Re: Smullyan Shmullyan, give me a real example
On Fri, 12 May 2006, Saibal Mitra wrote: > > Einstein seems to have believed in ''immortal observer moments''. > > In a BBC documentary about time it was mentioned that Einstein consoled a > friend whose son had died in a tragic accident by saying that relativity > suggests that the past and the future are as real as the present. > I'm sure Einstein would turn in his grave at your quoted expression. An immortal moment is a contradiction in terms, unless it implies a "second time" which passes as we contemplate "first time" embedded in 4-D space-time. Unfortunately a lot of popular discussion of space-time implicitly invokes this spurious second time, because it is hard to decouple the language of existence from the language of existence *in time*. To believe, with Einstein, that all points in space-time are equally real (because the relativity of simultaneity means that there is no unique "now") is quite the opposite of the nutty idea that all events exist "now" --- not even wrong, from Einstein's point of view. Einstein actually expressed this view in a letter of condolence to the widow of his old friend Michele Besso. His words are worth quoting accurately: "In quitting this strange world he has once again preceded me by just a little. That doesn't mean anything. For we convinced physicists the distinction between past, present, and future is only an illusion, however persistent." Later physicists, in particular John Bell, have pointed out that relativity doesn't *prove* that now is an illusion, it just makes it impossible to identify any objective "now". Not that any of this has anything to do with the sort of immortality contemplated by Everett, which is not at all enticing: like the Sibyl in classical myth, his immortality would not be accompanied by eternal youth... a rather horrible fate. --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Smullyan Shmullyan, give me a real example
On who invented quantum suicide, the following is from the biography of Hugh Everett by Eugene B. Shikhovtsev and Kenneth W. Ford, at http://space.mit.edu/home/tegmark/everett/ "Atheist or not, Everett firmly believed that his many-worlds theory guaranteed him immortality: His consciousness, he argued, is bound at each branching to follow whatever path does not lead to death --- and so on ad infinitum. (Sadly, Everett's daughter Liz, in her later suicide note, said she was going to a parallel universe to be with her father...)" The reference is to Everett's views in 1979-80, but there is no reason to suppose that Everett had only just thought of it at the time. On a personal note, some time in the '80s I met one of Everett's co-workers who told me that Everett used to justify his very unhealthy lifestyle on exactly these grounds. In our world, Everett died of a heart attack aged 52. I have always assumed that John Bell was thinking along these lines when he commented on Everett's theory: "But if such a theory was taken seriously it would hardly be possible to take anything else seriously." (1981, reprinted in _Speakable & Unspeakable in Quantum Mechanics). For that matter, this idea is implicit in Borges' story "The Garden of Forking Paths" (written before 1941), which provides the epigraph to the DeWitt & Graham anthology on The Many Worlds Interpretation. == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
Re: Questions on Russell's "Why Occam" paper
On Thu, 9 Jun 2005, Russell Standish wrote: Yes, if you think there is a concrete reality in which everything exists (your question of where does the observer live?), then the AP is a tautology. What I meant by "where does the observer live", in more formal language, is "how do you account for the (apparent) sense data we have?". I also have a strong preference for an account where the description of at least our "world" doesn't privilege one particular observer. In particular, this is hard to square with your insistence that "the observer provides the interpretation" of each of your bitstring universes. However, if you are prepared to allow for the possibility that observers exist "nowhere", then things are not quite so simple. One can always imagine being the brain-in-the-vat observer a reality which does not contain a body, or a brain, in a vat or anywhere else. Usually in this scenario, the observer will conclude that there must be a body somewhere else, and so concludes that it is inhabiting some kind of virtual reality. However, this implicitly assumes there has to a brain somewhere, and so implies a reality somewhere else for the brain to inhabit. But what if the brain is not required? Obviously, the last conclusion is full blown solipsism, but that is hardly a knock down argument. As both Hal and I keep trying to emphasise, we are interested in how, or whether, your theory can account for our own existence and the reality (or appearance, if you prefer) that we see around us. So the case of disembodied intelligences is a total a red herring. I don't really care whether these feature in your theory or not, but I do care whether you can account for (apparently) embodied intelligences. Instead, one can take the Anthropic Principle as an assertion of the reality we inhabit... Again, you are using a private language... the AP is not regarded as any such assertion by anyone else I've ever heard of. Most people regard their existence as proved by their own subjective experience, not some invented principle. If you don't agree I think we are just arguing about the meaning of "exist"... e.g. if we happen to be living in a computer simulation, or are just features of the solution of some set of equations, I would still say that we (really) exist. ... and experimentally test it. In all such cases is has been shown to be true, sometimes spectacularly. If we know "experimentally" "the reality we inhabit" (?!), which I guess I've just claimed that we do, why do we need a principle to assert it? Likely you mean something completely different, in which case please explain (with examples of said experiments!). Quoting me: Then you are implying that the observer can, in a finite time, read and attach meaning to a full (space-time) description of itself, including the act of reading this description and so on recursively. Not at all. Consistency is the only requirement. If the observer goes looking for erself, then e will find erself in the description. It doesn't imply the observer is doing this all the time. I think here we have run into the same inconsistency that you admitted in your discussion with Hal. In your first reply to Hal you assert that the observer O(x) attaches a unique meaning to the description string. Which would imply processing all bits of the string up to the start of the "don't care" region. A later reply suggest that we should in different contexts assume (a) this and (b) what your paper actually says, i.e. the meanings are updated as further bits are read. Now you have changed this again, and the observer is not (modelled by) a simple mapping but is a free agent who can choose to apply mappings to different "regions" of the bitstring at will. And even that doesn't actually answer my problem: let's assume the observer *does* "go looking for erself". You claim he will find himself, but if the description is *complete* my original problem remains: he will never finish reading his own description. Consequently the description will remain uninterpreted. In particular, he will never get to the part which would be interpreted as "himself now". So is there any sense in which *himself now* exists? This assumes that the string description contained a complete definition of the observer, which is a natural interpretation of your phrase: "Since some of these descriptions describe self aware substructures,..." But maybe you just meant that the string contains references to (tokens of) the observer. This would be consistent with your comment a couple of posts ago that both "observers" and "descriptions" are primary. This also seems to be consistent with your other recent post in response to Hal, in which the bitstrings are treated not so much as universes in the usual sense but as either the stream of sense data entering the observer's conciousness, or as a continuously-updated description of that concio
Re: collapsing quantum wave function
On Thu, 9 Jun 2005, Norman Samish wrote: Jonathan Colvin wrote: "If I take a loaf of bread, chop it half, put one half in one room and one half in the other, and then ask the question "where is the loaf of bread?", we can likely agree that the question is ill-posed." Depending on definitions, this may indeed be an ill-posed question. On the other hand, with appropriate definitions, the question might be answered by "The loaf is half in one room and half in the other," or "The loaf no longer exists." This reminds me of my problems trying to understand "the collapsing quantum wave function." I've heard of Schrödinger's Cat, which I'm told is half alive - half dead until the box is opened and the cat is observed. This observation "collapses the quantum wave function," and the cat at that point is either alive or dead. Here's a variation. Is my interpretation correct? Suppose we take ten apparently identical ball bearings and put stickers on each with the identifiers "1" through "10." We leave the room where the balls with stickers are, and a robot removes the stickers and mixes the balls up so that we don't which ball is which. However, the robot remembers which sticker belongs on which ball. We come back into the room and pick one ball at random to destroy by melting it in an electric furnace. If at this point we ask "What is the probability that the destroyed ball is ball '3'?" we can truthfully answer "My memory tells me that the destroyed ball has a one in ten probability of being '3.' " However, by reviewing the robot's record we can see that "6" was, in fact, the one destroyed. Does this mean that the quantum wave functions of all ten balls collapsed at the moment we viewed the record and observed what happened to "6"? Or did the wave function never exist, since the robot's record always showed the identity of the destroyed ball, irrespective of whether a human observed this identity or not? No this is not a quantum problem at all. The wavefunction does not encode ordinary lack-of-information uncertainty. Even if there was no robot, ball bearings are complicated enough that no two of them are genuinely identical, so there is always a fact of the matter about which was destroyed. Quantum "uncertainty" is better thought of as "both at once" rather than "either or". Here's a quantum analogue of your experiment. Take ten electrons held in a row of "Penning traps" (magnetic "bottles" that can hold single electrons) labelled 1 to 10 (the label is attached to the trap). Introduce an anti-electron into trap number 3, causing an annihilation, so we now have 9 electrons, held in traps 1, 2 and 4 to 10. Does this mean that electron number 3 was destroyed? No, because since electrons *are* genuinely identical, they are not individuals. The wavefunction for any group of electrons is always a perfect mixture of all possible "identity assignments", e.g. electron 1 in trap 1, 2 in trap 2 etc plus electron 2 in trap 1, 1 in trap 2 etc. This may sound ridiculous, but without this feature matter as we know it simply wouldn't exist, since it underlies the Pauli exclusion principle and hence the structure of atoms and all chemical properties. Paddy Leahy
