These vague ideas help point in certain directions but my focus is on ways in which purely algebraic structures can lead to geometric ones. It would be nice to have an answer in my lifetime so I will happily accept the hypothesis that multiple quantisation solves the path integral over the grand ensemble of theories and see where I can go with that.

I like the group field theory approach to forming spin metworks because it gives a manifold from an group by a process of quantisation. It has to be extended to higher dimensions and to manifolds with matter content too. That is not going to be a simple generalisation.

The information angle seems to have some baring on this. The relation between redundancy of information and redundancy in gauge theories seems useful and perhaps your ideas about information transfer also relate to something algebraic such as invariants as you mentioned.

I am not looking for something 3D or 4D specifically. I prefer to explore theories in any number of dimensions. I will look at the framework of field theories and worry about real physics later. The geometric process of dimensional reduction by compactification (Kaluza-Klein) would be related to taking an algebraic structure modulo some relations so it is a very natural process from an algebraic point of view and it enriches the physics on the geometric side.

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  • Post #22

Jochen

"I'm somewhat puzzled, since in your previous reply you said: "The actual spatial location of the rainbow does not alter.", which appears to presuppose there is such a thing...

Yes, because the next sentence was: "Its relative spatial location does, obviously, because its actual spatial location is being expressed with respect to a different spatial location." And then another sentence is (in the next post, but something similar I said after that sentence without the caveat): "We cannot know the absolute position of anything. Only its relative spatial position with respect to anything within our existentially closed system".

Absolute refers to where (and indeed what, etc) it might 'really' be. Which we cannot know, or to be more precise, we can never know that what we do know is the 'absolute' truth, because there is always the possibility of an alternative. So this is irrelevant. The corollary being that we can know a definite something, ie the form of existence potentially knowable to us. And within that closed system the rainbow, or indeed anything else, exists independently of us in a specific spatial location. But we can only calibrate that, because we cannot externalise ourselves from the closed system, by comparison and the identification of difference, ie its relative special position with respect to something else.

It's all about context. And science is not in the context of every possibility we can conceive and believe in, it is supposed to be considering what is potentially knowable, which is underpinned by a physical process.

"Regarding the position of the rainbow with respect to both observers, I again don't think there is such a thing: each observer observes the rainbow in a different position"

Of course there is. The rainbow and observers are all independently physically existent entities. They therefore have, by existing, a definitive spatial location. We just do not know where 'exactly' it is. Only where it is with respect to something else. Is is the observers who are in different spatial relationships vis a vis the rainbow, not the rainbow which is 'altering' position.

Re your last paragraph. Observation and existence are different. Observation involves the receipt of a physically existent representation of the existential sequence (in sight this is light). Light travels, which takes time. So the timing of the receipt of that representation (ie which is not the reality anyway) is different, and after a delay, from the timing of the actual occurrence. This is where Einstein went wrong, because he had no observation, he conflated occurrence and observation (note his in the "immediate proximity" caveat 1905 Part 1), as he had no observational light (just a ray, or lightening, etc). Just a constant in order to calibrate distance and duration, which happened to be an example of light. It could have been anything. This error was counterbalanced by his failure to understand the reference for timing, because he followed Poincaré's flawed concept of simultaneity. In short: he shifted the real timing differential from the receipt of light, which fundamentally is a function of spatial relationship, to deeming it to be a characteristic of reality (ie relativity). Which is what you are doing with the rainbow. The fact that physical existence can only occur as a sequence, one state at a time, is a supplementary point(!).

Paul

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  • Post #23

Guys

There can be nothing, vague, chaotic, uncertain, etc, etc about physical existence. In order to be existent (as we can know it, and this is science, not religion) there must be something in a physical state (what, why, etc, is for you lot to discover, I just do the easy bit!). And in order for it to occur in different states, there must be a physical cause which results in that different state. One state, the next different state, something caused that difference. It is all explainable, if we could 'bottom it out'.

The issue is about our ability to identify that, and not make up rules about how reality occurs in order to cover over our deficiencies.

When I was young (not that I'm that old) TV was quite new. Every now and then a message would show up saying: please do not adjust your sets we are having problems with the transmission.

