The Many Worlds of Hugh Everett
post by johnclark · 2011-04-22T15:26:05.968Z · LW · GW · Legacy · 50 commentsContents
50 comments
I've just finished this book and its one of the most enjoyable things I've read in a long time. Being a staple of science fiction and the only interpretation of quantum mechanics to enter the popular imagination it's a little surprising that "The Many Worlds of Hugh Everett" by Peter Byrne is the first biography of the originator of that amazing idea. Everett certainly had an interesting life, he was a libertarian and a libertine, became a cold warrior who with his top secret clearance was comfortable with the idea of megadeath, became wealthy by started one of the first successful software companies until alcoholism drove him and his company into the ground. Everett died of heart failure in 1982 at the age of 51, he was legally drunk at the time. He requested that his body be cremated and his ashes thrown into the garbage. And so he was.
Byrne had an advantage other potential biographers did not, the cooperation of his son Mark, a successful rock musician and composer whose music has been featured in such big budget movies as American Beauty, Hellboy, Yes Man, all three of the Shrek movies and many others. Mark gave Byrne full access to his garage which was full of his father's papers that nobody had looked at in decades.
Everett was an atheist all his life, after his death Paul Davies, who got 1,000,000 pounds for winning the Templeton religion prize, said that if true Many Worlds destroyed the anthropic argument for the existence of God. Everett would have been delighted. Nevertheless Everett ended up going to Catholic University of America near Washington DC. Although Byrne doesn't tell us exactly what was in it, Everett as a freshman devised a logical proof against the existence of God. Apparently it was good enough that one of his pious professors became very upset and depressed with "ontological horror" when he read it. Everett liked the professor and felt so guilty he decided not to use it on a person of faith again. This story is very atypical of the man, most of the time Everett seems to care little for the feelings of others and although quite brilliant wasn't exactly lovable.
Everett wasn't the only one dissatisfied with the Copenhagen Interpretation which insisted the measuring device had to be outside the wave function, but he was unlike other dissidents such as Bohm or Cramer in that Everett saw no need to add new terms to Schrodinger's Equation and thought the equation meant exactly what it said. The only reason those extra terms were added was to try to rescue the single universe idea, and there was no experimental justification for that. Everett was unique in thinking that quantum mechanics gave a description of nature that was literally true.
John Wheeler, Everett's thesis adviser, made him cut out about half the stuff in his original 137 page thesis and tone down the language so it didn't sound like he thought all those other universes were equally real when in fact he did. For example, Wheeler didn't like the word "split" and was especially uncomfortable with talk of conscious observers splitting, most seriously he made him remove the entire chapter on information and probability which today many consider the best part of the work. His long thesis was not published until 1973, if that version had been published in 1957 instead of the truncated Bowdlerized version things would have been different; plenty of people would still have disagreed but he would not have been ignored for as long as he was.
Byrne writes of Everett's views: "the splitting of observers share an identity because they stem from a common ancestor, but they also embark on different fates in different universes. They experience different lifespans, dissimilar events (such as a nuclear war perhaps) and at some point are no longer the same person, even though they share certain memory records." Everett says that when a observer splits it is meaningless to ask "which of the final observers corresponds to the initial one since each possess the total memory of the first" he says it is as foolish as asking which amoeba is the original after it splits into two. Wheeler made him remove all such talk of amoebas from his published short thesis.
Byrne says Everett did not think there were just an astronomically large number of other universes but rather an infinite number of them, not only that he thought there were a non-denumerable infinite number of other worlds. This means that the number of them was larger than the infinite set of integers, but Byrne does not make it clear if this means they are as numerous as the number of points on a line, or as numerous as an even larger infinite set like the set of all possible clock faces, or maybe an even larger infinity than that where easy to understand examples of that sort of mega-infinite magnitude are hard to come by. Neill Graham tried to reformulate the theory so you'd only need a countably infinite number of branches and Everett at first liked the idea but later rejected it and concluded you couldn't derive probability by counting universes. Eventually even Graham seems to have agreed and abandoned the idea that the number of universes was so small you could count them.
Taken as a whole Everett's multiverse, where all things happen, probability is not a useful concept and everything is deterministic. However for observers like us trapped in a single branch of the multiverse, observers who do not have access to the entire wave function and all the information it contains but only a small sliver of it, probability is the best we can do. That probability we see is not part of the thing itself but is just a subjective measure of our ignorance.
Infinity can cause problems in figuring out probability but Everett said his theory could calculate what the probability any event could be observed in any branch of the multiverse, and it turns out to be the Born Rule (discovered by Max Born, grandfather of Olivia Newton John) which means the probability of finding a particle at a point is the squaring of the amplitude of the Schrodinger Wave function at that point. The Born Rule has been shown experimentally to be true but the Copenhagen Interpretation just postulates it, Everett said he could derive it from his theory it "emerges naturally as a measure of probability for observers confined to a single branch (like our branch)". He proved the mathematical consistency of this idea by adding up all the probabilities in all the branches of the event happening and getting exactly 100%. Dieter Zeh said Everett may not have rigorously derived the Born Rule but did justify it and showed it "as being the only reasonable choice for a probability measure if objective reality is represented by the universal wave function [Schrodinger's wave equation]". Rigorous proof or not that's more than any other quantum interpretation has managed to do.
