I had an incredibly frustrating conversation this morning trying to explain the idea of quantum immortality to someone whose understanding of MWI begins and ends at pop sci fi movies. I think I've identified the main issue that I wasn't covering in enough depth (continuity of identity between near-identical realities) but I was wondering whether anyone has ever faced this problem before, and whether anyone has (or knows where to find) a canned 5 minute explanation of it.
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You think of your life as a branching tree of possibilities rather than a single line in time. And the immortality is supposed to happen because according to MWI, every possibility, no matter how improbable, actually happens, so there's always a branch of the tree that keeps growing - a possibility where the cancer goes away by itself, the bullet goes astray, and so on.
One possible attitude to quantum immortality would be that no matter what happens to you, there's always another copy of you who survives. But it seems that people who are really into quantum immortality want to identify with their copies - they have decided to think of the whole tree of possibilities as themselves, and not just the current branch. Wikipedia puts it like this: "Over time, the experimenter would therefore never perceive his or her own death." That's a little retarded, because if you're going to identify with the whole tree of possibilities, then you should identify with every death. So "quantum immortality" actually means that you die infinitely many times and in every possible way, but there's no final death.
Here's my explanation of quantum immortality:
Say you want to do something humans regard as important, like factoring large numbers. Lock yourself in a chamber that draws on a source of true quantum randomness to guess factors. Set it to kill you whenever it guesses incorrectly.
A short moment later, you will be alive, and have the correct factorization. This is because the only being you will identify with will be the being that decohered into the wavefunction branch that guessed correctly.
Seriously, this is the best way for humans to factor integers and solve other computationally-difficult problems. Give it a try.
Downvoted for urging people to commit suicide. Just because you define "you" to be "me who lives" does not mean that the colloquial you will not have death happen.
Look, maybe this would help: are you guys familiar with p-zombies? Let's say that I define myself to become a p-zombie if I measure a quantum spin to be up. This is exactly identical to quantum immortality, since "the only being I will identify with will be the being that decohered into the wavefunction that guessed correctly." Using this argument, you (notice the confusion caused by my use of the pronoun) should be able to measure a quantum spin to be down just by thinking that you'll become a p-zombie if you don't.
Okay, now replace "measure a quantum spin to be down" with "get superpowers." Aaaand GO! Congratulations, I have now turned you into a p-zombie.
Look, maybe this would help: are you guys familiar with p-zombies? Let's say that I define myself to become a p-zombie if I measure a quantum spin to be up.
I get what you mean (that this approach to problem solving relies on an odd definition of who "you" are), but I think using the term "p-zombie" here is unnecessarily confusing. A p-zombie is supposed to be a being who is identical to a human being in every physical way yet lacks qualia/consciousness. So when I read your post my immediate reaction was "Wait, what? How do you define yourself to become a p-zombie?"
Also, Clippy is a user roleplaying as a paperclip-maximizing AI. Don't take his suggestions of suicide too seriously.
Also, Clippy is a user roleplaying as a paperclip-maximizing AI. Don't take his suggestions of suicide too seriously.
Also, every User here is roleplaying a self-reflective apeyness-maximizing NI. Don't take their suggestions of catpersons too seriously.
I want to respond to that, but I am too amused by the idea of LW actually being composed of a bunch of AIs roleplaying as humans. That would certainly explain some things...
It could be worse, we could be AIs pretending to be humans pretending to be non-human animals pretending to be humans.
Oh, okay. Odd way to maximize (literal, I assume) paperclips, considering humans are currently the major producers of paperclips.
And yeah, perhaps I should have just said "you identify yourself as someone who consistently measures spin down," but it's always fun to make references and I wanted a slightly more intuitive analogue to death.
I'm curious, is your goal to maximizes the number of paper clips in a single part of the wavefunction or maximize the number across the entire wavefunction? If the first, you should use this strategy also.
Because it's obvious. Why is it obvious? Because I want to maximize the number of paperclips really existing in the universe, not the ones I directly observe existing -- so I can't just care about the paperclips in the branch that I will eventually experience decohering to.
Isn't it obivious?
Generally with this sort of thing it helps to be concrete.
Ask him to imagine that he's James Bond retiring at 60 years old. He's survived more lucky occasions then he can count. Why? Because he's the remaining lucky one.