Re: Questions on Russell's "Why Occam" paper
[Russell Standish wrote]: The AP is a statement that observed reality must be consistent with the observer being part of that reality. Famously, this can be interpreted as either a trivial tautology (Brandon Carter's original intention, I think), or an almost-obviously false principle of necessity (Barrow & Tipler's SAP). If you think there's a mystery here it suggests you go for the necessity version, but given your infinite ensemble the tautology would suffice perfectly well. You also said: >The observer _is_ the interpreter. There may well be more than one >observer in the picture, but they'd better agree! Why does this follow? It follows from the Anthropic Principle. If O_1 is consistent with its observed reality, and O_2 is consistent with its observed reality, and O_1 observes O_2 in its reality, then O_1 and O_2 must be consistent with each other (at least with respect to their observed realities). Ah. Just to be sure, do you mean that the string the observer "attaches meaning to" is the one which describes the very same observer? This seems to be implied by your comment above; but you don't say it or clearly imply it in your paper. Then you are implying that the observer can, in a finite time, read and attach meaning to a full (space-time) description of itself, including the act of reading this description and so on recursively. Which is impossible, of course. You also said: I'm not entirely sure I distinguish your difference between "external world" and "internal representation". We're talking about observations here, not models. I'm sure you can distinguish *my* mental representation of the world from your own. Hence if we share a world, and you can't distinguish between that world and your internal representation, then you are not granting equal status to other observers such as me. You also said (quoting me): My problem is that you are trying to make your observers work at two different levels: as structures within the universes generated (somehow!) by your bitstrings, but also as an interpretive principle for producing meaning by operating *on* the bitstrings. It's a bit like claiming that PCs are built by "The Sims". Yes it is a bit like that. Obviously, the Anthropic Principle (or its equivalent) does not work with "The Sims". Actually I don't see why not. The existence of The Sims implies a universe compatible with the existence of Sims. But granting this is not so for the sake of the argument, presumably the AP *will* apply to the Sims Mark VII which will be fully self-aware artificial intelligences. But it will still be absurd to claim that the Sims are responsible for construction of PCs (assuming they are not connected to robot arms etc, for which no analogs exist in your theory). Let alone for them to construct the actual PC on which they are running, as apparently implied by your last message... even robot arms wouldn't help there. Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
RE: Observer-Moment Measure from Universe Measure
On Tue, 7 Jun 2005, Hal Finney wrote: Jonathan Colvin writes: There's a question begging to be asked, which is (predictably I suppose, for a qualia-denyer such as myself), what makes you think there is such a thing as an "essence of an experience"? I'd suggest there is no such "thing" as an observer-moment. I'm happy with using the concept as a tag of sorts when discussing observer selection issues, but I think reifying it is likely a mistake, and goes considerably beyond Strong AI into a full Cartesian dualism. Is it generally accepted here on this list that a substrate-independent thing called an "observer moment" exists? Here's how I attempted to define observer moment a few years ago: Observer - A subsystem of the multiverse with qualities sufficiently similar to those which are common among human beings that we consider it meaningful that we might have been or might be that subsystem. These qualities include consciousness, perception of a flow of time, and continuity of identity. Observer-moment - An instant of perception by an observer. An observer's sense of the flow of time allows its experience to be divided into units so small that no perceptible change in consciousness is possible in those intervals. Each such unit of time for a particular observer is an observer-moment. So if you don't believe in observer-moments, do you also not believe in observers? Or is it the -moment that causes problems? Obviously, its the -moment. I'm pleased to see that Jonathan and Brent have the same problem with the concept that I do. Being an observer is a process. Slicing it into moments is OK mathematically, where a "moment" corresponds to a calculus dt (and hence is neither a particular length of time nor an instant). But to regard the "observer-state" at a particular moment as an isolated entity which is self-aware makes as much sense as regarding individual horizontal slices through a brain as being self-aware. It is the causal relation between successive brain states (incorporating incoming sense data) which constitutes intelligence, and self-awareness is just an epiphenomenon on top of intelligence, i.e. I would not agree that anything can be self-aware but have no intelligence. Paddy Leahy
Re: Questions on Russell's "Why Occam" paper
On Tue, 7 Jun 2005, Russell Standish wrote:
Hal dealt with this one already, I notice. 2^\aleph_0 = c. \aleph_1 is
something else entirely.
d'oh!
Now an observer will expect to find a SAS in one of the descriptions
as a corrolory of the anthropic principle, which is explicitly stated
as one of the assumptions in this work. I make no bones about this - I
consider the anthropic principle a mystery, not self-evident like
many people.
Very few supporters of the AP would "expect to find a SAS" in a bitstring.
Until you *specify* a way of interpreting the string, it contains nothing
but bits.
The observer specifies the interpretation.
But the observer is *generated* by the interpretation! Until you have an
interpretation, you have no observers. And until you have an observer, you
have no interpretation (at least that's how I read the sentence quoted
above).
How can structures which exist as some sort of pattern inside a bit string
(or am I supposed to say in the "meaning" integer output by some (other)
O(x)?) read a separate bitstring which exists as a parallel universe in
Platonia?
Why should an observer expect to see a token of erself
embedded in reality? That is the mystery of the AP.
What ARE you talking about? Observer's don't see tokens of
themselves...
I can see that I have a body - if I look in the mirror I can see a
face, eye etc, all of which appear to be under my control. This is a
token embedded in my reality that represents me.
So you find it a mystery that you have a face, eye etc?? If so, what does
the AP have to do with this mystery? Actually, maybe it would clarify
things if you said what you mean by the AP; it certainly doesn't seem to
be very like the AP that I know about.
I'm not sure whether your "my reality" refers to the external world or
your internal representation of it. I guess the latter, otherwise your
body would be you, not a token representing you.
In particular, any bitstring can be "interpreted" as any other bitstring
by an appropriate map. Hence until you specify an interpreter you are
simply not proposing a theory at all.
The observer _is_ the interpreter. There may well be more than one
observer in the picture, but they'd better agree!
Why does this follow? Your "observers" are maps O(x) from prefix strings
to the integers. Why can't you have two inconsistent maps... or rather,
how can you possibly avoid such? And since two different maps don't
interact at all (how can a mapping interact with another mapping?) each of
your observers seems to be sealed in his own little universe. In which
case having >1 observer appears to be an unverifiable speculation, which
is why I say it seems like solipsism.
All that is discussed in this paper is appearances - we only try to
explain the phenomenon (things as they appear). No attempt is made to
explain the noumenon (things as they are), nor do we need to assume
that there is a noumenon.
Most readers of your paper would take it that you are making a strong
ontological proposition, i.e. that the basis of reality is your set of
bitstrings.
This is the case.
Well, if you are making an ontological proposition, you are ipso facto not
just explaining appearances. In your model your "bitstrings" *are* the
noumenon (in Kant's terminology). Kant's point was that you can't infer
the nature of "things in themselves" from observation. He didn't say that
you can't speculate about their nature, and even guess right, by chance.
In effect, this mailing list discusses nothing but the nature of the
noumenon. (Kant would probably say this is a waste of time, of course).
I think either your terminology or you model has now got very confused.
Are your "observer" TMs the observers (SAS) whose experiences your theory
is trying to explain?
Yes.
In this case "where they live" is crucial because it
defines the environment the SAS find themselves in.
Why?
An intelligent system is "intelligent" by virtue of the way it interacts
with its environment. Think about the Turing test again: we conclude that
the computer is (not) intelligent because of the way it interacts with us.
To put it another way, you define these things as "observers". This
implies something observed. Obviously your model had better account for
people (observers) like us observing something like the world we see
("where we live"), and preferably interacting with other people who are
granted equal status your ontology.
It is not solipsism, if only for the reason that multiple observers
exist in our observed reality. They are all as real as our own consciousness.
Bruno Marchal calls this "shared dreaming". It seems apt.
If that's the *only* reason it's not solipsism, then I would say you just
don't have the courage of your convictions. Bruno's "shared dreaming"
sounds very like Leibniz's pre-established harmony, but that only works if
you believe in a provident deity (if it ever worked for an
Re: Can the arrow of time reverse?
On Mon, 6 Jun 2005, Norman Samish wrote: Norman Samish wrote: If the universe started contracting, its entropy would get smaller, which nature doesn't allow in large-scale systems. This seems to me an argument in support of perpetual expansion. Norman Samish writes: Thank you for your comments. My reasoning was that if a volume of gas contracts, its temperature must go up because particle collisions will occur more frequently. Since entropy is inversely proportional to temperature, the entropy must get smaller. If an entropy decrease upon contraction of our universe does not occur, does this mean that "the 'arrow of time' would reverse during the contraction"? Wouldn't this violate causality? No, it means that entropy is *not* inversely proportional to temperature. If a volume of gas is expanded or contracted adiabatically (i.e. with no heat exchanged with it's surroundings), its entropy stays constant. If it does exchange heat irreversibly, then entropy of (gas+surroundings) increases whether the gas expands or contracts (2nd law). Expansion of the universe (and re-collapse, if it happens) is roughly adiabatic, at least on very large scales (since there are no very large-scale temperature gradients that would drive heat transfer). Paddy Leahy
RE: where did the Big Bang come from?