Everett wrote to his friend Max Jammer:
"None of these physicists had grasped what I consider to be the major accomplishment of the theory- the "rigorous" deduction of the probability interpretation of Quantum Mechanics from wave mechanics alone. This deduction is just as "rigorous" as any deductions of classical statistical mechanics. [...] What is unique about the choice of measure and why it is forced upon one is that in both cases it is the only measure that satisfies the law of conservation of probability through the equations of motion. Thus logically in both classical statistical mechanics and in quantum mechanics, the only possible statistical statements depend upon the existence of a unique measure which obeys this conservation principle."
Nevertheless some complained that Everett did not use enough rigor in his derivation. David Deutsch has helped close that rigor gap. He showed that the number of Everett-worlds after a branching is proportional to the conventional probability density. He then used Game Theory to show that all these are all equally likely to be observed. Everett would likely have been delighted as he used Game Theory extensively in his other life as a cold warrior. Professor Deutsch gave one of the best quotations in the entire book, talking about many worlds as a interpretation of Quantum Mechanics "is like talking about dinosaurs as an interpretation of the fossil record".
Everett was disappointed at the poor reception his doctoral dissertation received and never published anything on quantum mechanics again for the rest of his life; instead he became a Dr. Strangelove type character making computer nuclear war games and doing grim operational research for the pentagon about armageddon. He was one of the first to point out that any defense against intercontinental ballistic missiles would be ineffectual and building an anti-balistic missile system could not be justified except for "political or psychological grounds". Byrne makes the case that Everett was the first one to convince high military leaders through mathematics and no nonsense non sentimental reasoning that a nuclear war could not be won, "after an attack by either superpower on the other, the majority of the attacked population that survived the initial blasts would be sterilized and gradually succumb to leukemia. Livestock would die quickly and survivors would be forced to rely on eating grains potatoes and vegetables. Unfortunately the produce would be seething with radioactive Strontium 90 which seeps into human bone marrow and causes cancer". Linus Pauling credited Evert by name and quoted from his pessimistic report in his Nobel acceptance speech for receiving the 1962 Nobel Peace prize.
Despite his knowledge of the horrors of a nuclear war Everett, like most of his fellow cold warrior colleagues in the 50's and 60's, thought the probability of it happening was very high and would probably happen very soon. Byrne speculates in a footnote that Everett may have privately used anthropic reasoning and thought that the fact we live in a world where such a war has not happened (at least not yet) was more confirmation that his Many Worlds idea was right. Incidentally this is one of those rare books where the footnotes are almost as much fun to read as the main text.
Hugh's daughter Liz Everett killed herself a few years after her father's death, in her suicide note she said "Funeral requests: I prefer no church stuff. Please burn be and DON'T FILE ME. Please sprinkle me in some nice body of water or the garbage, maybe that way I'll end up in the correct parallel universe to meet up with Daddy". And so she was.
John K Clark
50 comments
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comment by gwern · 2011-04-22T18:48:06.991Z · LW(p) · GW(p)
Not that I'm necessarily complaining since I enjoyed your previous review, but how could you have just finished it now?
Replies from: jwhendycomment by steven0461 · 2011-04-22T22:04:58.363Z · LW(p) · GW(p)
Good review, but it would be easier to read if you added some commas and changed some existing commas to semicolons.
Replies from: khafra↑ comment by khafra · 2011-04-25T19:04:31.575Z · LW(p) · GW(p)
For John's reference, the wikipedia article on the comma splice might help.
Also, consider reading while paying attention to your breath. Chopping apart sentences which cannot be completed in a single breath may improve readability. A more complete theory along those lines was linked on LW long ago; my cursory search failed to turn it up.
comment by RHollerith (rhollerith_dot_com) · 2011-04-22T22:36:10.315Z · LW(p) · GW(p)
Taken as a whole Everett's multiverse, where all things happen, probability is not a useful concept and everything is deterministic. However for observers like us trapped in a single branch of the multiverse, observers who do not have access to the entire wave function and all the information it contains but only a small sliver of it, probability is the best we can do.
I am unable to imagine an interpretation of this paragraph that makes it true.
Probability would be necessary for belief formation even if reality consisted of only a single world. More generally, the usefulness of probability to belief formation does not depend on any particular features or properties of the reality the belief-forming agent (or collection of agents, e.g., the people having this conversation) happens to find itself in (except for the trivial consideration that some realities cannot contain belief-forming agents).
Also, I am extremely skeptical that literally all things happen in the Everettian multiverse. For example, I would be extremely surprised if there exists or will ever exist a branch in which the law of the conservation of momentum is violated. The principle of charity demands that I assume that the OP (johnclark) knows that, but I have been in enough conversations on LW about many worlds to have strong evidence that some of the readers will take "all things happen" literally.