Paul

But consider how a rainbow works: light from the sun is refracted and retroreflected in a rain drop; if it is incident on, say, an observer's retina, this creates a virtual image, which we call the rainbow. But this is not anywhere in the world: there is no physical object that conforms to the rainbow at any point in space. How then there should be some absolute location to the rainbow seems wholly unclear to me. At best, you could argue that there's a set of rain drops, each of which contributes one 'pixel' to the image of the rainbow; but even this set will be different for different observers.

And I hear your point that observation and existence are different; in a sense, that's what I'm all about: everything we know and can make contact with (though precisely how is a problem in itself) is observation. Hence, I try to only consider observation, making no statement about 'what really exists'. Of course, this is also exactly what Einstein did, asking himself what the world would look like if he rode on a rainbow, etc. (and the functioning of my GPS confirms to me daily the correctness of his conclusions, at least in so far as there is any correctness in science).

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  • Post #25

Hi Jochen,

Very Interesting essay!

You wrote at the end of 4:

"In many worlds, both `the electron spin is up' and `the electron spin is down' are true; in the

relative-facts setting, only `the spin of the electron is (up/down) relative to the measurement

apparatus recording (up/down)' is true. A universe of relative facts is not a multiverse!"

I think there is one possibility that a universe of relative facts is a multiverse,

The possibility that the Big Bang produced fully symmetric and entangled anti-copy universes.

Then, there is at least one Charge Parity symmetrical anti-universe ( a mirror world) with equal time because all clocks over there are running (spinning) only in the opposite direction.

To make this a bit more redundant and symmetric, I propose that we live inside a 12 lobed dodecahedron raspberry shaped symmetric multiverse.

However then GOD PLAYS DICE with 12 entangled CP symmetric pinball machines where each pinball is instant connected to its anti-copy living in that other universe.

Perhaps we are even able to measure the number of these raspberry lobes if we are able to observe a definite number of neighbouring lobes in the CBradiation pattern, or even by repeating the well known Benjamin Libet experiments on reaction times and preplanning thoughts ( RPs) of test persons.

See attachments

Leo Vuyk.

http://vixra.org/author/leo_vuykAttachment #1: B.Libet_Preplanning_vs_Free_Will..jpg

    Philip, I generally try to keep as minimal as possible---so since three dimensions are all we can currently observe, that's what I'm trying to figure out (actually, that's not quite true---I'm fascinated by the apparent connection between the division algebras and the forces of the Standard Model, and in as much as the higher division algebras, the quaternions and octonions, are much more naturally connected to six- and ten-dimensional spacetime respectively, I've done a bit of thinking in that direction, too).

    I'm not very familiar with the group field theory approach, but I'd think that as far as getting geometry from algebra goes, noncommutative geometry has some interesting things to say (and of course, with my penchant for the division algebras, the recent foray into nonassociative geometry by Farnsworth and Boyle immediately caught my eye at least). About getting higher dimensions from GFT, with the close connection to spin networks, is that even possible? Generally, I thought that loop variables really only live in 4d space (though I've caught some papers by I think Thiemann generalizing them to higher dimensions out of the corner of my eye). But that stuff really is a bit out of my reach.

    Perhaps one comment about algebraically reducing the dimension to the observed 4: in an approach based on octonions, you'd naturally have the group SL(2,O) for spacetime symmetries, which is isomorphic to SO(9,1), i.e. the ten-d Lorentz group. Now it's always struck me as a curious (and perhaps deep) fact that if you 'fix' one of the imaginary octonions, you not only 'break' the automorphism group from G2 down to SU(3), but also SL(2,O) to SL(2,C), which is of course isomorphic to SO(3,1) (I think I probably got this from Baez). So we get the symmetries of spacetime as we know and love them, and the gauge group of the strong force. Now this maybe something like the large numbers (you mentioned them in the other thread): a cute coincidence that leads all too easily to premature conclusions. But I can't help wondering...

    -------------------

    Paul, to you, it seems to be axiomatic that there must be some underlying, fixed physical reality; however, our best theories appear to be telling us that this is not the case. Now, this as you rightly say is not sufficient grounds to abandon this assumption: however, these theories are, and continue to be, tested to greater precision than anything ever before. It is these experimental observations that are ultimately in conflict with the assumption of a fixed reality; every theory that should supersede quantum mechanics will include nonlocality and contextuality just as much, or else be experimentally not viable.