On Mon, 6 Jun 2005, Jesse Mazer wrote: Norman Samish wrote: > Norman Samish wrote: >> And where did this mysterious Big Bang come from? A "quantum >> fluctuation of virtual particles" I'm told. > On Mon, 6 Jun 2005, Jesse Mazer wrote: > Whoever told you that was passing off speculation as fact--in fact there > is no agreed-upon answer to the question of what, if anything, came before > the Big Bang or "caused" it. > Patrick Leahy wrote: Maybe Norman is confusing the rather more legit idea that the "fluctuations" in the Big Bang, that explain why the universe is not completely uniform, come from quantum fluctuations amplified by inflation. This is currently the leading theory for the origin of structure, in that it has quite a lot of successful predictions to its credit. Norman Samish writes: Perhaps I didn't express myself well. What I was referring to is at http://www.astronomycafe.net/cosm/planck.html, where Sten Odenwald hypothesizes that random fluctuations in "nothing at all" led to the Big Bang. "This process has been described by the physicist Frank Wilczyk at the University of California, Santa Barbara by saying, 'The reason that there is something instead of nothing is that nothing is unstable.' ". . . "Physicist Edward Tryon expresses this best by saying that 'Our universe is simply one of those things that happens from time to time.' " But as I said, this idea is pure speculation, there isn't any evidence for it and we'd probably need a fully worked-out theory of quantum gravity to see if the idea even makes sense. Even then it would beg the question, why do the rules of quantum gravity apply? I.e. these answers are a bit of a con trick. Back in 1984 when Odenwald composed his text, there were still quite a few physicists who really thought that it would turn out that one and only set of physical laws were logically possible. This is one of those ideas that seems obviously false to any but True Believers, but there you go. In defense of Odenwald, he does clearly flag his description of events before GUT era as highly speculative. (Actually he is overconfident on the GUT era: you don't hear much about "leptoquark bosons" and "X Higgs" these days.) Moreover, the idea that "our" big bang within the level-2 multiverse (Tegmark's notation) was produced by a quantum fluctuation is probably a loose but reasonable description if you believe in the level-2 multiverse at all (which is a fairly speculative thing to do). Paddy Leahy
Re: objections to QTI
On Mon, 6 Jun 2005, Jesse Mazer wrote: Norman Samish wrote: If the universe started contracting, its entropy would get smaller, which nature doesn't allow in large-scale systems. This seems to me an argument in support of perpetual expansion. From what I've read, if the universe began contracting this would not necessarily cause entropy to decrease, in fact most physicists would consider that scenario (which would mean the 'arrow of time' would reverse during the contraction) pretty unlikely, although since we don't know exactly why the Big Bang started out in a low-entropy state we can't completely rule out a low-entropy boundary condition on the Big Crunch. This is quite correct. The idea that there are future as well as past boundary conditions is an extreme minority one. And where did this mysterious Big Bang come from? A "quantum fluctuation of virtual particles" I'm told. Whoever told you that was passing off speculation as fact--in fact there is no agreed-upon answer to the question of what, if anything, came before the Big Bang or "caused" it. Jesse Maybe Norman is confusing the rather more legit idea that the *fluctuations* in the Big Bang, that explain why the universe is not completely uniform, come from quantum fluctuations amplified by inflation. This is currently the leading theory for the origin of structure, in that it has quite a lot of successful predictions to its credit. Paddy Leahy
Re: Questions on Russell's "Why Occam" paper
On Mon, 6 Jun 2005, Russell Standish wrote: I am beginning to regret calling the all descriptions ensemble with uniform measure a Schmidhuber ensemble. I think what I meant was that it could be generated by a standard dovetailer algorithm, running for 2^\aleph_0 timesteps. It can't! Timesteps are denumerable, hence this statement is just a contradiction in terms. You better postulate your ensemble without reference to any algorithm to generate it. However, as the cardinality of "my" ensemble is actually "c" (cardinality of the real numbers), it is quite probably a completely different beast. There you go again with your radical compression. Without the reading I've been doing in the last two weeks, I wouldn't have been able to decode this statement as meaning: 2^\aleph_0 = \aleph_1 (by definition) To assume c = \aleph_1 is the Continuum Hypothesis, which is unprovable (within standard arithmetic). Now an observer will expect to find a SAS in one of the descriptions as a corrolory of the anthropic principle, which is explicitly stated as one of the assumptions in this work. I make no bones about this - I consider the anthropic principle a mystery, not self-evident like many people. Very few supporters of the AP would "expect to find a SAS" in a bitstring. Until you *specify* a way of interpreting the string, it contains nothing but bits. Why should an observer expect to see a token of erself embedded in reality? That is the mystery of the AP. What ARE you talking about? Observer's don't see tokens of themselves... if anyone (God?) has a 3rd-person/bird's eye view, it is certainly not someone who is included in any particular reality. No way is anything like this implied by the AP. All the AP requires is that there *be* observers/SAS in (real) universes, which is true in our case at least. And now we find not only that the bit string is a description, but it is a complex enough description to describe SAS's? How does that work? The bitstrings are infinite in length. By reading enough bits, they can have arbitrarily complex meanings attached to them. In particular, any bitstring can be "interpreted" as any other bitstring by an appropriate map. Hence until you specify an interpreter you are simply not proposing a theory at all. All that is discussed in this paper is appearances - we only try to explain the phenomenon (things as they appear). No attempt is made to explain the noumenon (things as they are), nor do we need to assume that there is a noumenon. Most readers of your paper would take it that you are making a strong ontological proposition, i.e. that the basis of reality is your set of bitstrings. If this is *not* the case, and you think the bitstrings may be represented in some deeper reality (or maybe are just metaphors), then what is the motivation for your proposal? Why do we need to think about this intermediate layer of bitstrings? The original simplicity goes out the window. BTW I'm with Kant: you can't have an appearance without an underlying reality, even if that is unknowable. Bruno Marchal has a detailed discussion on this in his thesis, and concludes that he "has no need for this hypothesis" (what he calls the extravagant hypothesis). So the former statement is true :[the description strings are] things that "observer" TM's observe and map to integers. It is also true that descriptions of self aware observers will appear within the description by the Anthropic Principle. The phenomenon of observerhood is included. However where the observers actually live is not a meaningful question in this framework. I think either your terminology or you model has now got very confused. Are your "observer" TMs the observers (SAS) whose experiences your theory is trying to explain? In this case "where they live" is crucial because it defines the environment the SAS find themselves in. If you are not careful your theory becomes effectively that we are all "brains in bottles" or Leibnizian monads, which is solipsism by another name. Or are your "observers" the missing "interpreters" in your theory which give it meaning, and allow us to find (in principle) the SAS within the bitstrings that represent actual observers like us? In this case it's unhelpful to call these meta-entities "observers"; rather, in effect, they constitute the (meta-)laws of physics. Incidentally, a TM by itself can't generate meaning, as it is only a map from integers to integers. You still have to specify externally how to interpret the code as something more than a mere number. (E.g. in the Turing test the output bits have to be processed into English language text). The page then goes on to make some comments about measure applied to universes. Here again I am confused about how to relate it to all that has been descibed. What are the analogs of universes, in this model? Is it "descriptions", the infinite bit strings? From what h
Julian Barbour (was: Re: objections to QTI)
I read his book a year or so ago, so may be a bit hazy, but: Pour Bruno: he definitely does not want to talk about space-time capsules. Partly this is motivated by his metaphysical ideas about time, partly by the technicalities of the 3+1 (i.e. space+time, not persons!) approach to GR and the Wheeler-De Witt equation which he advocates. This leads him into severe difficulties, and he has not successfully described how this can be reconciled with the relativity of simultaneity, which he also wants to assert. Barbour regards this as an open question within his theory; others regard it as a fatal objection. Of course when Barbour says that "time is an illusion" he really means that the *flow* of time is an illusion, or rather a category error, which is a pretty standard position (e.g. forcefully argued by Deutch in his book). Although he sometimes speaks as though he denies it, I think if push came to shove he would have to admit that there is an identifiable, objective, structural feature in his (or anybody's) theory of physics which corresponds to time. Reminds me of the opening of a history book: "There was no such thing as the Scientific Revolution, and this is a book about it." Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
Re: Plaga
As an exercise I've been trying to pinpoint exactly what is wrong with Plaga's paper. For anyone who doubts that it *is* wrong, note that it proposed 10 years ago an experiment which he said was feasible with what was then state-of-the-art equipment. This technology has now massively advanced. The experiment would guarantee a Nobel prize for anyone who performed it successfully. In that time the paper has been cited in the published literature only 3 times, and never by an experimental physicist. And this is not because the paper was unnoticed by the community at the time, e.g. it was publicised by John Baez, whose writings are widely read. On careful reading, the paper is just littered with confusions and errors. I guess this explains why no-one bothered to publish a rebuttal; this falls in the class of "not even wrong". Probably the root problem is a confusion about the true nature of decoherence. Decoherence is often presented using the maths of density matrices, so I better explain this briefly: Density matrices allow you to handle the case when you don't know the exact quantum state. The procedure is to divide your description into a measurable "system" and a complex, not-measurable-in-detail "environment". One can then define the density matrix of the combined system, and "trace out" the uncertain state of the environment, giving a density matrix for the system alone in the absence of information about the environment. A test to see if the system has been decohered by its interaction with the environment is that the off-diagonal terms in the system-only density matrix go to zero. Plaga clearly accepts the usual position that irreversible branching in MWI occurs when decoherence is (FAPP) total. If you follow this through in Plaga's example, you do indeed find that the density matrix for the states of his trapped ion, |A1> and |A2>, is diagonal, confirming the obvious that once a macroscopic measurement has taken place, we have total decoherence. But what Plaga does in his Eq 8 is to reverse the roles of system and environment (he actually does the algebra wrong but the numerical answer is unaffected). Because at this stage the ion knows nothing about the rest of the lab, he gets a density matrix *for the lab* with large off-diagonal terms, corresponding to a "pure" state: (|W1> + |W2>) / sqrt(2). So far, so correct (after all, in MWI the state is *always* pure). But he now concludes that decoherence has not yet occurred. *WRONG*. The condition "off-diagonal terms go to zero" is just a sufficient condition for decoherence. It is only necessary if the "system" itself is so simple that it could not decohere without the help of the environment. But Plaga is treating the complex, macroscopic lab as the "system" and that certainly can decohere without the help one more ion. The more basic definition is that decoherence has occured once the states are permanently orthogonal, so you cannot demonstrate quantum interference. Plaga correctly states that |W1> and |W2> *are* permanently orthogonal, but does not realise that this means that decoherence *is* complete, contrary to what he says. Another way to put this is that the observer "Silvia" doesn't need the density matrix in Eq. (8) because she knows for sure already whether she detected the original photon or not, hence whether she is in branch |W1> or |W2>. Given this, the rest of Plaga's argument is just irrelevant. But he should have noticed that his process blatantly violates the linearity of time evolution, which is one of the fundamental assumptions of MWI QM. This is manifest in his Eq. 6 which associates an excited ion with the |P2> term in which no excitation took place (if you start with a photon in state |P2>, when the photon is guaranteed not to be detected, the ion is never excited). Hence Eq 6 is not a linear superposition of the two possible histories. Hence, if we saw what he predicted, we would actually *disprove* MWI QM, not confirm it as he thinks. Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