Replies from: rhollerith_dot_com, johnclark, johnclark, johnclark↑ comment by RHollerith (rhollerith_dot_com) · 2011-04-22T23:00:26.619Z · LW(p) · GW(p)
Clarification of parent:
I consider it true that probability is in the mind and that when taken as a whole, reality does not contain probabilities. In that sense, John Clark's statement, "Taken as a whole Everett's multiverse . . . probability is not a useful concept and everything is deterministic," is true. What I am extremely skeptical of is whether that truth depends somehow on the fact that our reality consists of branches that are constantly splitting or depends somehow on that fact that Schroedinger's equation applies to our reality.
↑ comment by johnclark · 2011-04-23T05:26:09.820Z · LW(p) · GW(p)
Some of the laws of physics could change from universe to universe, but there must be some laws that remain invariant across the entire multiverse because without rules it would behave chaotically and if the multiverse behaved that way so would all the universes in it, including ours. However there is order in our universe, but what is fundamental and what is not? I think we probably all agree that purely mathematical things like pi or e would remain constant in all universes, but consider some of the physical things that might change:
The Planck constant. The speed of light. The gravitational (big G) constant. The mass of the electron, proton, and neutron. The electrical charge on the proton and electron. The inverse square law of gravity and electromagnetism. The conservation of Mass-energy, momentum, angular momentum, spin and electrical charge.The relative strength of the 4 forces of nature. The number of large dimensions in a universe. The Hubble constant. The ratio of baryonic matter to dark matter and dark energy.
It seems to me that the speed of light and Planck's constant may be more fundamental than other "constants" and the basic structure of the laws of physics may be more fundamental than the constants they use. But I could be wrong, perhaps the things that always remain the same are none of the above and we haven't even discovered them yet.
John K Clark
Replies from: rhollerith_dot_com↑ comment by RHollerith (rhollerith_dot_com) · 2011-04-23T05:47:18.637Z · LW(p) · GW(p)
Some of the laws of physics could change from universe to universe
I will assume that by "universe" here you mean "Everett branch".
I agree that they could (with low probability), just as they could slowly change over time or be different billions of light years away. This is a possibility because our reality has a property that physicists call locality. What I object to is the belief, which more than a handful of LW and SL4 participants hold, that there is something about many worlds (or about quantum mechanics for that matter) that increases the probability of stuff like that happening above what it would be if our reality had this locality property but no Everett branching.
should increase the probability we assign to stuff like that happening.
There is also the
↑ comment by johnclark · 2011-04-23T05:23:34.930Z · LW(p) · GW(p)
I understand that some of the laws of physics could change from universe to universe, but there must be some laws that remain invariant across the entire multiverse because without rules it would behave chaotically and if the multiverse behaved that way so would all the universes in it, including ours. However there is order in our universe, but what is fundamental and what is not? I think we probably all agree that purely mathematical things like pi or e would remain constant in all universes, but consider some of the physical things that might change:
The Planck constant.
The speed of light.
The gravitational (big G) constant.
The mass of the electron, proton, and neutron.
The electrical charge on the proton and electron.
The inverse square law of gravity and electromagnetism.
The conservation of Mass-energy, momentum, angular momentum, spin and electrical charge.
The relative strength of the 4 forces of nature.
The number of large dimensions in a universe.
The Hubble constant.
The ratio of baryonic matter to dark matter and dark energy.
It seems to me that the speed of light and Planck's constant may be more fundamental than other "constants" and the basic structure of the laws of physics may be more fundamental than the constants they use. But I could be wrong, perhaps the things that always remain the same are none of the above and we haven't even discovered them yet.
John K Clark
↑ comment by johnclark · 2011-04-23T05:19:56.582Z · LW(p) · GW(p)
Some of the laws of physics could change from universe to universe, but there must be some laws that remain invariant across the entire multiverse because without rules it would behave chaotically and if the multiverse behaved that way so would all the universes in it, including ours. However there is order in our universe, but what is fundamental and what is not? I think we probably all agree that purely mathematical things like pi or e would remain constant in all universes, but consider some of the physical things that might change:
The Planck constant.
The speed of light.
The gravitational (big G) constant.
The mass of the electron, proton, and neutron.
The electrical charge on the proton and electron.
The inverse square law of gravity and electromagnetism.
The conservation of Mass-energy, momentum, angular momentum, spin and electrical charge.
The relative strength of the 4 forces of nature.
The number of large dimensions in a universe.
The Hubble constant.
The ratio of baryonic matter to dark matter and dark energy.
It seems to me that the speed of light and Planck's constant may be more fundamental than other "constants" and the basic structure of the laws of physics may be more fundamental than the constants they use. But I could be wrong, perhaps the things that always remain the same are none of the above and we haven't even discovered them yet.
John K Clark
comment by Cyan · 2011-04-22T21:52:36.026Z · LW(p) · GW(p)
I enjoyed the review; but there's one comment I believe to be in error.
