Apr 26, 2008
Continuation of: Where Physics Meets Experience
When we last met our heroes, the Ebborians, they were discussing the known phenomenon in which the entire planet of Ebbore and all its people splits down its fourth-dimensional thickness into two sheets, just like an individual Ebborian brain-sheet splitting along its third dimension.
And Po'mi has just asked:
"Why should the subjective probability of finding ourselves in a side of the split world, be exactly proportional to the square of the thickness of that side?"
When the initial hubbub quiets down, the respected Nharglane of Ebbore asks: "Po'mi, what is it exactly that you found?"
"Using instruments of the type we are all familiar with," Po'mi explains, "I determined when a splitting of the world was about to take place, and in what proportions the world would split. I found that I could not predict exactly which world I would find myself in—"
"Of course not," interrupts De'da, "you found yourself in both worlds, every time -"
"—but I could predict probabilistically which world I would find myself in. Out of all the times the world was about to split 2:1, into a side of two-thirds width and a side of one-third width, I found myself on the thicker side around 4 times out of 5, and on the thinner side around 1 time out of 5. When the world was about to split 3:1, I found myself on the thicker side 9 times out of 10, and on the thinner side 1 time out of 10."
"Are you very sure of this?" asks Nharglane. "How much data did you gather?"
Po'mi offers an overwhelming mountain of experimental evidence.
"I guess that settles that," mutters Nharglane.
"So you see," Po'mi says, "you were right after all, Yu'el, not to eliminate 'subjective probability' from your worldview. For if we do not have a 4/5 subjective anticipation of continuing into the thicker side of a 2:1 split, then how could we even describe this rule?"
"A good question," says De'da. "There ought to be some way of phrasing your discovery, which eliminates this problematic concept of 'subjective continuation'..."
The inimitable Ha'ro speaks up: "You might say that we find ourselves in a world in which the remembered splits obey the squared-thickness rule, to within the limits of statistical expectation."
De'da smiles. "Yes, excellent! That describes the evidence in terms of recorded experimental results, which seems less problematic than this 'subjective anticipation' business."
"Does that really buy us anything...?" murmurs Yu'el. "We're not limited to memories; we could perform the experiment again. What, on that next occasion, would you anticipate as your experimental result? If the thickness is split a hundred to one? Afterward it will be only a memory... but what about beforehand?"
"I think," says De'da, "that you have forgotten one of your own cardinal rules, Yu'el. Surely, what you anticipate is part of your map, not the territory. Your degree of anticipation is partial information you possess; it is not a substance of the experiment itself."
Yu'el pauses. "Aye, that is one of my cardinal rules... but I like my partial information to be about something. Before I can distinguish the map and the territory, I need a concept of the territory. What is my subjective anticipation about, in this case? I will in fact end up in both world-sides. I can calculate a certain probability to five decimal places, and verify it experimentally—but what is it a probability of?"
"I know!" shouts Bo'ma. "It's the probability that your original self ends up on that world-side! The other person is just a copy!"
A great groan goes up from the assembled Ebborians. "Not this again," says De'da. "Didn't we settle this during the Identity Wars?"
"Yes," Yu'el says. "There is no copy: there are two originals."
De'da shakes his head in disgust. "And what are the odds that, out of umpteen billion split Ebbores, we would be the originals at this point?"
"But you can't deny," Bo'ma says smugly, "that my theory produces good experimental predictions! It explains our observations, and that's all you can ask of any theory. And so science vindicates the Army of Original Warriors—we were right all along!"
"Hold on," says Yu'el. "That theory doesn't actually explain anything. At all."
"What?" says Bo'ma. "Of course it does. I use it daily to make experimental predictions; though you might not understand that part, not being a physicist."
Yu'el raises an eye. "Failure to explain anything is a hard-to-notice phenomenon in scientific theories. You have to pay close attention, or you'll miss it. It was once thought that phlogiston theory predicted that wood, when burned, would lose phlogiston and transform into ash; and predicted that candles, burning in an enclosed space, would saturate the air with phlogiston and then go out. But these were not advance predictions of phlogiston theory. Rather, phlogiston theorists saw those results, and then said 'Phlogiston did it.' Now why didn't people notice this right away? Because that sort of thing is actually surprisingly hard to notice."