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    • Post #28

    Hi Leo, thanks for your comments and for finding my essay interesting. You make great leaps and bounds in your reply, and I'm unable to follow you at that speed (I've always been a bit of a slow one), so I'm not sure I can cogently reply to them... In general, I'm not a particular fan of multiverses: in particular, I have never seen the term 'universe' defined well enough in order to judge whether there's one or more of them around. It might sound facetious, but, you and me, are we in the same universe? There's things that definitely exist in your universe, but not in mine: your thoughts, feelings, experiences, etc. Furthermore, you have been influenced by events that I have not yet come in causal contact with, i.e. our past lightcones do not overlap---our celestial spheres differ. Then, you experience everything in the world from your unique vintage point---an experience I never could share.

    So in the end, what makes a universe? I couldn't claim to know. And much less so in the case of any supposed multiverses.

    Just posting to say that the above was me---I must've gotten logged out somehow.

    Jochen, you have a good collection of references at your fingertips. I had not seen the non-associative geometry development. There are connections between necklace Lie algebras and non-commutative geometry http://arxiv.org/abs/math/0010030 but I am not chasing that angle. The division algebras are also interesting. The relationships to dimensions are important and probably part of a bigger picture, but I am skeptical about connections to the low energy gauge groups. I think if we knew how much physics there was between 1 TeV and the Planck scale we would laugh at the idea.

    With all these interconnected threads the trick will be to find which part is the key concept after which everything else will fall into place. The combinatorial necklaces that are associated with Lyndon words are also related to irreducible polynomials over finite fields which are in turn related to cyclic linear codes and exceptional structures such as division algebras, exceptional groups and lattices. My goal is therefore to understand the Necklace Lie algebras first and then see how these exceptional structures appear within them and how that relates to special properties of different dimensions. Of course the whole program is ambitious and I have no idea if I know enough mathematics to make any progress even assuming there is really something to discover, but it is all fun to think about.

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    • Post #31

    Thank you Jochen,

    i realize that my proposals for the idea of a Big bang symmetry and that you and I are dealing with one or more other you's and I's living in an anti matter universe coupled by instant entanglement is too much to grasp.

    sorry.

    Jochen

    How the rainbow works, and that it just happens to involve light, is irrelevant. It is a physically existent entity. This argument has the structure of Einstein where there is always some form of light, but it is actually just a constant, which happens to be light (eg lightening on his train). Indeed, this always happens. Turn to Cox & Forshaw, a good explanation of the argument, and there we have a clock that is a light beam. The light entity is a constant against which to calibrate duration and distance, not observational light.

    Whatever the entity, to observe it involves an interaction between it as the existent sequence progresses, and something else, which creates a physically existent entity that is representational of it, ie observational light. This then travels, and some of it interacts with eyes.

    Put another way, the rainbow is not inside people's heads. As you then say, because it is to do with water and light.

    "everything we know and can make contact with (though precisely how is a problem in itself) is observation"

    No. Everything we could potentially know is determined by all the sensing systems of all sentient organisms. Which would include an alien if he/she landed here and explained a new sensory system to us. And that encompasses hypothesising so long as it is effected in accordance with the operational rules of the sensory system it is supplementing. Because, obviously, not everything can be sensed directly, therefore it is perfectly legitimate to establish what could have been so had some identifiable problem not prevented it. But this does not involve belief, which is the assertion of reality on the basis of no experienceability whatsoever. In other words, physical existence is whatever we can potentially know as enabled by the physical processes (which are part of physical existence) which feed the sensory systems, or hypothetical equivalents.

    "Hence, I try to only consider observation, making no statement about 'what really exists'."

    Indeed. But leaving aside the point that it is more than just observation, direct or indirect, if you do not start with the premise which corresponds to the form of existence we can know, then you are likely to be considering alternatives (ie what possibly really exists), albeit inadvertently. It is considered intellectually correct to presume physical existence as an abstract concept, ie presume nothing. But it is not abstract, generically it has a definitive form and modus operandi.