As to continuity of identity, this is actually a tricky part. Ask him if he identifies with himself from 1 millisecond ago. Ok, but another version of him that had had that 1 millisecond go slightly differently would also identify with the one from 1 millisecond ago. And there identification is just as reasonable. So you should identify with the one who did the last millisecond slightly differently.
This doesn't quite work out (because presumably there's some sufficiently different version of me that doesn't have a goatee and thus is a goody-two shoes, and I'm not going to identify with that one.) But this detail isn't too relevant because the real notion is observers.
I'd skip the issue of continuity completely if you can and simply explain that the idea is that at any given point if you accept MWI then there will always be an observer who shares an identical history up to you at the current time.
Frankly, unless both you and your interlocutor have a firm mathematical understanding of quantum theory, discussing topics like these is like a debate about subtleties of engine tuning between people who have never opened a car hood. It's simply impossible for anyone to reach any real understanding or insight in such a situation.
The first question is whether you understand the point you were trying to make. Maybe your argument doesn't hold any water, and so difficult to profess. You didn't clearly explain your position in the post. (What about "continuity of identity"? What did you try to explain about "quantum immortality"?)
Maybe your argument doesn't hold any water, and so difficult to profess.
It's entirely possible. I don't claim a complete understanding of MWI and its implications yet. But my current understanding is useful for my purposes, so I'm ok to stick with it.
What did you try to explain about "quantum immortality"
One of the weirder implications of MWI that I've read here is that quantum immortality implies that it's literally impossible for me to die. Which means that if I was to attempt to kill myself the outcome would be large numbers of universes in which I hurt everyone I care about by killing myself, while from my perspective it wouldn't have succeeded and I would then have to live with the fallout from a failed attempt. I'm not sure how much I believe that this would actually happen but as a form of Pascal's Wager for suicide it's extremely effective at making me not want to test it.
You seem to be identifying yourself with only those beings like you that exist in universes like this one. I don't think that's necessarily valid, and removing the requirement adds more interesting options.
Let's say you exist in distribution of laws of physics, not just locations in this one. Then, attempting to kill yourself could change that distribution for those of you that survive.. in particular, it could vastly increase the measure of those extremely complex worlds that care about intelligent life.
For reasons that have been thoroughly explained previously, this would not necessarily be a good thing..
Another way to put it, then: If quantum immortality holds, even for non-quantum deaths, then there's a good chance that "death" really means "having your soul eaten by cthulhu", even if cthulhu has to invent it first. Fun stuff.
I'm a MWI cynic*, so here is my approach.
There is an interpretation of quantum mechanics called the many-world interpretation that rejects the wave-function collapse of the classical interpretation.
This interpretation leads some people to the belief that all possible alternative histories are real. Everything that can happen does happen. They assume that if you die in this world, you would also continue living on in another world. Since there must always be a chance you won't die, then there must be a world in which you live forever.
The problem with this belief is that at best it applies only for simple quantum systems, and generally not to events as large and complex as a person's death. To avoid death, a very large number of quantum level alternatives have to be simultaneously selected for. The probability of this simultaneous selection will be effectively zero.
In MWI terms this means that you die in all worlds, except the impossible ones.
Value the life you have and don't depend on quantum immortality.
* I like the MWI on lack of wave-function collapse and on quantum decoherence, but I don't think that the idea of separate worlds is necessary.
Eh not impossible... just very improbable (in a given world) and certain across all worlds.
I would have thought the more conventional explanation is that the other versions are not actually you (just very like you). This sounds like the issue of only economists acting in the way that economists model people. I would suspect that only people who fixate on such matters would confuse a copy with themselves.
I suspect that people who are vulnerable to these ideas leading to suicide are in fact generally vulnerable to suicide. There are lots of better reasons to kill yourself that most people ignore. If you think you're at risk of this I recommend you seek therapy, thought experiments should not have such drastic effects on your actions.
Eh not impossible... just very improbable (in a given world) and certain across all worlds.
"Very improbable" is the typical assumption with MWI, but I think that it is mistaken in most cases dealing with complex systems.
Each wave-function sets limits on what can occur. Wave-functions don't have infinite extents, there are areas with zero amplitude. Each additional wave-function that must meet specific requirements further restricts the possible outcomes. In general, the likelihood of failing to meet the simultaneous condition grows exponentially as the system size grows linearly.