RE: White Rabbit vs. Tegmark
On Thu, 26 May 2005, Brent Meeker wrote: I agree with all you say. But note that the case of finite sets is not really any different. You still have to define a measure. It may seem that there is one, compelling, natural measure - but that's just Laplace's principle of indifference applied to integers. The is no more justification for it in finite sets than infinite ones. That there are fewer primes than non-primes in set of natural numbers less than 100 doesn't make the probability of a prime smaller *unless* you assign the same measure to each number. Brent Meeker I'll answer both Brent and Hal (m6556) here. Yup, I hadn't thought through the measure issue properly. Several conclusions from this discussion: * As Brent says, you always have to assume a measure. Sometimes a measure seems so "natural" you forget you're doing it, as above. Another example is in an infinite homogeneous universe, where equally a uniform measure seems natural, and also limiting frequencies over large volumes are well defined, as per Hal's message. * As Hal points out, it *is* possible to assign probability measures to countably-infinite sets. * Alternatively, you can assign a non-normalizable measure (presumably uniform) and take limiting frequencies. But then as per Cantor the answer does depend on your ordering, which is something extra you are adding to the definition of the set (even for numerical sequences). * Different lines of argument can easily lead to different "natural" measures for the same set, e.g. Hal's "Universal Distribution" vs. Laplacian indifference for the integers. * For me, the only way to connect a measure with a probability in the context of an "everything" theory is for the measure to represent the density of universes (or observers or observer-moments if you factor that in as well). * Since the White Rabbit^** argument implicitly assumes a measure, as it stands it can't be definitive. * But the arbitrariness of the measure itself becomes the main argument against the everything thesis, since the main claimed benefit of the thesis is that it removes arbitrary choices in defining reality. Paddy Leahy ^** "This is a song about Alice, remember?" --- Arlo Guthrie
Re: White Rabbit vs. Tegmark
On Thu, 26 May 2005, Alastair Malcolm wrote: An example occurs which might be of help. Let us say that the physics of the universe is such that in the Milky Way galaxy, carbon-based SAS's outnumber silicon-based SAS's by a trillion to one. Wouldn't we say that the inhabitants of that galaxy are more likely to find themselves as carbon-based? Now extrapolate this to any large, finite number of galaxies. The same likelihood will pertain. Now surely all the statistics don't just go out of the window if the universe happens to be infinite rather than large and finite? Alastair Well, it just does, for countable sets. This is what Cantor showed, and Lewis explains in his book. Cantor defines "same size" as a 1-to-1 pairing. Hence as there are infinite primes and infinite non-primes there are the same number (cardinality) of them: (1,3), (2,4), (3,6), (5,8), (7,9), (11,12), (13,14), (17,15), (19,16) etc and so ad infinitum You might say there are obviously "more" non-primes. This means that if you list the numbers in numerical sequence, you get fewer primes than non-primes in any finite sequence except a few very short ones. But in another sequence the answer is different: (1,2,4) (3,5,6) (7,11,8) (13,17,9) etc ad infinitum. In this infinite sequence, each triple has two primes and only one non-prime. Hence there seem to be more primes than non-primes! For the continuum you can restore order by specifying a measure which just *defines* what fraction of real numbers between 0 & 1 you consider to lie in any interval. For instance the obvious uniform measure is that there are the same number between 0.1 and 0.2 as between 0.8 and 0.9 etc. Why pick any other measure? Well, suppose y = x^2. Then y is also between 0 and 1. But if we pick a uniform measure for x, the measure on y is non-uniform (y is more likely to be less than 0.5). If you pick a uniform measure on y, then x = sqrt(y) also has a non-uniform measure (more likely to be > 0.5). A measure like this works for the continuum but not for the naturals because you can map the continuum onto a finite segment of the real line. In m6511 Russell Standish describes how a measure can be applied to the naturals which can't be converted into a probability. I must say, I'm not completely sure what that would be good for. Paddy Leahy
Re: Induction vs Rubbish
On Wed, 25 May 2005, Benjamin Udell wrote: The induction-friendly universe with so much detectable rubbish that a wide variety of phenomena cannot be unified into a simple theory sounds like a universe where induction works but surmise, or inference to the simplest explanation, faces grave difficulties and too often fails. In other words, in difficult cases, efforts toward surmise -- i.e., "rambling speculations about half-formed ideas that probably won't pan out to anything" -- really will lead too often too far astray to be practicable, and cogent everyday surmises would be few and far between -- not everyday or quotidian at all. A greatly increased difficulty in the formation of explanatory hypotheses would, it seems, hamper not only science but SASs in general. Would intelligence and commonsense perception tend, on balance, to be useful in such a world? It sounds like a world which would allow vegetable-like systems (i.e., essentially mindless in the usual sense) but be severely punitive toward SASs inclined to try to be shrewd or clever and to try, for instance, to infer particular entities or events or universal laws (as opposed to prolonged tendencies) as explanatory reasons, or to try to play architect instead of subsisting on the continuation of tendencies. It also sounds like the evolution or "natural architecting" of even merely vegetable-like systems would likely be under pressure to play it a lot safer than it does in our world, so that the systems thus evolved would tend to be not only vegetable-like but also a lot more "generic" than those which we see. I guess I'm trying to argue (unconfidently) or suggest, for what it's worth, that induction-friendly but much-detectable rubbish universes with SASs are induction-friendly but surmise-unfriendly universes with SASs, and that their measure would be rather small. Best regards, Ben Udell It's a question of degree, again. There is surely a level of "noise" which doesn't cause the problems you mention (although I would say that surmise, common sense etc are basically inductive reasoning from past experience, including past experience genetically encoded by natural selection which is one big inductive experiment). For most of history, the world has seemed a pretty random place to people (probably still does to most people), but they managed to survive without understanding how QM unifies the structure of matter, Natural selection explains so much about living things, etc. If the rubbish was there, we'd get used to it. Only scientists would be frustrated that they couldn't make any kind of sense of it. But they would be able to isolate the features of their world which did show regularity, so it wouldn't prevent science, either. Paddy
Re: Induction vs Rubbish
On Wed, 25 May 2005, Russell Standish wrote: On Tue, May 24, 2005 at 10:10:19PM +0100, Patrick Leahy wrote: Lewis also distinguishes between inductive failure and rubbish universes as two different objections to his model. I notice that in your articles both you and Russell Standish more or less run these together. I'm interested in this. Could you elaborate please? I haven't had the advantage of reading Lewis. If what you mean by by the first is why rubbish universes are not selected for, it is because properties of the selected universe follow a distribution with well defined probability, the universal prior like measure. This is dealt in section 2 of my paper. If you mean by failure of induction, why an observer (under TIME) continues to experience non-rubbish, then that is the white rabbit problem I deal with in section 3. It comes down to a "robustness" property of an observer, which is hypothesised for evolutionary reasons (it is not, evolutionarily speaking, a good idea to be confused by hunters wearing camouflage!) In that case, how am I conflating the two issues? If I'm barking up the wrong tree, I'd like to know. It's the second point where I think you conflate two problems. My distinction is a little different from Lewis' anyway. From my pov, it's a matter of degree, but one which makes a qualitative difference: * Failure of induction: the past fails to predict the future. This occurs in universes a la Hume where physical laws only appear to have been followed by some massive fluke. Also in universes which always had no, or very little, regularity. I claim that as soon as regularity breaks down to this extent, SAS cease to exist, so no matter how common these cases are, we never observe them. No problem. (Lewis' defence is different). * White Rabbit: cognizable universes require a high degree of regularity for the survival of SAS (not to mention evolution), as above. Hence induction in any cognizable universe will work most of the time (which is all it does anyway), for a sufficient set of properties of the world. The key point is that this is not *every* property, and not all of the time. Hence there should be universes in which SAS can survive pretty well, but contain a wide variety of phenomena which cannot be unified into a simple theory. An extreme case is the "rubbish" universe proposed against Lewis, in which the extra phenomena are completely undetectable. Lewis takes this as a serious objection and counters by arguing that it is not possible to say that such universes are "more likely". As scientists, I guess we would only take seriously detectable rubbish. NB: whatever the measure you use, unless extremely artificial, the rubbish almost certainly would have much higher entropy than talking White Rabbits. Think of reality has having "snow", like a badly-tuned TV. Of course on objective state-reduction models of QM, our universe does have "snow" in the form of random quantum jumps. But this is a very regular form of snow, which does "unify" into the basic physical laws. The argument is that for some plausible measures (not yours, obviously), even macro-scale snow is much more likely than not. Paddy Leahy
RE: White Rabbit vs. Tegmark
On Wed, 25 May 2005, Stathis Papaioannou wrote: Consider these two parallel arguments using a version of the anthropic principle: (a) In the multiverse, those worlds which have physical laws and constants very different to what we are used to may greatly predominate. However, it is no surprise that we live in the world we do. For in those other worlds, conditions are such that stars and planets could never form, and so observers who are even remotely like us would never have evolved. The mere fact that we are having this discussion therefore necessitates that we live in a world where the physical laws and constants are very close to their present values, however unlikely such a world may at first seem. This is the anthropic principle at work. (b) In the multiverse, those worlds in which it is a frequent occurence that the laws of physics are temporarily suspended so that, for example, talking white rabbits materialise out of thin air, may greatly predominate. However, it is no surprise that we live in the orderly world that we do. For in those other worlds, although observers very much like us may evolve, they will certainly not spend their time puzzling over the curious absence of white rabbit type phenomena. The mere fact that we are having this discussion therefore necessitates that we live in a world where physical laws are never violated, however unlikely such a world may at first seem. This is the *extreme* anthropic principle at work. If there is something wrong with (b), why isn't there also something wrong with (a)? --Stathis Papaioannou Good point, this is a fundamental weakness of the AP. If you take it to extremes, we should not be surprised by *anything* because the entire history of our past light-cone to date, down to specific microscopic quantum events, is required in order to account for the fact that you and I are having this particular exchange. To give the AP force, you have to work on the most general possible level (hence it was a big mistake for Barrow & Tipler to restrict it to "carbon-based life forms" in their book, certainly not in line with Brandon Carter's original thought). Paddy Leahy
Re: Observables, Measurables, and Detectors
It looks as though you advocate a role for each of these:
observables
measurements
detectors
and for all I know
observers
It seemed to me that MWI allowed me to get away with a considerable
simplification. Gone were observers and even observations. Even
measurements, I discard. (After all, who can say that a measurement
occurs in the middle of a star? And yet things do go on there, all
the time.)