...Byrne does not make it clear if this means they are as numerous as the number of points on a line, or as numerous as an even larger infinite set like the set of all possible clock faces...
I'm pretty sure that the set of all possible clock faces has the same cardinality as the set of points in a line (see space-filling curve).
Replies from: Sniffnoy, gjm, johnclark↑ comment by Sniffnoy · 2011-04-22T22:37:56.794Z · LW(p) · GW(p)
What's even meant by a "clock face" in this context?
(BTW, note that you don't need a space filling curve to show that R has the same cardinality as R x R; what's surprising about a space filling curve is that it's additionally continuous.)
Replies from: Cyan↑ comment by Cyan · 2011-04-22T23:26:39.647Z · LW(p) · GW(p)
Good point. I just assumed that "clock face" meant something isomorphic to the unit square.
(I prefer linking space-filling curves as opposed to just the bare statement because space-filling curves are constructive.)
Replies from: Sniffnoy↑ comment by Sniffnoy · 2011-04-22T23:31:27.469Z · LW(p) · GW(p)
I just assumed that "clock face" meant something isomorphic to the unit square.
Like, what, a subspace of R^2 homeomorphic to it? OK, yeah, so only continuum-many.
(I prefer linking space-filling curves as opposed to just the bare statement because space-filling curves are constructive.)
Yeah I guess there isn't anything obvious to link to that shows how to do R x R constructively the direct way, huh?
Replies from: Cyan↑ comment by Cyan · 2011-04-23T00:56:58.270Z · LW(p) · GW(p)
When I read the anecdote about how Peano was inspired to give an explicit construction because Cantor's original proof didn't, I decided to not bother looking up info on Cantor's proof.
ETA: Re-reading my own link, I find that I misremembered what inspired Peano to search for space-filling curves -- he was looking specifically for continuous mappings, and my link was more specific than necessary, just as you pointed out in your first reply to me in this thread.
Replies from: Sniffnoy↑ comment by Sniffnoy · 2011-04-23T01:10:23.341Z · LW(p) · GW(p)
Huh, did Cantor do it by well-ordering or something? I wouldn't know. In any case it's pretty easy to explicitly put 2^N x 2^N in bijection with 2^N, because the former is just 2^(2N), and 2N is in bijection with N. What this cashes out to is, if you have two elements of 2^N, and want to make one that encodes them both, you just interleave them. Note also this works for any 2^S when you have an explicit bijection between S and 2S. If you want it for R all you need is an explicit bijection between R and 2^N. It's a more general consequence of well-ordering that S x S is of the same cardinality as S for any infinite S, and that is necessarily nonconstructive, but for "practical" infinite sets (by which I basically mean ℶ_n for n finite) many bijections can be made explicitly that would in general require choice.
↑ comment by gjm · 2011-04-23T00:03:12.564Z · LW(p) · GW(p)
No. Each clock face has the same cardinality as the points of a line, but the set of possible clock faces (which I take to mean something like "ways of colouring a disc", where how many colours you're allowed doesn't really matter) has the same cardinality as the set of ways of colouring a line, or equivalently the set of subsets of the line, in other words 2^continuum.
Replies from: Sniffnoy↑ comment by Sniffnoy · 2011-04-23T00:35:22.417Z · LW(p) · GW(p)
Voted down for starting off with "no" before it was established what what was being talked about. This all depends on what he meant by "clock face". Until that's clear it's simply impossible to say.
Replies from: gjm↑ comment by gjm · 2011-04-23T10:31:18.326Z · LW(p) · GW(p)
I'd been going to reply and say "butbutbutbut the only possible interpretation of 'clock face' that would make the cardinality not be 2^c would be that 'clock face' means 'pair of clock-hand positions', and that's just ridiculous" ... and then I looked at the rest of the thread and saw that that was apparently what JKC meant. My apologies.
EDITED to add: er, and actually there are perfectly reasonable interpretations that make the cardinality only the same as that of R, such as anything that can be described completely by a finite set of simple closed curves, or an RGB colouring where the colour at each point is a continuous function of position. So: total fail. Sorry again.
↑ comment by johnclark · 2011-04-23T05:10:45.700Z · LW(p) · GW(p)
The cardinality of the number of points on a line is the same as the number of points in a square or the number of points in a cube or the number of clock faces a properly operating clock will display. However the position of the clock hands where the short hand is exactly at 12 and the long hand is exactly at 6 is NOT included in the set of all valid clock faces, or just turn a clock upside-down and you will see a clock face that a proper clock will never display when it is right-side up. Thus the number of all possible clock faces must be have a higher cardinality than the number of valid clock faces or the number of points on a line; it is the same larger cardinality as the set of all 2 dimensional curves, which is the highest cardinality I can give a simple example of. The boring cardinality of valid clock faces is just the same as the points on a line.