    Einstein did not do this, please find me an example of actual observation, ie where there is observational light and it is being received. You can't. In the AB example it is a ray of light, in the train there is another ray of light (which takes the place of the man walking-which it does not) and there is lightening.

    To try and settle the Einstein 'angle', I will post on my essay site the first 24 paras of another paper. This is only 4 pages and is easy to read, it gets slightly more complex after that.

    Paul

    Jochen

    "so since three dimensions are all we can currently observe"

    Although this is not really a function of the way observation works, as you are hinting here, the issue is how many dimensions are there in physical reality.

    In establishing what constitutes dimension, distance and space in our reality, we are using a reference which conceives of any given physical reality being 'divided' into a grid of spatial positions. And in order to 'locate' that conceptual grid it is associated with one physical entity. Any constituent physically existent states of that physical reality have definitive dimension/size/shape (ie spatial footprint), which can be defined as spatial positions 'occupied' on the grid at the given time of existence.

    'Mapping' other states that were existent at the same given time, would reveal not only both the spatial footprint of those states and their comparability with each other, but also, distance. Which is usually measured between the two nearest dimensions of the existent states, but could include any combination of dimensions. And depending on the spatial relationship of the states involved, distance could involve a relationship in terms of separation of the states, or one within another, that again being with respect to specified dimensions.

    Dimension is a specific aspect of spatial footprint, relating to the distance along any possible axis of that 'occupation'. So, three is the minimum number of spatial dimensions that is ontologically correct at the highest level of conceptualisation of any given physical reality (ie up/down, side/side, back/forth). But is not what is physically existent. At that existential level, the number of possible dimensions is half the number of possible directions that the smallest substance in physical existence could travel from any single spatial point.

    Paul

    Jochen

    "Paul, to you, it seems to be axiomatic that there must be some underlying, fixed physical reality..."

    Science is concerned with that form of existence which we can potentially know, not alternative we can believe in. Knowing is underpinned by a physical process. Based on input received, we can identify that the form of physical existence we can know has two fundamental characteristics:

    -what occurs, does so, independently of the processes which detect it

    -it involves difference, ie comparison of inputs reveals difference, and therefore that there is change/alteration.

    Therefore, the physical existence we can know must be existential sequence. The entirety of whatever comprises it can only exist within that sequence in one definitive physically existent state at a time, as the predecessor must cease to exist so that the successor can exist. To be physically existent, by definition, entails no form of change or indefiniteness in whatever is existent at any given time. And so physical existence is a spatial phenomenon, which occurs differently over time.

    Look at the bush in your garden. Over time it manifest different, colour, size, leaves/no leaves, etc. These different physically existent states do not co-exist. The same principle applies if we drill down to what physically constitutes 'bush'. One definitive physically existent state of whatever constitutes it at a time.

    Paul

    Regarding the rainbow, I don't think it's right to consider it as a 'physical entity'. Ultimately, the physical entities it derives from are a set of rain drops, each of which reflects light in a particular way. For each observer, this set of drops changes. There is no further independent entity that you could call the rainbow anywhere.

    In any case, you seem to have fixed your assumptions---and as you say over on Mikalai's essay thread, even if experiment disagrees with these assumptions, you would not be willing to revisit them. But this is too dogmatic for me: there's no reason for nature to be the way we'd like it to be. If it tells us via experiment that the assumption of a definite physical state for a system at all times isn't right, we have to accept that, no matter how little we might like it.

    Philip, yes, if one knows little oneself, one has to try to compensate by at least knowing where to find things. ;-) And you're right that perhaps the most reasonable assumption is not a 'great desert' from here to the Planck scale, and it's very well possible that near-future developments will completely destroy this possibility. But for the time being, there's a sort of mini-revision centered mainly around the work of Shaposhnikov and Wetterich, who in 2009 predicted the observation of a 126 GeV Higgs from the assumptions of asymptotic safety for gravity, and no new physics up to the Planck scale. Building somewhat on this, there's been the proposal of a 'nuMSM', a standard model minimally extended with massive right handed neutrinos to account for dark matter. Whether or not something comes out of this remains to be seen, but it aligns well enough with my prejudices to at least merit some attention (for me, that is). And I think I'm really going to have to look at necklace algebras (is there some canonical reference?).