Since quantum survival (avoiding death in some worlds, in some meaningful context) will usually require a very large number of quantum level alternatives to be simultaneously selected for, quantum survival will almost always be impossible.
A person who experiences quantum survival once is very lucky, but almost certainly won't survive the next time. A person who fails to experience quantum survival never gets another chance.
So my conclusion is that quantum immortality is impossible, not just very improbable.
Your logic here makes no rational sense. Your saying things which can be proved to be false.
Firstly I accept your premise that some things have zero probability. The wave-function doesn't mean literally anything can happen
BUT
I strongly disagree with you when you start saying that simultaneously selecting for possible (but improbable things) makes them impossible because this makes no rational sense. Quantum events are independent of each other the fact that 1 radioactive atom decays doesn't mean that the next is more or less likely to (unless they interact of course but I am ignroring that) nor is there a cutoff as to how many atoms can decay.
If a random event has a non zero probability of occurring, then a googleplex number of those events has a nonzero probability of occurring, in ANY COMBINATION.
I am a mathematical biologist, have a think about how many random mutations had to occur to create you.
Wait, is there proof for wave-functions not having infinite extents?
I always thought that the idea was similar to a normal distribution curve, where any value could be possible, just some are extremely unlikely.
The (very brief) explanation I normally use:
So, according to quantum physics, any time something can happen more than one way, it actually happens all those ways, but in different universes. So let's say you flip a coin. In one universe, it comes up heads, but in another it comes up tails. However, it's not really that simple, because there are all kinds of crazy things that can happen but are really unlikely. So it's more like, in almost half the universes, the coin lands heads, and in the other almost half of the universes, the coin lands tails, but in a tiny fraction of the universes the coin actually lands on its side. Anyway, quantum immortality holds that, since you can't perceive universes in which your perceptive abilities no longer exist, you'll always end up in a universe where you're still alive. So imagine you push a button that has a 50% chance of killing you. From an observer's perspective, you will die 50% of the time. However, from your perspective, you'll only perceive those universes in which you don't die, so you'll never actually die, no matter how many times you push the button. This applies to everything, since there's never a true 100% chance of death, no matter how crazy the situation.
This is indeed the standard thought, but be careful not to make a mountain out of a molehill.
"You'll never actually die" only holds if you define "you" to be "you who doesn't die." If you include "you who does die" as also you, then you do die! In short, both the mind-blowing and confusifying powers of quantum immortality are related to the fact that it is a tautology in disguise. It's a question of definitions and not a property of the universe because the universe doesn't really care whether you're alive or dead; dead bodies make great quantum observers, they interact with their environment just fine. But we choose to divide up the possible universes into "us alive" and "us dead," and then only look at what happens in one of those groups, making the answer predetermined by our fairly arbitrary choice of which universes to look at.
The standard response to that fact is "yes, but we choose to care about the universes where we live for a very special reason: you can't think about yourself being dead when you're dead. Cogito ergo sum."
And then I would say: "This is true, but just be careful. Though by definition we won't perceive ourselves as dead, that doesn't mean we can't die, unless we redefine 'we.' Pronouns beyond here become confusing; for a clearer view try to think of things following fuzzy worldlines in spacetime, with events being points in spacetime."
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To make quantum immortality not confusing is to make the mountain back into a molehill, relatively speaking at least. "The big complicated space that we exist in can be thought of like parallel universes, and in some of those parallel universes you'll live forever." Cool!
So it's more like, in almost half the universes, the coin lands heads, and in the other almost half of the universes, the coin lands tails, but in a tiny fraction of the universes the coin actually lands on its side.
Much like David Allen, I'm not sure this is truly the case. The path of the coin is completely determined, if you knew all starting conditions perfectly you could predict how it would end up with complete accuracy. And the universe knows all starting conditions perfectly.
Quantum fluctuations would tend to cancel each other out (much like all the air molecules in a room are never only in one corner, leaving the rest of the room in a vacuum, even though that's not strictly impossible), and thus never exert any significant pressure on the coin in any direction, and have no net impact on it's path. I would think that the coin would land on one particular side in nearly 100% of universes, with maybe the tiniest fraction containing a truly extraordinary confluence of random fluctuations all in the same direction that made it land on the other side.
I'm almost sure that it's not actually the case. I only use the example of a coin flipping because most people consider that random and it's easier than having to explain Schrodinger's Cat.