Here is another way of seeing this. Cantors theorem says that for any set X the set of all subsets of X (called the Power Set of X) has a greater cardinality than X. For example 1,2,3,4,5.... would be one way of arranging all the integers, 2,1,3,4,5.... would be another. The set of all possible ways of arranging the integers is the Power Set of the integers, and it would have a higher cardinality than the integers (in this case the cardinality of the real numbers), and thus cannot be put into a list.
If C is the set of all clock faces a working clock can produce, then it is equivalent to an infinite set where the elements are a pair of real numbers (one for each hand) between 0 and 12; so {12,12} {3,12} and {.5,6} would all be members of this set, but {12,6} and {3,9} and {.5,12} would not be. This set C may be infinite but it's just ONE way the real numbers between 0 and 12 can be paired up. The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
Therefore the set of all possible clock faces has a higher cardinality than C the set of real numbers.
John K Clark
Replies from: Sniffnoy, steven0461↑ comment by Sniffnoy · 2011-04-23T05:40:17.762Z · LW(p) · GW(p)
Firstly, thank you for stating what you meant by clock faces. You should really have stated that explicitly, though, as it's not a standard term. Also I had to read that twice to notice you were making a distinction between "clock faces" and "valid clock faces".
But this is simply wrong:
However the position of the clock hands where the short hand is exactly at 12 and the long hand is exactly at 6 is NOT included in the set of all valid clock faces, or just turn a clock upside-down and you will see a clock face that a proper clock will never display when it is right-side up. Thus the number of all possible clock faces must be have a higher cardinality than the number of valid clock faces or the number of points on a line; it is the same larger cardinality as the set of all 2 dimensional curves, which is the highest cardinality I can give a simple example of.
If S is strictly contained in T, and S is a finite set, then T necessarily has strictly larger cardinality than S. The same does not hold for infinite sets - this is just the old "Galileo's paradox"; Z has the same cardinality as N despite strictly containing it.
The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
Therefore the set of all possible clock faces has a higher cardinality than C the set of real numbers.
EDIT: Sorry, I wrote something wrong here before due to misreading! Thanks to steven0461 for catching the real problem.
You seem to be equivocating between C and the power set of C. C is in bijection with R, its power set is not. (And since C is in bijection with R, its introduction was really unnecessary - you could have just used the power set of R.) (You also seem to be using unordered pairs when you want ordered pairs, but that's a more minor issue.)
In short this has a number of errors (fortunately they seem to be discrete, specifically locatable errors) and I suggest you go back and reread your basic set theory.
Replies from: johnclark, johnclark↑ comment by johnclark · 2011-04-23T06:33:07.934Z · LW(p) · GW(p)
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces. And I thought I was clear, I don't know what you mean about me equivocating between a set and it's power set; the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
You also say an entire paragraph is "simply wrong" but you don't say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don't you like in my statement?
John K Clark
Replies from: None, Sniffnoy, Cyan, JoshuaZ↑ comment by [deleted] · 2011-04-23T07:37:51.355Z · LW(p) · GW(p)
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces.
The set of all clock faces a working clock can produce - call this the set of all valid clock faces - has the same topology (and cardinality) as a circle. The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
However, the cardinality of a 2-dimensional torus is the same as the cardinality of a square, which is the same as the cardinality of a line (as you yourself recognize), which is the same as the cardinality of a circle.
Therefore the set of all valid clock faces has the same cardinality as the set of all possible clock faces.
the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
A power set indeed has a larger cardinality than the set it is a power set of. However, the set of all possible clock faces is not the power set of the set of all valid clock faces.
Replies from: johnclark↑ comment by johnclark · 2011-04-23T17:47:30.566Z · LW(p) · GW(p)
The set of all clock faces a working clock can produce - call this the set of all valid clock faces - has the same topology (and cardinality) as a circle.
Yes.
The set of all possible clock faces has the same topology (and cardinality) as a 2-dimensional torus.
Show me.
John K Clark
Replies from: None↑ comment by [deleted] · 2011-04-23T18:07:36.798Z · LW(p) · GW(p)
There are two hands, an hour hand and a minute hand. The set of all possible positions that the hour hand can take describes a circle. The same is true of the minute hand: its set of all possible positions describes a circle. Consequently, the set of all ordered pairs of possible positions (h,m), where h is the position of the hour hand and m is the position of the minute hand, is the Cartesian product of the two individual sets, and thus the Cartesian product of two circles. This is a two-dimensional torus.
Replies from: AdeleneDawner↑ comment by AdeleneDawner · 2011-04-24T00:29:32.502Z · LW(p) · GW(p)
Did you take into account that the positions of the two hands are not independent? When the hour hand of a given clock is at its 12:00:00 position, there's only one possible location of the minute hand for that clock, and this is true for any position of the hour hand.
Replies from: Cyan, wedrifid, None↑ comment by Cyan · 2011-04-24T00:55:03.856Z · LW(p) · GW(p)
If you read elsewhere in the thread, you'll see that johnclark draws a distinction between all possible clock faces and "valid clock faces", i.e., those that obey the constraint you describe. Constant is addressing the former, not the latter.