    There is not really a canonical reference for Necklace Lie akgebras. If you search you may find three things, (1) references to the free lie algebra as a necklace lie algebra. This was what I described in my essay but I had to keep it short so it may not be clear (2) references to a Hopf algebra studied by Le Bruyn, Ginzburg, Bocklandt. This is a little messy and I dont read those papers much but it was seeing this that made me start calling my structures Necklace Lie Algebras because they fit the same idea. (3) comments by me referring to the algebras I defined e.g. here http://vixra.org/abs/0907.0033 The structures are quite elegant but I dont know how well I explained them. Recently I have been getting to know these things better and feel the need to write a new paper about them. The Yangian lie algebras that appear in scattering theory are also kind of similar.

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    • Post #38

    I thought a second time about your statement:

    you wrote:

    "In many worlds, both `the electron spin is up' and `the electron spin is down' are true; in the relative-facts setting, only `the spin of the electron is (up/down) relative to the measurement apparatus recording (up/down)' is true"

    However In my raspberry multiverse with until now still entangled anti-copy universes created in the Big Bang CP symmetry means that UP is UP and DOWN is DOWN in the Stern Gerlach apparatus.

    Over there the Noth is South and the electron is a positron.

    I realize that my proposal has the disadvantage of the huge distance between thse CP symmetrical universes, But untill know there NO distance boundary observed for entanglement experiments

    I would be shocked if there ever were a limiting distance discovered for entanglement---I don't want to be making any of those 'I would quit physics if'-pronouncements, since they seem to come back to haunt people, but it's close. So at least as far as I'm concerned, you're quite safe distance wise.

    Jochen

    Most things are comprised of other things.

    It is not a matter of fixed assumptions, etc. If you can explain how physical existence occurs in any other manner, please tell me. The experimentation is being interpreted on the basis of flawed presumptions.

    I have put up posts which clearly argue the points, if you can find fault with these, please do so.

    Paul

    Jochen

    "I would be shocked if there ever were a limiting distance discovered for entanglement"

    This is another one of those statements which invokes a 'how can that correspond with physical existence' reaction. Please note this is not a personal comment, I know this is the 'common understanding', you just happen to be the one saying it now!

    As with all of what I am saying, forget the actuality manifest, as such, etc, and think in generic terms. Entanglement is some form of physical interaction. Now, how can a physical interaction involve anything except one spatial relationship, ie 'next to', and be at one given time? Considering anything except direct involvement is pointless, since everything is ultimately interrelated. So once beyond the 'direct' status of involvement, where does one stop including factors, and why? Therefore, when considering any given physical interaction, we must only consider what was directly involved in that particular interaction (forget whether that is possible, practically).

    Whatever was involved cannot be in anything but a 'next to' spatial relationship, and cannot contravene sequence order, ie whatever was involved must have been existent at the same time Being in a spatial position is something, being in a different spatial position at the same time is something else. So whatever was at a distance was not whatever was directly involved in the interaction. And to be able to exert physical influence means that whatever was doing so must have been physically existent. Physical influence cannot 'jump' physical circumstance in terms of spatial position and sequence order, neither can physical influence be a function of 'something' which is not physical.

    My point here being that if one drops all these labels and theory references and considers physical existence as must be, then the 'fog' clears. Being physically existent cannot involve any form of indefiniteness or alteration. It can only occur in a sequence in one physically existent state at a time. Neither is it impacted upon by sensing.

    We know all this because existence occurs independently of whatever detects it, ie physical input is received, and there is alteration, ie when inputs are compared that reveals difference. Depictions of existence which contravene these rules are belief, because we can only know of, what may or may not 'really be, a particular form of existence. That is, we can only know what it is potentially possible for us to know. That being the function of a physical process (valid hypothesising is not extrinsic to this, it is confined by the same rules, it is just a way of establishing what could have been sensed had it been practically possible to do so). And if we establish that knowledge properly, then we can deem it to be the equivalent of physical existence. We cannot externalise ourselves from the existentially closed system. We exist, we are part of what we are trying to explain.