Now *some* of that language perhaps returns when decoherence is
discussed. I mean, I'll grant that *something* significant starts
off a new branch, and so it's okay for it to have a name. :-)
But here is what I'd like to be able to say:
A new branch starts, or decoherence obtains, or an irreversible
transformation occurs, or a record is made. They all seem the
same to me. Why not?
My main motivation is to get as far away from Copenhagen as possible,
and so thereby get free of observers and observations, and anything
else that seems to afford some pieces of matter a privileged status.
Do you think that such simplified language leaves out anything important?
I don't think we disagree much about the physics. The trouble is, the
physics is even simpler than you suggest. Branching is not something
special in the theory, it is a macroscopic description that we apply to
what emerges from the theory. If you simplify your language too much, all
that happens is you have to define all those useful approximate terms from
scratch.
Just for fun, here's how it would go:
The framework of QM in the MWI is that
(1) The state of the "system" (universe) can be represented by a
time-dependant, normalized vector, say |S>, in a Hilbert space.
(2) Time evolution of |S> is linear.
That's it! (1) implies that time evolution is also unitary, so the vector
stays normed. (1) + (2) imply the Schrodinger equation, including the fact
that the generator of time evolution ("Hamiltonian") is a Hermitian
operator. (2) causes all the trouble.
A full (non-framework) description requires you to (a) specify the Hilbert
space (b) specify the Hamiltonian (c) specify the initial state. None of
which are known exactly for the universe. (And in fact for the universe as
a whole we had better adapt this description to relativity somehow, since
you can't just take time as a given.)
Now to introduce some more specific terms so we can relate the theory to
everyday reality.
"Observable": In a simple system, the set of values of an observable are
simply the labels we attach to elements of a basis, i.e. a set of
orthogonal unit vectors (defining a "coordinate system"), in Hilbert
space. We can freely choose any basis we like, but some are more useful
than others because they relate to the structure and symmetries of the
Hamiltonian. Let's call a basis {|o>} where o is our variable label. The
set might be finite, denumerable, or continuous, depending on the size of
the Hilbert space. For convenience, and to make the transition to
classical physics as seamless as possible, the labels are usually chosen
to be real numbers.
To put my previous answer to Serafino into this context, note that
observables (e.g. position) play a very different role in the theory from
time.
For each basis, we can construct a linear operator on Hilbert-space
vectors whose eigenvectors are the basis vectors and whose eigenvalues are
our "observable" labels. If our labels are real, the operator will be
Hermitian. With suitable choice of labels, the algebra of some of these
operators approximately maps onto the algebra of variables in classical
physics, which explains why classical physics works, and also how QM was
discovered. (In particular, since the Hamiltonian itself is hermitian it
has a set of real eigenvalues which we call "Energy").
"Wave Function": The inner product of a basis vector with the state
vector, written , is "geometrically" the length of the projection of
the state onto that basis vector, and so the "cartesian coordinate" along
the axis defined by |o>. In conventional QM it is the probability
amplitude for "observing" o. If the basis is continuously infinite, as in
position or momentum, is a continuous function of the real variable
(observable) o. This is what we call the "wave function" in o-space. (e.g.
o = position, or momentum).
"Subsystems": In a complex system, we have to be a bit more careful. What
physicists call observables certainly don't parameterize a complete basis
for the universe. Such a complete basis would be characterised by a
"complete set of commuting observables". Commuting because their
characteristic operators commute. In effect, we factorize the Hilbert
space into subspaces (corresponding to quasi-independent subsystems).
Practical observables correspond to bases on some subspace.
"Branching": In *some* bases of sufficiently complex systems (appropriate
basis and needed complexity depending again on the Hamiltonian), the
time-structure of the wavefunction appr
Re: Hamel Basis
I know this one! I had a friend who published a magazine called "Zorn" printed on pale yellow paper... ;) Paddy Leahy
Re: White Rabbit vs. Tegmark
On Tue, 24 May 2005, Alastair Malcolm wrote: Perhaps I can throw in a few thoughts here, partly in the hope I may learn something from possible replies (or lack thereof!). - Original Message - From: Patrick Leahy <[EMAIL PROTECTED]> Sent: 23 May 2005 00:03 . This is not a defense which Tegmark can make, since he does require a measure (to give his thesis some anthropic content). I don't understand this last sentence - why couldn't he use the 'Lewisian defence' if he wanted - it is the Anthropic Principle (or just logic) that necessitates SAS's (in a many worlds context): our existence in a world that is suitable for us is independent of the uncountability or otherwise of the sets of suitable and unsuitable worlds, it seems to me. (Granted he does use the 'm' word in talking about level 4 (and other level) universes, but I am asking why he needs it to provide 'anthropic content'.) You have to ask what motivates a physicist like Tegmark to propose this concept. OK, there are deep metaphysical reasons which favour it, but the they arn't going to get your paper published in a physics journal. The main motive is the Anthropic Principle explanation for alleged fine tuning of the fundamental parameters. As Brandon Carter remarks in the original AP paper, this implies the existence of an ensemble. Meaning that fine tuning only ceases to be a surprise if there are lots of universes, at least some of which are congenial/cognizable. But this bare statement is not enough to do physics with. But suppose you can estimate the fraction of cognizable worlds with, say the cosmological constant Lambda less than its current value. If Lambda is an arbitrary real variable, there are continuously many such worlds, so you need a measure to do this. This allows a real test of the hypothesis: if Lambda is very much lower than it has to be anthropically, there is probably some non-anthropic reason for its low value. (Actually Lambda does seem to be unnecessarily low, but only by one or two orders of magnitude). The point is, without a measure there is no way to make such predictions and the AP loses its precarious claim to be scientific. There are hints that it may be worth exploring fundamentally different approaches to the White Rabbit problem when we consider that for Cantor the set of all integers is the same 'size' as that of all the evens (not too good on its own for deciding whether a randomly selected integer is likely to come out odd or even); similarly for comparing the set of all reals between 0 and 1000, and between 0 and 1. The standard response to this is that one *cannot* select a real (or integer) in such circumstances - but in the case of many worlds we *do* have a selection (the one we are in now), so maybe there is more to be said than that of applying the Cantor approach to real worlds, and also on random selection. This is very reminiscent of Lewis' argument. Have you read his book? IIRC he claims that you can't actually put a measure (he probably said: you can't define probabilities) on a countably infinite set, precisely because of Cantor's pairing arguments. Which seems plausible to me. Lewis also distinguishes between inductive failure and rubbish universes as two different objections to his model. I notice that in your articles both you and Russell Standish more or less run these together. A final musing on finite formal systems: I have always considered formal systems to be a provisional 'best guess' (or *maybe* 2nd best after the informational approach) for exploring the plenitude - but it occurs to me that non-finitary formal systems (which could inter alia encompass the reals) may match (say SAS-relevant) finite formal systems in simplicity terms, if the (infinite-length) axioms themselves could be algorithmically generated. This would lead to a kind of 'meta-formal-system' approach. Just a passing thought... I think this is the kind of trouble you get into with the "mathematical structure" = formal system approach. If you just take the structure as mathematical objects, you are in much better shape. For instance, although there are aleph-null theorems in integer arithmetic, and a higher order of unprovable statements, you can just generate the integers with a program a few bits long. And the integers are the complete set of objects in the field of integer arithmetic. Similarly for the real numbers: if you just want to generate them all, draw a line (or postulate the complete set of infinite-length bitstrings). No need to worry about whether individual ones are computable or not. Paddy Leahy
RE: Sociological approach
On Tue, 24 May 2005, aet.radal ssg wrote: "See http://decoherence.de "? It was good for a laugh, not much else. Funnily enough, that was my thought about your friend Plaga, whose paper is rubbish because he doesn't know the first thing about decoherence, and fails to notice that his proposed solution violates linearity of the Schrodinger equation. Whereas the articles on the above web site are by people actively involved in research on decoherence, including the person who invented it (Zeh). Paddy Leahy
RE: Sociological approach
On Mon, 23 May 2005, Brent Meeker wrote: -Original Message- From: Patrick Leahy [mailto:[EMAIL PROTECTED] NB: I'm in some terminological difficulty because I personally *define* different branches of the wave function by the property of being fully decoherent. Hence reference to "micro-branches" or "micro-histories" for cases where you *can* get interference. Paddy Leahy But in QM different branches are never "fully decoherent". The off axis terms of the density matrix go asymptotically to zero - but they're never exactly zero. At least that's standard QM. However, I wonder if there isn't some cutoff of probabilities such that below some value they are necessarily, exactly zero. This might be related to the Bekenstein bound and the holographic principle which at least limits the *accessible* information in some systems. I'm talking about standard QM. You are right that my definition of macroscopic branches is therefore slightly fuzzy. But then the definition of any macroscopic object is slightly fuzzy. I don't see any need for a cutoff probability... the probabilities get so low that they are zero FAPP (for all practical purposes) pretty fast, where, to repeat, you can take FAPP zero as meaning an expectation of less than once per age of the universe.