Replies from: None, AdeleneDawner↑ comment by AdeleneDawner · 2011-04-24T00:56:44.999Z · LW(p) · GW(p)
Ah. Okay.
↑ comment by [deleted] · 2011-04-24T01:03:54.635Z · LW(p) · GW(p)
You're talking about what I've been calling valid clock faces, the faces of a working clock. That set forms a circle. Right here I'm talking about what we've been calling the set of possible clock faces, where we no longer assume the gears of the clock are constraining the positions of the hands. This set forms a two-dimensional torus, the surface of a donut.
↑ comment by Sniffnoy · 2011-04-23T07:51:28.859Z · LW(p) · GW(p)
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces. And I thought I was clear, I don't know what you mean about me equivocating between a set and it's power set; the faces a working clock can produce is just ONE way all real numbers can be paired together, the power set is ALL the ways 2 real numbers can be paired together, it has a larger cardinality than the points on a line and is the number of all possible clock faces.
Evidently you are more confused than I realized. OK, last attempt at explaining this. The power set of R would be the set of all subsets of R, not just the set of all size-2 subsets of R. (I will ignore for now that you are talking about pairs and what you want here is ordered pairs.) The set of pairs of reals is in bijection with R. And any clock face, valid or not, can be described by an ordered pair of reals; there is no such clock face, valid or not, as {1, 3, 5} or {n in N | 2n+1 is prime}. Your conclusion that the set of clock faces has a higher cardinality than R does not follow, and in fact is false - as I pointed out in a cousin comment, R x R is in bijection with R, and as the set of pairs of reals injects into this, the result follows by Schroeder-Bernstein.
You seem to be equivocating between "the set of all clock faces (valid or not)" and "the set of all ways of pairing up members of R" (which could mean any of several different things, but for now we'll leave it unspecified as the distinction is irrelevant - they'd have the same cardinality). The latter does indeed have cardinality greater than that of R, but this is an entirely different set than the former.
You really need to be more precise with your language. "The set of all ways 2 real numbers can be paired together" would usually be interpreted to mean "the set of all 2-element subsets of R", not the much larger set above.
Earlier you wrote:
The Power Set of C would be the set of all subsets of paired numbers between 0 and 12, all the ways a pair of 2 real numbers can be arranged, all the ways 2 clock hands can be arranged not just the ways a properly operating clock will produce them.
It's really not clear what sets you're referring to here. "All subsets of paired numbers between 0 and 12?" "All the ways a pair of 2 real numbers can be arranged?" I can guess at what you mean but I can't be certain I'm right - especially because you are using these as if they are self-evidently the same, while my best guesses for what you mean by each of them, if they were taken in isolation, would be very different sets! Please go back and learn the standard terminology so people have some idea what you're saying.
You also say an entire paragraph is "simply wrong" but you don't say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don't you like in my statement?
I did not just say "the laws concerning finite sets and infinite sets are different"; I pointed out specifically which principle you appeared to be attempting to use that is not valid. Downvoted.
Replies from: johnclark↑ comment by johnclark · 2011-04-23T17:37:02.605Z · LW(p) · GW(p)
The power set of R would be the set of all subsets of R, not just the set of all size-2 >subsets of R.
I know that, but I'm not talking about R, I'm talking about the set a working clock could produce, call it VC for valid clock, the elements of this set consist of 2 real numbers. VC has the same number of points as there is on a line or in a square or in a cube. VC is one way all the real numbers can be put into pairs to form a set, but it is not the only way, there are infinitely many other ways and other sets. It's easy to find a mapping between the points on a line and all the clock faces a working clock can produce:
Every single point on the circular rim a clock is associated, without exception, to the face a working clock could display. Every single point. There is no room for a single extra association, much less the infinite number of them that would be needed. You could pick a point on the rim and say it is associated with the small hand being exactly at 12 and the large hand exactly at 6 but that would be untrue, that point has already been associated with a working clock face as can be seen just by moving the hour hand to point to that point, so now the same point is associated with 2 very different clock faces and that is a invalid mapping.
It's impossible to find a mapping between the points on a line (or on a circular rim) and all possible clock faces, so it must have a higher cardinality
The set of pairs of reals is in bijection with R.
No, one (not "the") infinite set whose elements are pairs of real numbers is in bijection with R, the set VC; but there are an infinite number of other infinite sets whose elements are pairs of real numbers, the set of all possible clock faces. This has a larger cardinality than R just like the set of all curves.
You seem to be equivocating between "the set of all clock faces (valid or not)" and >"the set of all ways of pairing up members of R"
I'm confused that you're confused.
The latter does indeed have cardinality greater than that of R
Thank you.
but this is an entirely different set than the former. You really need to be more precise >with your language. "The set of all ways 2 real numbers can be paired together" would >usually be interpreted to mean "the set of all 2-element subsets of R", not the much >larger set above.
How is "the set of all ways of pairing up members of R" different from "the set of all 2-element subsets of R" different from "the set of all ways 2 real numbers can be paired together"??