    The 'trick' lies in comprehending what 'one state at a time' really means. Because apart from the sheer complexity and speed of turnover involved, which has practical implications, this is contrary, somewhat obviously as we need to live, to the way we conceive of existence. Even our language embodies that misconception. Fundamentally this can be characterised as existence being 'things' and there are 'changes' to these. Just stop for a second there and spot the inherent contradiction. If something exhibits any form of difference, then it is something else. That bush in your garden does not change colour, size, etc. It is a sequence of discrete definitive physically existent states. It only appears to persist in existence because we are defining its existence at a superficially higher level than what actually occurs, ie via certain physical features which we deem constitutes 'bush'. Indeed, we continue to assert the persistence of its existence even when, at the superficial level, it has changed!

    Paul

    PS: don't quit, but don't spend your life worrying or ultimately discovering that, there is 'something wrong here'

      Paul, you seem to be making a lot of demands on nature that it has to fulfill in order to make sense to you. But nature may not actually have that goal.

      Regarding entanglement, it's not really an interaction---it is produced in an interaction, but it is only the resulting correlation (say, between the spins of two particles, to be precise). Finding out one spin then gives you information about the other. That's not prima facie different from the classical word: if I have a red and a green ball, and place each in a box, shuffle them, and give you one, if you open your box and find the red ball, you will immediately know that mine contains the green one, no matter our spatial distance. The difference is just that in classical mechanics, you can meaningfully infer that the red ball was in your box before you looked; but in quantum mechanics, that inference leads to observable contradictions. This is encapsulated in the fact that quantum correlations are 'stronger' than classical ones.

      About your assertion that there is 'one state at a time', this too is something in direct conflict in quantum mechanics. Leggett and Garg have shown that it leads to inequalities that any theory in which physical evolution proceeds according to the maxim 'one state at a time' must obey, but which are again experimentally violated in quantum mechanics.

      So while the world might make more sense to us all if it worked the way you want it to, it does not seem that the world cares terribly much about that (and of course, that'S a good thing, as it is well known that no purely classical world would be physically coherent---you could not have stable atoms, for instance).

      Yes, in fact, almost all things are composed of other things. But the case of the rainbow is different: it is not composed of the same things for two different observers. So, if for one observer it has a location---given by the locations of all the drops of rain, their center of mass, or whatever---, for another observer, that location is different, and there is no further relevant concept of location involved.

      In fact, that's why I argue that it's better to consider the rainbow to be 'made of' relations rather than things: the relation between observer, rain and sun that gives rise to the rainbow is the same in all cases.

      Jochen

      What you are referring to is the physical cause of the rainbow. I am not sure that what gives rise to the rainbow is in different positions. But even if it is, then it is in different locations for different observers because of what it is, which is irrelevant to the real point you were making. Because it is still in a spatial position. The observer does not cause it to be there. And that position (or positions) can be calibrated by comparison and identification of difference.

      All 'things' are a function of 'relations', whatever constitutes the 'thing' at any given time being that physically existent state in the sequence. As I keep saying. There is nothing different about the rainbow in this respect. Ultimately, there must be some thing, or various types of thing, because there is existence. And this has nothing whatsoever to do with the observer. This, due to the subsequent evolution of sight, just has the ability to detect and hence become aware of it.

      Put another way. If all sentient organisms were wiped out, physical existence would continue, there would just be no awareness of it. Physical input is received, ie whatever constitutes existence is exists independently. Subsequent to reception it is processed, resulting in a perception of that input. This is not a physical process.

      Paul

      Jochen

      I am making no demands of nature, just stating, in generic terms, how it must be. Indeed, I would not attribute it with a "goal". How this manifests in practice is a different matter, but that cannot contravene the rules of its existence.

      Another way of putting this is that you need to find the flaw in my argument (which is a simple statement after all), and/or (really it is the same thing) explain an alternative. Remember, I am not saying what happens, just how it must happen. The point being that it is there, it is not an abstract concept. If you intended to do some wood working, then you would take recognisance of the nature of wood before starting. Because a brush and a lawnmower, which are tools, will be of no use.

      "Regarding entanglement, it's not really an interaction---it is produced in an interaction"

      I am not sure this counters my "Entanglement is some form of physical interaction" because you are referring to the result of an interaction, ie "Finding out one spin then gives you information about the other".