Isn't that an argument against quantum immortality though? As the event that kills you in any given universe is not going to be a random quantum event, but a hard-to-affect deterministic event that kills you in (nearly?) 100% of universes.
It shouldn't matter, since even the most infinitesimal chances are guaranteed to come up somewhere.
I think it is important to clarify the meaning of "chance", as you refer to it.
If I say that the behavior of a flipped coin is almost certainly deterministic, the remaining uncertainty is not part of the system, it is caused by my inability to predict the outcome. This is not the kind of "chance" that you are referring to.
The type of "chance" related to quantum immortality is the probability attached to non-zero quantum wave-function amplitudes.
It is not enough for there to be a conceptual "chance" that quantum wave-functions could influence the outcome of a coin toss. There must be actual reachable sequences of quantum state sets, all with non-zero wave-function amplitudes, that result in alternate outcomes.
It may also not be enough to utilize a hypothetical model of the quantum wave-functions. It may be possible that real low probability wave-functions don't result in universe splits. For example, those world-lines might merge with higher probability world lines, or there might be resolution limits set by the holographic universe, or by quantum foam noise.
With these restriction and granting (just for this argument) that the MWI is the right way to think about the universe, I'll agree with your statment:
"even the most infinitesimal chances are guaranteed to come up somewhere."
That's pretty much the explanation I gave her. I think the weakness in it is that if you're already confused about MWI it sounds almost like you teleport between universes, rather than there being other universes where someone you identify as yourself never stopped existing in the first place. At least, I think that's one of the parts she wasn't getting.
FWIW, I understand the above explanation, and also that one does not teleport, but that a single instance of, say, me, will avoid all opportunities to die. I can't speak for your friend, but the part that remains unclear to me is, since I am still only consciously aware of the one instance of me, and statistically it's very unlikely to be the immortal one, why I should care that some other one is. :P
How much background do you have in the relevant nerdy stuff though? This is someone who can basically be described as the polar opposite of a nerd. She'd never heard of the idea of multiple timelines/realities except really vaguely from mainstream pop culture, has no idea about quantum or any other types of physics, and afaik has never played a computer game in her life. I would also bet that she's never watched any hard sci fi.
Basically I am curious as to whether it's a problem of inferential distance or whether I just didn't explain it clearly enough :p
As for why you should care, depends on how you view continuity of identity.
How much background do you have in the relevant nerdy stuff though?
More than your friend, but maybe not a lot more. I'm one of the least nerdy nerds I know, and probably the least educated if you normalize age. Computer games yes, hard sci fi rarely, quantum physics pretty much none.
As for why you should care, depends on how you view continuity of identity.
True. I have trouble imagining anything being "me" which does/did/will not share the same stream of consciousness which is doing the imagining.
So do you view the 'you' that wakes up in the morning as the same as the 'you' that went to bed?
DSimon woke up this morning. From that point on, there have been a lot of diverging universes, one of which involved DSimon going to work, one of which involved DSimon getting hit by a car, and one of which involved DSimon founding a new town called "Shinypants" where the president lands Air Force One the very same day and makes him emperor of the world on a whim. There's also many many other divergent DSimons, but let's not worry about them.
So we'll call those derivatives DSimonWork, DSimonDead, and DSimonEmperor. All three of these DSimons feel that they are the same as the DSimon that woke up this morning. However, they do not feel that they are the same as each other.
So actually "the same as" isn't quite the right concept here, because then I would be saying that A = B and B = C but A != C. "Identify with" might be better. The main point though is that this feeling of identification proceeds backwards up the divergence tree, but not sideways across it.
erratio:
How much background do you have in the relevant nerdy stuff though?
[Edit: Oops, I misread the above question as asking how much background you have to have to be able to understand this stuff. I certainly don't have much background in physics myself. What I wrote below is true regardless, though, so I'm leaving the comment as it is.]
You have to have a deep mathematical grasp of quantum theory, both formal and intuitive, before you can even start to understand what the controversial issues such as MWI really are about, let alone to form any reasonably grounded opinions about them. Otherwise, you can only fool yourself that you understand anything about these topics, and any beliefs you form about them can be based only on faith in authority or arbitrary whim, not sound reasoning.