Re: White Rabbit vs. Tegmark
On Mon, 23 May 2005, Hal Finney wrote: I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. The cardinality of such functions is c, the same as the continuum. The existence of the constant functions alone shows that it is at least c, and my understanding is that continuous, let alone differentiable, functions have cardinality no more than c. Oops, mea culpa. I said that wrong. What I meant was, what is the cardinality of the data needed to specify *one* continuous function of the continuum. E.g. for constant functions it is blatantly aleph-null. Similarly for any function expressible as a finite-length formula in which some terms stand for reals.
Re: White Rabbit vs. Tegmark
On Mon, 23 May 2005, Bruno Marchal wrote: Concerning the white rabbits, I don't see how Tegmark could even address the problem given that it is a measure problem with respect to the many computational histories. I don't even remember if Tegmark is aware of any measure relating the 1-person and 3-person points of view. Not sure why you say *computational* wrt Tegmark's theory. Nor do I understand exactly what you mean by a measure relating 1-person & 3-person. Tegmark is certainly aware of the need for a measure to allow statements about the probability of finding oneself (1-person pov, OK?) in a universe with certain properties. This is listed in astro-ph/0302131 as a "horrendous" problem to which he tentatively offers what looks suspiciously like Schmidhuber's (or whoever's) Universal Prior as a solution. (Of course, this means he tacitly accepts the restriction to computable functions). So I don't agree that the problem can't be addressed by Tegmark, although it hasn't been. Unless by "addressed" you mean "solved", in which case I agree! Let's suppose with Wei Dai that a measure can be applied to Tegmark's everything. It certainly can to the set of UTM programs as per Schmidhuber and related proposals. Obviously it is possible to assign a measure which solves the White Rabbit problem, such as the UP. But to me this procedure is very suspicious. We can get whatever answer we like by picking the right measure. While the UP and similar are presented by their proponents as "natural", my strong suspicion is that if we lived in a universe that was obviously algorithmically very complex, we would see papers arguing for "natural" measures that reward algorithmic complexity. In fact the White Rabbit argument is basically an assertion that such measures *are* natural. Why one measure rather than another? By the logic of Tegmark's original thesis, we should consider the set of all possible measures over everything. But then we need a measure on the measures, and so ad infinitum. One self-consistent approach is Lewis', i.e. to abandon all talk of measure, all anthropic predictions, and just to speak of possibilities rather than probabilities. This suited Lewis fine, but greatly undermines the attractiveness of the everything thesis for physicists. more or less recently in the scientific american. I'm sure Tegmark's approach, which a priori does not presuppose the comp hyp, would benefit from category theory: this one put structure on the possible sets of mathematical structures. Lawvere rediscovered the Grothendieck toposes by trying (without success) to get the category of all categories. Toposes (or Topoi) are categories formalizing first person universes of mathematical structures. There is a North-holland book on "Topoi" by Goldblatt which is an excellent introduction to toposes for ... logicians (mhhh ...). Hope that helps, Bruno Not really. I know category theory is a potential route into this, but I havn't seen any definitive statements and from what I've read on this list I don't expect to any time soon. I'm certainly not going to learn category theory myself! You overlooked a couple of direct queries to you in my posting: * You still havn't explained why you say his system is "too big (inconsistent)". Especially the inconsistent bit. I'm sure a level of explanation is possible which doesn't take the whole of category theory for granted. Also, if you have a proof that his system is inconsistent, you should publish it. * Is it correct to say that category theory cannot define "the whole" because it is outside the heirarchy of the cardinals? And another mathematical query for you or anyone on the list: I've overlooked until now the fact that mathematical physics restricts itself to (almost-everywhere) differentiable functions of the continuum. What is the cardinality of the set of such functions? I rather suspect that they are denumerable, hence exactly representable by UTM programs. Perhaps this is what Russell Standish meant. I must insist though, that there exist mathematical objects in platonia which require c bits to describe (and some which require more), and hence can't be represented either by a UTM program or by the output of a UTM. Hence Tegmark's original everything is bigger than Schmidhuber's. But these structures are so arbitrary it is hard to imagine SAS in them, so maybe it makes no anthropic difference. Paddy Leahy
Re: Sociological approach
On Mon, 23 May 2005, scerir wrote: Do you agree we can have branches (or histories) in space (in a space) but also branches (or histories) in time? I guess there is an implicit "not only" in this question :) You have an atom, excited (ie by a laser). This atom can radiate a photon in two different ways (I mean two different transitions), having the same life-time. Of course the photon has the energy E1, or the (different) energy E2. You can write the amplitude as [a exp(-iE1 t/h) + b exp(-iE2 t/h)] e^(-k t) The probability, for a transition to occur, at a specific time, with emission of one photon (it does not matter, here, if its energy is E1 or E2), is given by two terms, plus an interference term. Yes. The two terms in your equation correspond to the "micro branches" I mentioned. Hence, I gather, you have demonstrated interference effects in time. For those who have not seen this before, it means that you get oscillations in the detection probability with time. You can, of course, introduce decoherence in this sort of 'interference in time' quantum experiment (quantum beat), by filtering for energies, ie for E1 or for E2. In that case the extra interference term disappears. Yes, if by "filtering" you mean putting in physical filters so that one photon is absorbed. The filter then provides the "environment" refered to in the theory of decoherence. To answer your initial question: interference effects are not branches. Actually they imply the absence of effective branching. You don't get branching in time because time is a parameter, not an observable: this means that there is no quantum uncertainty about what the time is. (At least in the non-relativistic theory. Frankly, I don't know how to handle the relativistic case). You might say: we don't know what time the particle will be detected. Yes, but the theory doesn't consider the detection event as *one thing* with an uncertain time. In the MWI there are many (a continuum) of detection events, each of which happens at a well defined time and each of which starts off its own branch. And the act of detection changes the detector physically, which is to say that its particles are re-arranged. Hence the slogan "every measurement is a position measurement". Of course they are all momentum measurements as well, etc. Paddy Leahy
Re: Decoherence and MWI
On Mon, 23 May 2005, Hal Finney wrote: I'd like to take advantage of having a bona fide physicist on the list to ask a question about decoherence and its implications for the MWI. If this is true, then how can a physicist not accept the MWI? Beats me... Isn't that just a matter of taking this decoherence phenomenon to a (much) larger degree? Either you have to believe that at some point decoherence stops following the rules of QM, or you have to believe that the mathematics describes physical reality. And the mathematical equations predict the theoretical existence of the parallel yet unobservable branches. The physicists who I really respect, but who do not support MWI, do indeed believe (or hope) that the rules of QM break down at some point. E.g. Roger Penrose, or Tony Leggett. They would point out that to date every theory we have has only been an approximation. Penrose would point out additionally that QM is inconsistent with GR, and there is no reason to suppose that only GR has to be modified. I support these people to the extent that I think it is tremendously important to keep doing experiments, especially ones that can test well-formulated alternatives to QM. I'm just not very hopeful that any discrepancy will show up. Of course, given that they are in practice unobservable, a degree of agnosticism is perhaps justifiable for the working physicist. He doesn't have to trouble himself with such difficult questions, in practice. But still, if he believes the theory, and he applies it in his day to day work, shouldn't he believe the implications of the theory? For most physicists the Copenhagen interpretation (in some half-understood way) works perfectly well at the lab bench. There are also those who have thought very carefully about the issue and have come to a hyper-sophisticated philosophical position which allows them to fudge. I'm thinking particularly of the consistent-histories gang, including Murray Gell-Mann. I particularly liked Roland Omnes' version of this: "quantum mechanics can account for everything except actual facts". He thinks this is a *good* thing! To me, it almost requires believing a contradiction to expect that decoherence experiments will follow the predictions of QM, without also expecting that the more extreme versions of those predictions will be true as well, which would imply the reality of the MWI. You either have to believe that a sufficiently accurate decoherence experiment would find a violation of QM, or you have to believe in the MWI. Don't you? Yes. Paddy Leahy PS: this is an endorsement of the MWI of QM, not of any "everything" theory.