      "but in quantum mechanics, that inference leads to observable contradictions" then links to "About your assertion that there is 'one state at a time', this too is something in direct conflict in quantum mechanics".

      Yes, the point being that there is something wrong with the model of existence in QM, which of course then determines the results of experimentation. Ironically, I think, QM is attempting to depict existence at the physically existent state level. The difference you are alluding to is non-existent. Any given physical state is the function of other states, and assuming it is practically possible, then that relationship can be discerned. No physically existent state has some form of indefiniteness or whatever about it, existence does not involve vagueness. But as I have said many times, this is probably impossible to discern, and needs a mental exercise in eradicating the normal mode of conceptualisation to realise what this actually implies. This is why one gets conclusions such as: "it leads to inequalities that any theory in which physical evolution proceeds according to the maxim 'one state at a time' must obey". Er, no, it is, if understood properly, the only way that existence can occur.

      Instead of all these people saying this, that, and the other, just look out the window and ask yourself, how can that bush occur. In doing so do not slide into beliefs which are extrinsic to physical existence as knowable to us by virtue of a physical process.

      Paul

      How is saying how nature 'must be' not making demands on it? For one, this assumes that your intellect is capable of even grasping the relevant concepts for understanding how nature is, which is not necessarily a given---consider a giraffe: just as it would be fruitless to try and teach it calculus, it might be fruitless for a human to try and understand how nature works. You therefore can't claim to know how nature 'must be': it is always possible that you're wrong.

      Given this possibility, the best guide to finding out how nature works is experiment. So I'm going to try and describe to you a very simple experiment to check your idea that there is 'one state at a time', independently of QM, without assuming anything other than that we can make measurements that tell us at least something about those states (otherwise, this whole science thing would be for the dogs, anyway).

      So consider you have some physical system, and you can perform two measurements on it, say C and C'. These measurements are what is called dichotomic: depending on the state of the system, they'll come out either +1 or -1 (you can thus consider them as a primitive comparison: a value of +1 says the physical system is 'like' some reference system, and -1 that it is 'unlike' the reference; but this is somewhat immaterial).

      Now, consider you measure the system at two points in time, t1 and t2. Your hypothesis says that at each of these times, there is one physical state, and thus, if measurements can tell us anything about these states, both C and C' will produce one outcome or another. Now consider the quantity C1(C2 + C'2) + C'1(C2 - C'2), where the indices refer to the times the measurements are carried out. It is easy to see that this can be maximally 2: either both C and C' yield +1 at t2, then the first term is equal to 2, but the second vanishes; or C is equal to +1, but C' is equal to -1, then the second term equals 2, but the first vanishes. So we can state the following inequality, arrived at merely by stipulating that 1) there is a definite physical state at all points in time, and 2) that measurements yield information about that state:

      [math]C_1C_2^\prime + C_1C_2 + C_1^\prime C_2 - C_1^\prime C_2^\prime \leq2[/math]

      Now, this doesn't yet help us much, because we can't necessarily measure both C and C' at the same point in time. But we can prepare many copies of the physical system we wish to investigate---or, if you're not prepared to grant this, which I believe you may not be, then we can at least prepare many physical systems that, if measured, produce statistically the same measurement outcomes (this is something we can check independently: just prepare batches of candidate systems, and label those that yield the same statistics 'the same', which effectively is how we always decide similarity between systems). Then, of course, we must average over the outcomes, but this does not change the inequality; so we get (the angle bracket just represent the averaging):

      [math]\langle C_1C_2^\prime\rangle + \langle C_1C_2\rangle + \langle C_1^\prime C_2\rangle - \langle C_1^\prime C_2^\prime\rangle \leq2[/math]

      This actually allows us to further relax assumption 2): Measurements only need to statistically give us information about the underlying physical state.

      Now, the key is to note that no quantum mechancal elements went into the derivation of this inequality: it is theory independent, and rests only on the assumptions that there is some underlying physical state, definite at all times, and that measurement gives us at least some information about this state. And if it didn't, this would be a very strange conception of physical state: let's say that the state is s; then, typically a measurement---or any interaction---would be some function C(s) of that state. If now the measurement did not depend on that state, this function would not depend on s at all---but then, this so-called physical state would be completely without consequences on the physical world, at least as we perceive it.