You say in the above that you agree with me that "the set of all ways of pairing up members of R" has a higher cardinality than the real numbers, and you certainly must agree that some of those number pairings a working clock would never produce, and you must agree that it would be easy to find a mapping between that set and the set of all possible clock faces. So what are we arguing about?
John K Clark
Replies from: Sniffnoy, komponisto↑ comment by Sniffnoy · 2011-04-23T21:10:13.284Z · LW(p) · GW(p)
Every single point on the circular rim a clock is associated, without exception, to the face a working clock could display. Every single point. There is no room for a single extra association, much less the infinite number of them that would be needed. You could pick a point on the rim and say it is associated with the small hand being exactly at 12 and the large hand exactly at 6 but that would be untrue, that point has already been associated with a working clock face as can be seen just by moving the hour hand to point to that point, so now the same point is associated with 2 very different clock faces and that is a invalid mapping.
This sort of reasoning only works with finite sets. I'm not going to bother to address the rest of your comment, because it's full of confusion and it's clear you really need to go back and relearn basic set theory. It would be a waste of all our time to continue this argument further.
↑ comment by komponisto · 2011-04-23T21:55:41.764Z · LW(p) · GW(p)
Every single point on the circular rim [of] a clock is associated, without exception, to the face a working clock could display.
And also, via a different association, to a face any clock (working or not) could display.
You are the victim of a very common misunderstanding, which is to forget that mappings between sets are allowed to vary when we use them for the purpose of comparing cardinalities.
↑ comment by Cyan · 2011-04-23T15:23:38.470Z · LW(p) · GW(p)
You can set up a one to one correspondence between all the points on a line (or in a square or in a cube) and all the clock faces a working clock can produce but you cannot do the same with all possible clock faces.
Perhaps I can help explain why this is wrong by giving a constructive counter-example. Correct me if I'm wrong, but by "the set of all clock faces" you mean the set of all positions the two hands of a clock could take. You can specify the position of the clock hands by stating the angles they make relative to any fixed position -- say the 12 position for concreteness. Suppose the angles are expressed in radians and take values in the set [0, 2*pi). Multiply each angle by the conversion factor "1 rotation per 2*pi radians" to map the angles into the set [0, 1). Now you can express any specific "clock face" by a point in the unit square. Then you can return to my original point about space-filling curves showing that this has the same cardinality as a line segment.
↑ comment by JoshuaZ · 2011-04-23T16:13:27.735Z · LW(p) · GW(p)
You also say an entire paragraph is "simply wrong" but you don't say what you object to other than to note that the laws concerning finite sets are different than those concerning infinite sets and obviously I agree. But what don't you like in my statement?
Sniffnoy's next remark was relevant. What he was saying was simply wrong was the idea that if one set A contains another set B then A must have higher cardinality than B. It seems that you have some confusion about how cardinality of infinite sets behaves. It might help to read the relevant Wikipedia entries starting with the basic one on cardinality or look at a standard textbook on set theory. Some of these issues will also be handled by a real analysis textbook.
↑ comment by johnclark · 2011-04-24T04:00:07.857Z · LW(p) · GW(p)
How did Cantor prove that there were more real numbers than integers? He set up a mapping between every single integer and a unique real number and then showed that there were still some real numbers not associated with an integer; this proved that the real numbers had a larger cardinality than the integers.
In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
It's incontrovertible that every number on the real number line is associated with unique clock face and it's also incontrovertible that not every clock face is associated with a unique number on the real number line; this is the very method one uses to determine the cardinality of infinite sets, it worked for Cantor and the logic is ironclad.
John K Clark
Replies from: JoshuaZ, Sniffnoy, Cyan↑ comment by JoshuaZ · 2011-04-24T04:09:26.543Z · LW(p) · GW(p)
In the same way I can show you a mapping that associates every single real number with a unique clock face (all the clock faces a properly working clock can produce in this case) but I can also show you clock faces (an infinite number of them in fact) that are not involved in this mapping; I can show you clock faces not associated with a real number, thus the number of all possible clock faces must have a larger cardinality than the real numbers.
This is not sufficient to show that you have a larger cardinality. This is essentially claiming that if I have sets A and B, and a bijection between A and a proper subset of B, then A and B must have different cardinality. This is wrong. To see a counterexample, take say the map from the positive integer to the positive integers from n -> n+1. In this example, A and B are both the positive integers. Since A=B they must be the same cardinality. But we have a 1-1, onto map from A into a proper subset of B since the map only hits 2,3.4... and doesn't hit 1.
What Cantor did is different. Cantor's proof that the reals have a larger cardinality than the natural numbers works by showing that for any map between the positive integers and the reals, there will be some reals left over. This is a different claim than exhibiting a single map where this occurs.