"In this case," continues Yu'el, "you have given us a rule that the original Ebborian has a probability of ending up in a world-side, which is proportional to the squared thickness of the side. We originally had the mystery of where the squared-thickness rule came from. And now that you've offered us your rule, we have the exact same mystery as before. Why would each world have a squared-thickness probability of receiving the original? Why wouldn't the original consciousness always go to the thicker world? Or go with probability directly proportional to thickness, instead of the square? And what does it even mean to be the original?"
"That doesn't matter," retorts Bo'ma. "Let the equation mean anything it likes, so long as it gives good experimental predictions. What is the meaning of an electrical charge? Why is it an electrical charge? That doesn't matter; only the numbers matter. My law that the original ends up in a particular side, with probability equaling the square of its thickness, gives good numbers. End of story."
Yu'el shakes his head. "When I look over the raw structure of your theory—the computer program that would correspond to this model—it contains a strictly superfluous element. You have to compute the square of the thickness, and turn it into a probability, in order to get the chance that the original self goes there. Why not just keep that probability as the experimental prediction? Why further specify that this is the probability of original-ness? Adding that last rule doesn't help you compute any better experimental predictions; and it leaves all the original mysteries intact. Including Po'mi's question as to when exactly a world splits. And it adds the new mystery of why original-ness should only end up in one world-side, with probability equal to the square of the thickness." Yu'el pauses. "You might as well just claim that all the split world-sides except one vanish from the universe."
Bo'ma snorts. "For a world-side to 'just vanish' would outright violate the laws of physics. Why, if it all vanished in an instant, that would mean the event occurred non-locally—faster than light. My suggestion about 'originals' and 'copies' doesn't postulate unphysical behavior, whatever other criticisms you may have."
Yu'el nods. "You're right, that was unfair of me. I apologize."
"Well," says Bo'ma, "how about this, then? What if 'fourth-dimensional thickness', as we've been calling it, is actually a degree of partial information about who we really are? And then when the world splits, we find out."
"Um... what?" says Yu'el. "Are you sure you don't want to rephrase that, or something?"
Bo'ma shakes his head. "No, you heard me the first time."
"Okay," says Yu'el, "correct me if I'm wrong, but I thought I heard Nharglane say that you had to do things like differentiate the fourth-dimensional density in order to do your experimental calculations. That doesn't sound like probability theory to me. It sounds like physics."
"Right," Bo'ma says, "it's a quantity that propagates around with wave mechanics that involve the differential of the density, but it's also a degree of partial information."
"Look," Yu'el says, "if this 4D density business works the way you say it does, it should be easy to set up a situation where there's no possible 'fact as to who you really are' that is fixed in advance but unknown to you, because the so-called 'answer' will change depending on the so-called 'question'—"
"Okay," Bo'ma says, "forget the 'probability' part. What if 4D thickness is the very stuff of reality itself? So how real something is, equals the 4D thickness—no, pardon me, the square of the 4D thickness. Thus, some world-sides are quantitatively realer than others, and that's why you're more likely to find yourself in them."
"Why," says Yu'el, "is the very stuff of reality itself manifesting as a physical quantity with its own wave mechanics? What's next, electrical charge as a degree of possibility? And besides, doesn't that violate -"
Then Yu'el pauses, and falls silent.
"What is it?" inquires Po'mi.
"I was about to say, wouldn't that violate the Generalized Anti-Zombie Principle," Yu'el replies slowly. "Because then you could have a complete mathematical model of our world, to be looked over by the Flying Spaghetti Monster, and then afterward you would need to tell the Flying Spaghetti Monster an extra postulate: Things are real in proportion to the square of their fourth-dimensional thickness. You could change that postulate, and leave everything microphysically the same, but people would find... different proportions of themselves?... in different places. The difference would be detectable internally... sort of... because the inhabitants would experience the results in different proportions, whatever that means. They would see different things, or at least see the same things in different relative amounts. But any third-party observer, looking over the universe, couldn't tell which internal people were more real, and so couldn't discover the statistics of experience."
De'da laughs. "Sounds like a crushing objection to me."
"Only," says Yu'el, "is that really so different from believing that you can have the whole mathematical structure of a world, and then an extra fact as to whether that world happens to exist or not exist? Shouldn't that be ruled out by the Anti-Zombie Principle too? Shouldn't the Anti-Zombie Principle say that it was logically impossible to have had a world physically identical to our own, except that it doesn't exist? Otherwise there could be an abstract mathematical object structurally identical to this world, but with no experiences in it, because it doesn't exist. And papers that philosophers wrote about subjectivity wouldn't prove they were conscious, because the papers would also 'not exist'."
