeterminate information conveyed to the past will be "smeared out" over the possible range of times, and that further backwards-time travel is limited by the closest possible value. So, if Amelia came from 3 hours in the future and related a piece of information that leaves it ambiguous if she came from 1 to 5 hours in the future, Dumbledore should still be able to travel 5 hours into the past -- provided he is not also in possession of information that lets him narrow Amelia's possible departure time.
Additional additional consequence: given the above is true, time turners can be used to empirically expose one's possession of such side information. I can imagine this being used for some clever deductive feat.
EDIT: To address some confusion about indistinguishability: 1) This is in the context of a specific point in space-time. 2) A given piece of information can only distinguish Scenario A and Scenario B if it's plausibly consistent with Scenario A but not Scenario B, or vice versa. So the paths of single neutrinos or configurations of air molecules aren't going to be able to do this. However, if there was a leak in a canister of a gas (let's say helium) at the time of Scenario A (4 hours in the future) but not at Scenario B's time, then there would be additional data available in the form of an implausibly large number of helium atoms in Amelia Bone's clothes.