Re: Sociological approach
QM is a well-defined theory. Like any theory it could be proved wrong by future experiments. My point is that R. Miller's suggestions would definitely constitute a replacement of QM by something different. So would aet.radal's (?) suggestion of information tunnelling between macroscopic branches. The crucial point, which is not taught in introductory QM classes, is the theory of Quantum decoherence, for which see the wikipedia article and associated references (e.g. the Zurek quant-ph/0306072). This shows that according to QM, the decay time for quantum decoherence is astonishingly fast if the product ((position shift)^2 * mass * temperature) is much bigger than the order of a single atom at room temperature. Moreover, the theory has been confirmed experimentally in some cases. Since coherence decays exponentially, after say 100 decay times there is essentially no chance of observing interference phenomena, which is the *only* way we can demonstrate the existence of other branches. "No chance" meaning not once in the history of the universe to date. No existing animal is small enough or cold enough to participate directly in quantum interference effects (i.e. to perceptibly inhabit different micro-branches simultaneously), hence my claim that your "behaviour system", whatever it is, must be in the fully-decohered regime. I have to backpedal some though, because by definition an intelligent quantum computer would be in this regime (in practice, by being very cold). I certainly don't want to imply that this goal is known to be impossible. NB: I'm in some terminological difficulty because I personally *define* different branches of the wave function by the property of being fully decoherent. Hence reference to "micro-branches" or "micro-histories" for cases where you *can* get interference. Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
Re: White Rabbit vs. Tegmark
On Sun, 22 May 2005, Hal Finney wrote: Regarding the nature of Tegmark's mathematical objects, I found some old discussion on the list, a debate between me and Russell Standish, in which Russell argued that Tegmark's objects should be understood as formal systems, while I claimed that they should be seen more as pure Platonic objects which can only be approximated via axiomatization. I see that Russell is right, and that Tegmark does identify mathematical structures with formal systems. His chart at the first link above shows "Formal Systems" as the foundation for all mathematical structures. And the discussion in his paper is entirely in terms of formal systems and their properties. He does not seem to consider the implications if any of Godel's theorem. Actually he does both. Most of the time he implies that universes are formal systems, e.g. formal systems can be self-consistent but objects just exist: only statements about objects (e.g. theorems) have to be consistent. But he also says that *our* universe corresponds to the solution of a set of equations, i.e. a mathematical object in Platonia. This is definitely muddled thinking. Specifying a universe by a set of statements about it seems to be a highly redundant way to go about things, whether you go for all theorems or the much larger class of true statements. (Of course, many statements, including nearly all the ones interesting to mathematicians, apply to whole classes of objects). Hence despite all the guff about formal systems, I think any attempt to make sense of Tegmark's thesis has to go with the object = universe. Paddy Leahy
Re: White Rabbit vs. Tegmark
Now I'm really confused! I took Russell to mean that real numbers are excluded from his system because they require an infinite number of axioms. In which case his system is really quite different from Tegmark's. But if Bruno is correct and reals only need a finite number of axioms, then surely Russell is wrong to imply that real-number universes are covered by his system. Sure, they can be modelled to any finite degree of precision, but that is not the same thing as actually being included (which requires infinite precision). For instance, Duhem pointed out that you can devise a Newtonian dynamical system where a particle will go to infinity if its starting point is an irrational number, but execute closed orbits if its starting point is rational. On Mon, 23 May 2005, Bruno Marchal wrote (among other things): Le 23-mai-05, à 06:09, Russell Standish a écrit : Hence my interpretation of Tegmark's assertion is of finite axiomatic systems, not all mathematic things. I don't think Tegmark would agree. I agree with you that "the whole math" is much too big (inconsistent). Since Tegmark defines "mathematical structures" as existing if self-consistent (following Hilbert), how can his concept be inconsistent? But there may be an inconsistency in (i) asserting the identity of isomorphic systems and (ii) claiming that a measure exists, especially if you try both at once. It is mainly from a logician point of view that Tegmark can hardly be convincing. As I said often, physical reality cannot be a mathematical reality *among other*. The relation is more subtle both with or without the comp hyp. I have discussed it at length a long time ago in this list. Category theory and logic provides tools for defining big structure, but not the whole. As I understand it, this is because "the whole" is unquantifiably big, i.e. outside even the heirarchy of cardinals. Correct? The David Lewis problem mentionned recently is not even expressible in Tegmark framework. It might be illuminating if you could explain why not. On the face of it, it fits in perfectly well, viz: for any given lawful universe, there are infinitely many others in which as well as the observable phenomena there exist non-observable "epiphenomenal rubbish". The only difference from the White Rabbit problem is the specification that the rubbish be strictly non-observable. As a physicist, my reaction is that it is then irrelevant so who cares? But this can be fixed by making the rubbish perceptible but mostly harmless, i.e. "White Rabbits". Paddy Leahy
Re: Sociological approach
On Sun, 22 May 2005, rmiller wrote:
I'm approaching this as a sociologist with some physics background so I'm
focusing on what the behavior system perceives ("measures"). If all possible
worlds exist in a superpositional state, then the behavior system should
likewise exist in a superpositional state.
First, it looks like you are confusing the multiverse of QM with the
plenitude of "all theories" or all UTM programs (Level 3 with Level 4
multiverse in Tegmark's terminology). Different level 4 worlds do not
superpose, they don't relate to each other in any way, by definition.
Second, in QM you need to distinguish between two kinds of superposition:
those which cause interference effects (e.g. 2-slit experiment), and those
which don't (because the wave functions of the superposed "worlds" don't
overlap or are incoherent). "Behaviour systems" are complicated enough
that it is a mathematical certainty that they fall in the second class. In
which case there is no way to detect that the superposition is happening;
for all practical purposes each world goes its own sweet way.
If there are say, 10 possible
"worlds" available to the behavioral state (percipient) but each world
differs from the other by elements that are not observed by the percipient,
then the behavior system is under the assumption that interaction is taking
place with a single, unified environment.
Recalling the Copenhagen interpretation: does Chicago exist if you happen to
be by yourself in a hotel room in Des Plaines, IL? The answer is irrelevant
until the behavior system begins to experience some aspect of Chicago.
The superposition properties depend on the information available in the
whole system (e.g. your hotel room), not just the mind of the observer.
The world is constantly in close touch with itself. For instance, if
Chicago vanished in a large quantum fluctuation, photons which would
otherwise have been reflected from its streets to the clouds would be
different. Hence photons leaving the clouds that land in fields 40 miles
away would be different and so on. Very soon (within microseconds) the
photons comint through your hotel window are affected, and you become 100%
correlated with the state of Chicago. From your point of view, Chicago is
either there or not.
What if Deutsch is incorrect about contact between the various worlds?
i.e., what if quantum theory is wrong and a different theory applies?
But the only reason we have to believe this stuff is the evidence in
favour of QM (which is pretty overwhelming).
Suppose the behavior system normally exists across a manifold of
closely-linked probabilities, with the similarities forming a central
tendency and the differences existing at each edge of the distribution?
Again, QM makes definitive (but difficult-to-understand) predictions about
this. Yes there is a manifold of possibilities, in fact an infinite
number of them, for instance "configuration space" which is the manifold
of all particle positions (3N dimensional for N particles), or "momentum
space", the manifold of all particle momenta (also 3N dim). According to
QM, the probability distributions in these manifolds are not independent,
e.g. config and momentum wave functions are related by a Fourier
transform.
Your "central tendency" is just the wave function, which is peaked around
some configuration of particles in any given branch. What people generally
don't factor into this is just how *BIG* 3N dimensional spaces are, when N
is macroscopic. Even apparently minor differences, such as the presence or
absence of a speck of dust, correspond to enormously large separations in
configuration space. Although technically there is some (usually
infinitesimal) amplitude for all configurations, the only way you can get
a useful amplitude for two macroscopically different "worlds" is to
amplify some quantum behaviour, in which case the wave function splits
into 2 or more branches, each of which behaves more or less according to
classical physics. The "width" of the distribution for a single branch
corresponds to ordinary microscopic quantum fuzziness. Hence the branches
don't overlap in configuration space (or in the space of any other
macroscopic variable), and so can't communicate.