"Um..." says an Ebborian in the crowd, "correct me if I'm mistaken, but didn't you just solve the mystery of the First Cause?"
"You are mistaken," replies Yu'el. "I can tell when I have solved a mystery, because it stops being mysterious. To cleverly manipulate my own confusion is not to dissolve a problem. It is an interesting argument, and I may try to follow it further—but it's not an answer until the confusion goes away."
"Nonetheless," says Bo'ma, "if you're allowed to say that some worlds exist, and some worlds don't, why not have a degree of existence that's quantitative? And propagates around like a wave, and then we have to square it to get an answer."
Yu'el snorts. "Why not just let the 'degree of existence' be a complex number, while you're at it?"
Bo'ma rolls his eyes. "Please stop mocking me. I can't even imagine any possible experimental evidence which would point in the direction of that conclusion. You'd need a case where two events that were real in opposite directions canceled each other out."
"I'm sorry," says Yu'el, "I need to learn to control my tendency to attack straw opponents. But still, where would the squaring rule come from?"
An Ebborian named Ev'Hu suggests, "Well, you could have a rule that world-sides whose thickness tends toward zero, must have a degree of reality that also tends to zero. And then the rule which says that you square the thickness of a world-side, would let the probability tend toward zero as the world-thickness tended toward zero. QED."
"That's not QED," says Po'mi. "That's a complete non-sequitur. Logical fallacy of affirming the consequent. You could have all sorts of rules that would let the reality tend toward zero as the world-thickness tended toward zero, not just the squaring rule. You could approach the limit from many different directions. And in fact, all our world-sides have a thickness that 'tends toward zero' because they keep splitting. Furthermore, why would an indefinite tendency in the infinite future have any impact on what we do now?"
"The frequentist heresy," says Yu'el. "It sounds like some of their scriptures. But let's move on. Does anyone have any helpful suggestions? Ones that don't just shuffle the mystery around?"
Ha'ro speaks. "I've got one."
"Okay," Yu'el says, "this should be good."
"Suppose that when a world-side gets thin enough," Ha'ro says, "it cracks to pieces and falls apart. And then, when you did the statistics, it would turn out that the vast majority of surviving worlds have splitting histories similar to our own."
There's a certain unsettled pause.
"Ha'ro," says Nharglane of Ebbore, "to the best of my imperfect recollection, that is the most disturbing suggestion any Ebborian physicist has ever made in the history of time."
"Thank you very much," says Ha'ro. "But it could also be that a too-small world-side just sheds off in flakes when it splits, rather than containing actual sentient beings who get to experience a moment of horrified doom. The too-small worlds merely fail to exist, as it were. Or maybe sufficiently small world-sides get attracted to larger world-sides, and merge with them in a continuous process, obliterating the memories of anything having happened differently. But that's not important, the real question is whether the numbers would work out for the right size limit, and in fact," Ha'ro waves some calculations on a piece of paper, "all you need is for the minimum size of a cohesive world to be somewhere around the point where half the total fourth-dimensional mass is above the limit -"
"Eh?" says Yu'el.
"I figured some numbers and they don't look too implausible and we might be able to prove it, either from first-principles of 4D physics showing that a cracking process occurs, or with some kind of really clever experiment," amplifies Ha'ro.
"Sounds promising," says Yu'el. "So if I get what you're saying, there would be a completely physical explanation for why, when a typical bunch of worlds split 2:1, there's around 4 times as many cohesive worlds left that split from the thicker side, as from the thinner side."
"Yes," says Ha'ro, "you just count the surviving worlds."
"And if the Flying Spaghetti Monster ran a simulation of our universe's physics, the simulation would automatically include observers that experienced the same things we did, with the same statistical probabilities," says Yu'el. "No extra postulates required. None of the quantities in the universe would need additional characteristics beyond their strictly physical structure. Running any mathematically equivalent computer program would do the trick—you wouldn't need to be told how to interpret it a particular way."
Ha'ro nods. "That's the general idea."
"Well, I don't know if that's correct," says Yu'el. "There's some potential issues, as you know. But I've got to say it's the first suggestion I've heard that's even remotely helpful in making all this seem any less mysterious."
Part of The Quantum Physics Sequence
Next post: "On Being Decoherent"
Previous post: "Where Physics Meets Experience"