↑ comment by Sniffnoy · 2011-04-24T04:20:49.636Z · LW(p) · GW(p)
...screw it, I'll reply to this one just to point out what you should be looking up. That is not Cantor's proof, Cantor used (invented) the diagonal argument[0]. Nor is that a correct proof; if it were, it would prove that Q would have a larger cardinality than Z. You may remember this surprise? Q has the same cardinality as Z but R has strictly larger cardinality? If this doesn't sound familiar to you, you need to relearn basic set theory. If this does sound familiar to you but you don't see why it applies, you need to better develop your ability to analyze arguments, and relearn basic set theory (going by your previous statements).
EDIT: JoshuaZ points out a clearer counterexample to your argument in a brother comment.
Here. Here are some Wikipedia links to get you started.
http://en.wikipedia.org/wiki/Hume%27s_principle
http://en.wikipedia.org/wiki/Galileo%27s_paradox
http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
http://en.wikipedia.org/wiki/Equinumerosity
http://en.wikipedia.org/wiki/Bijection
http://en.wikipedia.org/wiki/Dedekind-infinite_set
http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
http://en.wikipedia.org/wiki/Cantor%27s_theorem
http://en.wikipedia.org/wiki/Cardinality
http://en.wikipedia.org/wiki/Cardinal_number
http://en.wikipedia.org/wiki/Injective_function
http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein%E2%80%93Schroeder_theorem
That should do for a start, though a more organized textbook may be preferable. Now you have at least something to read and I will spend no more time addressing your arguments myself as the linked pages do so plenty well.
[0]Yes, I know this was not his original proof. That is not the point.
↑ comment by Cyan · 2011-04-24T04:28:13.040Z · LW(p) · GW(p)
it's also incontrovertible that not every clock face is associated with a unique number on the real number line;
We can do a constructive counter-example for this one too, if you don't like space-filling curves. Take any real number in [0, 1) and construct two real numbers each of which is also in [0,1) by concatenating the first, third, fifth, etc., digits to make one real number and the second, fourth, sixth, etc., digits to make the second real number. Treat those real numbers as specifying fractions of a revolution for the two clock hands, as in my previous comment. Now every clock face is associated with a unique number in a subset of the real number line and vice versa.
↑ comment by steven0461 · 2011-04-23T05:29:15.233Z · LW(p) · GW(p)
But what you're talking about at the end is not the set of all possible clock faces. It's the set of all possible ways you could divide that set into "valid" and "invalid" clock faces, that is, the set of all possible sets of clock faces.
comment by waveman · 2013-05-13T02:13:15.180Z · LW(p) · GW(p)
I also enjoyed the book immensely and would commend it to anyone interested in the foundations of QM. Everett called his theory 'relative state' though he did not object to others calling it 'many worlds'. In E's theory there is one world, the wave function. We macroscopic beings are in a 'slice' of that world.
E's theory solves a lot of problems: derives the Born probability rule and removes the need to postulate it, explains Schrodinger's cat and Wigner's friend, fits perfectly with decoherence, allows partial measurements, removes the need to postulate a classical measurement apparatus, works with relarivistic QFT and QCD, has no spooky action at a distance, is deterministic at a fundamental level, and has no ill defined 'collapse'. Not bad.
The price is to give away the intuition that 'this' version of me is the only 'real' one. Just as we gave away the intuition that the earth is not moving post Copernicus.
The sociology is also interesting - Bohr was totally dominant and crushed the Everett heresy. John Wheeler, Everett's supervisor, was utterly cowed by Bohr.
Replies from: shminux↑ comment by Shmi (shminux) · 2013-05-13T03:36:49.800Z · LW(p) · GW(p)
Everett's model does not even explain simple state transitions. let alone anything more interesting, like radioactive decay.
Replies from: waveman↑ comment by waveman · 2014-03-19T05:03:28.210Z · LW(p) · GW(p)
Having read his thesis I think it does. Why not?
Edit: original thesis not thesis. There were two versions, a longer version with more proofs and detail and a shorter version ordered by Bohr which is sadly truncated and watered down.
Replies from: shminux↑ comment by Shmi (shminux) · 2014-03-19T15:48:07.026Z · LW(p) · GW(p)
How many worlds are created when an atom emits a photon when going from an excited state to the ground state?
Replies from: waveman↑ comment by waveman · 2015-08-12T03:08:46.936Z · LW(p) · GW(p)
I suspect you have not read his thesis.
The number of worlds is really measure not count.
Replies from: shminux↑ comment by Shmi (shminux) · 2015-08-12T05:21:16.315Z · LW(p) · GW(p)
The issue is not whether the number of worlds in finite or infinite, but when the worlds come into existence. When do you think it happens?
Replies from: waveman↑ comment by waveman · 2015-11-16T07:22:09.189Z · LW(p) · GW(p)
This is one of those "unask that question grasshopper!" situations.
There is according to Everett, just one world, consisting of the wave function.
As macroscopic beings we experience a projection of that wave function onto a lower dimensional space. One can think of one projection as one 'world' but really there is just one world. You have to read the whole thing unfortunately to really see it.
As quantum measurements occur the projections 'split'. This creates the illusion of indeterminacy because we only see part of the world.