The same holds for most other topics in physics. In some areas for science, non-technical pop-science explanations can lead to correct understanding, but in physics, they are worse than useless. (The reason why physicists often display a positive view of pop-science physics books even when they don't stand to profit from their sales is that they raise their status.)
I think that's much too egalitarian. As best as I can tell people vary tremendously in their ability and inclination to grasp certain topics with minimal preparation. What you say is true of the majority but not of everyone. The value to the student of popular accounts also varies tremendously. What you say seems to be true if you pair the worst students with the worst books.
Hm.. this is the second time today that I posted a comment based on a misreading (see the correction above). Makes me wonder about the quality of the work I did during the rest of the day.
Regarding the pop-science approach to physics, I read a lot of such books as a teenager. Since then, I've studied several areas of physics with real mathematical rigor, some in university courses and others just out of curiosity, and in retrospect I must say that the popular books had failed to give me any accurate understanding whatsoever. All I got was either confusion or a somewhat coherent but in fact completely misleading picture of the real thing.
The only cases where physical theories can be explained with some accuracy in layman non-mathematical terms are those that deal with people's everyday experiences, and don't involve any phenomena that are outside of that. When it comes to relativity, quantum theory, cosmology, let alone more advanced and esoteric areas of modern physics, to me it seems impossible to convey any accurate understanding without the mathematics. I have yet to see a pop-science book that would successfully do it, no matter how smart the reader. (I liked this recent LW comment made by a physics about this topic.)
If you subscribe to the view that an illusion of understanding is worse than nothing at all (as I do), it's really hard to find anything positive to say about pop-science physics books. Yet scientists will rarely criticize them, sometimes because they profit by writing them (or hope to do so one day), but more often because they raise the status of physicists and scientists in general in the eyes of the general public. Ultimately, these works allow readers to signal their smarts and sophistication, famous physicists to make lots of money, and other physicists to enjoy the high status brought this way to their profession, but certainly nobody gains any (scientific) knowledge in the process.
A couple of my own experiences. As a kid I devoured Lewis Carroll Epstein's Thinking Physics, which so easy and fun and filled with pictures it is like a comic book, and I devoured it like one (though I always - always - answered each puzzle before seeing his explanation). Talk about popularizations, this was a frickin comic book. I think it took a week of afternoons to get through. In retrospect I don't think it misled me. On the contrary, I think I learned a lot from it. My major as an undergraduate was mathematics but I took two semesters of physics along with the regular physics majors in my freshman year, classical mechanics and electromagnetism, so I think I am in a position to judge that book in light of a "real" education.
Second experience. I never took a course in special or general relativity per se, but I touched on both in a graduate course on differential geometry (I think it was). In light of my graduate level understanding of the mathematics of spacetime, I would not say that I had been misled by popularizations of it (though I could not at this point name you which exact popularizations of it I read - Epstein wrote one but I never read it). It has been many years but what I learned in that graduate course regarding special relativity was a certain metric, a simple, trivial metric, called I think minkowski metric (it's been a while). That metric made the geometric aspects of special relativity utterly trivial, all the familiar amazing geometric predictions of SR regarding lengths and timing just fell right out of it, everything, the flashlight on the train, everything. It was a course on geometry so it didn't integrate the minkowski geometry with the laws of mechanics, force, acceleration, etc. But my point is this: it did not reveal the popularizations that I had read as a teenager to be a load of bull. On the contrary, it confirmed and completed them.
To compare, I have read some of the "best" popularizations of string theory. I do not feel very enlightened by them. There is no illusion of understanding because there is no belief that I understand. In contrast, I find Feynman's QED (the thin book, not the theory) enlightening - as far as it goes. The book has modest goals. It does not pretend to arm you with the ability to carry out real-world predictions. So I find popular books to vary tremendously in their ability to convey knowledge.
As for why you should care, depends on how you view continuity of identity.
Before you get to exotic* applications of quantum mechanics, you should cover the basics. Quantum mechanics forces a view of personal identity that treats the many words equally. This is true even if you endorse collapse, as long as people can occasionally be put in superposition.
In a classical world with randomness, you could imagine that there are lots of parallel copies of people, each with a serial number ("indexical uncertainty") who march in parallel until random events occur whose outcome is determined by the serial number. In such a setting, you could imagine that the invisible serial number is a core part of the person's consciousness and there is no continuity of identity between them. But QM says that quantum sources of randomness are not like that. It is not meaningful to talk of which parallel person gets which result.