If the behavior system can perceive only a small chunk of information at
a time, then it may be possible that each percipient really does live in
his or her own little world---a small island of similar probabilities
made"real" from the larger cloud of probabilities.
We are all in our own little worlds, but in an objective sense; the same
is true for "non-behavior" systems, e.g. rocks.
If we quantify a behavior system in terms of elements and interactions
between elements, we arrive at a complex, but definable state. If that
behavior system exists across multiple worlds that differ in minute details
(i.e. a unobserved kitchen saucer moved an inch to the side) th
Re: White Rabbit vs. Tegmark
On Mon, 23 May 2005, Russell Standish wrote: I think most of us concluded that Tegmark's thesis is somewhat ambiguous. One "interpretation" of it that both myself and Bruno tend to make is that it is the set of finite axiomatic systems (finite sets of axioms, and recusively enumerated theorems). Thus, for example, the system where the continuum hypothesis is true is a distinct mathematical system from one where it is false. Such a collection can be shown to be a subset of the set of descriptions (what I call the Schmidhuber ensemble in my paper), and has some fairly natural measures associated with it. As such, the arguments I make in "Why Occam's razor paper" apply just as much to Tegmark's ensemble as Schmidhuber's. Hmm, my lack of a pure maths background may be getting me into trouble here. What about real numbers? Do you need an infinite axiomatic system to handle them? Because it seems to me that your ensemble of digital strings is too small (wrong cardinality?) to handle the set of functions of real variables over the continuum. Certainly this is explicit in Schmidhuber's 1998 paper. Not that I would insist that our universe really does involve real numbers, but I'm pretty sure that Tegmark would not be happy to exclude them from his "all of mathematics". Conversely, if you wish to stand on the phrase "all of mathematics exists" then you will have trouble defining exactly what that means, let alone defining a measure. I don't wish to, but this concept has been repeated by Tegmark in several well publicised articles (e.g. the Scientific American one). Again, lack of mathematical background forbids me from making definitive claims, but I suspect that it could be proved impossible even to define a measure over *all* self-consistent mathematical concepts. In which case Lewis was right and Tegmark's "level 4 multiverse" is essentially content-free, from the point of view of a physicist (as opposed to a logician). Paddy Leahy
White Rabbit vs. Tegmark
I looked into this mailing list because I thought I'd come up with a
fairly cogent objection to Max Tegmark's version of the "everything"
thesis, i.e. that there is no distinction between physical and
mathematical reality... our multiverse is one particular solution to a set
of differential equations, not privileged in any way over other solutions
to the same equations, solutions to other equations, and indeed any other
mathemetical construct whatsoever (e.g. outputs of UTMs).
Sure enough, you came up with my objection years ago, in the form of the
"White Rabbit" paradox. Since usage is a bit vague, I'll briefly re-state
it here. The problem is that worlds which are "law-like", that is which
behave roughly as if there are physical laws but not exactly, seem to
vastly outnumber worlds which are strictly "lawful". Hence we would expect
to see numerous departures from laws of nature of a non-life-threating
kind.
This is a different objection to the prediction of a complete failure of
induction... it's true that stochastic universes with no laws at all (or
where laws abruptly cease to function) should be vastly more common still,
but they are not observed due to anthropic selection.
A very similar argument ("rubbish universes") was put forward long ago
against David Lewis's modal realism, and is discussed in his "On the
plurality of worlds". As I understand it, Lewis's defence was that there
is no "measure" in his concept of "possible worlds", so it is not
meaningful to make statements about which kinds of universe are "more
likely" (given that there is an infinity of both lawful and law-like
worlds). This is not a defense which Tegmark can make, since he does
require a measure (to give his thesis some anthropic content).
It seems to me that discussion on this list back in 1999 more or less
concluded that this was a fatal objection to Tegmark's version of the
thesis, although not to some alternatives based exclusively on UTM
programs (e.g. Russell Standish's Occam's Razor paper).
Is this a fair summary, or is anyone here prepared to defend Tegmark's
thesis?
Paddy Leahy
==
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics & Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618
Re: WHY DOES ANYTHING EXIST
I find this a very odd question to be asked on this list. To me, one of the main attractions of the "everything" thesis is that it provides the only possible answer to this question. Viz: as Jonathan pointed out, mathematical objects are logical necessities, and the thesis (at least in Tegmark's formulation) is that physical existence is identical to mathematical existance. Despite this attractive feature, I'm fairly sure the thesis is wrong (so that there is just no answer to the big WHY?), but that's another story. Paddy Leahy == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
Re: Many Pasts? Not according to QM...
On Wed, 18 May 2005, Hal Finney wrote: Does anybody believe that this is consistent with the many-worlds interpretation of QM? First, welcome to the list. Thanks! However, particularly as we look to larger ensembles than just the MWI, it becomes attractive to define observers and observer-moments based solely on their internal information. I wondered if that's what was meant... hence the last para of my message, and my comments in my follow-up to Quentin Anciaux. But you explain it better (in a bit I snipped!). Mind you, I don't understand why you find your definition "attractive". It would be pretty confusing for physicists to say "there's only one electron", even though they all are absolutely identical. And also, as I mentioned to Quentin, if you are going for such a radical first-person perspective, an OM really *has* no outside so it is a bit misleading to talk of "pasts" at all. regards, Paddy
Re: Many Pasts? Not according to QM...
On Wed, 18 May 2005, Quentin Anciaux wrote: Le Mercredi 18 Mai 2005 17:57, Patrick Leahy a écrit : SNIP Of course, many of you (maybe all) may be defining pasts from an information-theoretic point of view, i.e. by identifying all observer-moments in the multiverse which are equivalent as perceived by the observer; in which case the above point is quite irrelevant. (But you still have to distinguish the different branches to find the total measure for each OM). Hi, I thought of Observer Moment as containing the observer... What is the meaning of an OM (the same) which spread accross branches ? If you start by the assumption that OM are fundamental, then a "branch" is an OM. Or a branch is a consistent succession of OM ? I'm also learning a new "language" here as well, so forgive me if I got it wrong. I was trying to put the best "spin" I could on the idea of multiple pasts. Personally I'm not sympathetic to the OM concept in the first place, except as a useful device for anthropic calculations. By a "branch" I mean a branch of the wave function (Psi for short), which in MWI does literally have a branching structure in (configuration space + time). This is absolutely not an OM: for one thing, a branch is extended in time. Also, each branch of Psi describes a history for all the observers in the universe (not to mention all the non-self-aware bits), and hence contains (>>?) billions of OM at any given time. And of course a different OM for each observer at each moment. If the split forever is correct, then does a consciousness spread accross all those branch where the OM is in ? or just in one branch, and in other branches with the same OM, this is not the same consciousness ? This is really a matter of definition, I think. Is there a distinction between "consciousness" and "OM" ? I would say yes but I suspect many here would disagree. From my point of view, I'd prefer to say that each observer (and her consciousness) inhabits a specific branch and has only one past, even if it is indistinguishably different from that of a copy in another branch. If the later, why can it be said that it is in fact the same OM ? I'm with you. But if you take OM as fundamental, as some here do, you might prefer to re-sort the OMs scattered throughout the multiverse so that all identical OMs go into one "pot"; then you can choose to call this pot a single OM with a greater or lesser weight. In which case it is probably legitimate to talk about these having multiple pasts, though in another sense they have no past (they are self-contained moments!), only a memory of one (which is *not* multiple, by definition).
Many Pasts? Not according to QM...
I've recently been reading the archive of this group with great interest
and noted a lot of interesting ideas. I'd like to kick off my contribution
to the group with a response to a comment made in numerous posts that a
single observer-moment can have multiple pasts, including macroscopically
distinct pasts, e.g. in one memorable example, pasts which differ only
according to whether a single speck of dust was or was not on a
confederate soldier's boot in 1863.
Does anybody believe that this is consistent with the many-worlds
interpretation of QM? If so, please think again! Even such an apparently
minor change is sufficient to split the universal wave function into two
distinct branches (i.e. branches peaking in vastly-separated regions of
configuration space), which can recombine with probability effectively
zero. The reason for this is "decoherence" in the technical sense used by
Zurek and others.
To counter one obvious rejoinder, I'm not denying that micro-histories can
recombine, as in the two-slit experiment. Rather, decoherence ensures that
states with macroscopic (or even mesoscopic) entropy spread their
information so effectively that it is practically impossible to erase it
("practically" in the sense that even the entire resources of the universe
would be insufficient, as emphasised by Omnes).
Of course, many of you (maybe all) may be defining pasts from an
information-theoretic point of view, i.e. by identifying all
observer-moments in the multiverse which are equivalent as perceived by
the observer; in which case the above point is quite irrelevant. (But you
still have to distinguish the different branches to find the total measure
for each OM).
==
Dr J. P. Leahy, University of Manchester,
Jodrell Bank Observatory, School of Physics & Astronomy,
Macclesfield, Cheshire SK11 9DL, UK
Tel - +44 1477 572636, Fax - +44 1477 571618
JOINING
Hi, I'm Paddy Leahy. I'm an astrophysicist and observational cosmologist with a long-standing interest in the foundations of QM. == Dr J. P. Leahy, University of Manchester, Jodrell Bank Observatory, School of Physics & Astronomy, Macclesfield, Cheshire SK11 9DL, UK Tel - +44 1477 572636, Fax - +44 1477 571618