* a euphemism for "wrong."
I don't have any answers to you question, but I have my own questions. Maybe someone will answer them here.
Just yesterday I had a long discussion with wedrifid about quantum suicide. It came to a halt when wedrifid got very offended because I claimed that he misrepresented my position. Or something like that. I am still not sure. Anyway, in spite of this, I think it was an interesting discussion mostly relevant here. There I enumerated several reasons why committing quantum suicide is a bad idea. But I am still very confused about the real question: How would it feel to commit quantum suicide? I wrote:
My common sense tells me that if I put a (quantum or other) gun in my mouth right now, and pull the trigger many times, then the next thing I will feel is not that I am very lucky. Rather, I will not feel anything at all because I will be dead. I am quite sure about this instinct, and let us assume for a minute that it is indeed correct. This can mean two things. One possible conclusion is that MWI must be wrong. Another possible conclusion is that MWI is right but we make some error when we try to apply MWI to this situation. I give high probability to both of this possibilities, and I am very interested in any new insights.
A potential error for the second conclusion is that we have incorrectly predicted the nature of consciousness, and the true solution is that one is somehow able to perceive without a physical avatar functioning in the way we expect of a human capable of perception. Thus, "you" are able to experience the branches of the MWI where everyone else perceives you to be dead.
Can I ask a related question? Is there a physical model available that allows for immortality (eternally stable structure) in a cyclic model of the universe only (limited space with finite time between cycles)?
MWI and other parallel universe models seem to allow for suitable ways of replication and escape - but I never found anything related for a cyclic model. There is talk of surviving the Heat death (superconductor based computers) and Big Crunch/Big bang (using suitable black holes, etc..) - but there is one specific problem I haven't seen addressed: Particle decay.
If everything else works as planned and a future stable structure is created - in a cyclic model - Is there any way to prevent it from catastrophically desintegrating through particle decay (which is bound to happen in enough finite time)?
The best way to get at the idea behind quantum immortality might be to explain that if the universe is infinite there must be in infinite number of yous because there are only so many ways of arranging the atoms in a galaxy so all possible galaxies exist in infinite number. No matter what happens there must always be galaxies in which a you somehow escapes death so there will always be a you that experiences life.
Pretend you undergo a dangerous operation in which you have a 99% chance of "dying". Before the operation you can correctly say to yourself that "I will wake up from this operation" because some versions of you will survive. Also, in some galaxies death will be cured before you age too much and therefore yous will be alive in 1,000 years.
Two relatively simple rebuttals to your premise:
(1) One can easily create a number with a non-terminating decimal expansion which makes use of a finite quantity of the digit "5". Therefore it is conceivable that one could also exist in an infinite universe which makes use of only a finite quantity of atomic structures identical to "you".
Similarly, and working in the opposite direction (complex-to-simple as opposed to the former simple-to-complex extrapolation), it is strongly believed that we exist in a universe with fixed universal constants, whose values are just-so, such that no universe similar to ours could be produced with any variation on those constants. If we may accept that as an argument that no such "universe" exists that we can perceive, then it should be just as easy to accept that no such arrangement of particles exists in our current perceivable universe which is similar in form and history to "you".
(2) Even if there exists an infinite number of yous in this universe comprising the entire scope of conceivable future selves (or, more succinctly, if the Mathematical Universe Hypothesis is correct), this argument still does not answer the question: "How do you know that we exist in that 1% of the galaxies where you're correct about surviving the operation?"
DSimon woke up this morning. From that point on, there have been a lot of diverging universes, one of which involved DSimon going to work, one of which involved DSimon getting hit by a car, and one of which involved DSimon founding a new town called "Shinypants" where the president lands Air Force One the very same day and makes him emperor of the world on a whim. There's also many many other divergent DSimons, but let's not worry about them.
So we'll call those derivatives DSimonWork, DSimonDead, and DSimonEmperor. All three of these DSimons feel that they are the same as the DSimon that woke up this morning. However, they do not feel that they are the same as each other.
So actually "the same as" isn't quite the right concept here, because then I would be saying that A = B and B = C but A != C. "Identify with" might be better. The main point though is that this feeling of identification proceeds backwards up the divergence tree, but not sideways across it.