      So let's go on. As has been shown, all theories in which there is 'one physical state at a time' must necessarily obey the above inequality. So this has become something we can test! One possibility would simply be to try and measure all systems we encounter in all the ways we can think of. We would find that the majority of physical systems---cars, horses, bushes, shoes etc.---indeed fulfill the inequality. But eventually, we would find physical systems that don't: those which are, nowadays, described by quantum mechanics. These need not be 'small' systems: the magnetic flux in a Josephson junction can be quite macroscopic, and has been shown to indeed violate the inequality. Then, driven by this experimental discrepancy with our preconceptions, we might draw up a theory to explain them---this would then be quantum mechanics, or something very much like it.

      But importantly, what this theory can't include, as we have seen, is the assumption that nature works according to the principle 'one state at a time'; and thus, whatever reasoning we have used to come up with this requirement, was flawed.

      Jochen

      "How is saying how nature 'must be' not making demands on it?"

      Because that is how it must occur. Please note (though I think you do) that I am not saying what happens.

      The first point to recognise is that the physical existence we can examine is possibly a limited form of existence. It might not be, it might be what 'really' is out there, but we can never know. We are trapped in an existentially closed system. We know courtesy of a physical process (supplemented by hypothesis, but that must still abide by the processes, ie it is not belief). In simple language: we can know what is potentially possible for us to know. This is where most people go wrong. They deem, and intellectually it appears correct, physical existence to be an abstract concept, ie 'it could be anything and we can find out what by examination in accord with due process'.

      Now, having established that it is, for us, a particular form of existence, that then means certain facts can be established about its fundamental nature, because we are not now, incorrectly, considering potential alternatives, ie possible forms of existence which we cannot know, either directly or by hypothesis, and are therefore belief.

      So, consider that bush in the garden. Even at the superficial, high level of observational conceptualisation, we know it involves difference. Put under an electron microscope, much more difference would be revealed. So, the key question is: how does difference exist? And the answer is sequence, because no two consecutive states can exist at the same time, but there is a discrete state at any time. To be physically existent must involve definitive presence, there can be no 'vagueness' or change in it. But for difference to occur there must be a sequence of such states. This is incredibly differentiated. Even if one grasps the circumstance mentally, we could never identify it in practice.

      "So consider you have some physical system... depending on the state of the system..."

      So there are different states which are different realities.

      "Now, consider you measure the system at two points in time, t1 and t2..."

      This is a different time, and hence it is very likely that there is a different reality.

      "Your hypothesis says that at each of these times, there is one physical state, and thus, if measurements can tell us anything about these states, both C and C' will produce one outcome or another"

      It is not a case of 'one or another'. They will produce one outcome in each case, which is different, because they are different realities (ie physically existent states). There is no such thing as 'physical system', or 'bush' or 'White House'. This is a conceptualisation we invoke, in order to make sense of reality, which is ontologically incorrect. It is based on us differentiating reality, via superficial physical characteristics, at a higher level than what occurs. In other words, we are deeming realities which are actually sequences of realities (ie physically existent states). It might be that your timing of these measurements is of such a duration that the two realities are consecutive. That is, there has only been enough time for 'one degree' of alteration in whatever, and that could be more than one type of alteration (ie different forms of alteration), if they have a rate of change which is the same. Put another way, if some form of alteration was faster, then these two states are not consecutive, ie you have missed one, two, however many, states out of the sequence.

      I will stop at that point (ie I am not ignoring what you have said), otherwise the basic message will be lost

      Paul

      I think we're somewhat talking past each other, or at any rate I don't see how your arguments impinge on the point I made. Yes, there are different physical states---different 'realities' if you want to put it that way---at different times. This was exactly the starting supposition of the arguments in my last post. And the measurements produce one outcome depending on what the state or reality is; I only say 'one or another' because we don't know what outcome that'll be.

      If your criticism is that I don't really make the same measurements at the same points in time, i.e. that say my point t2 is not always the same time, then first of all, the averaging procedure ought to get rid of any discrepancies here, based on the plausible assumption that I'm as likely to measure 'too early' as I am to measure 'too late'; but if you're not willing to grant that, then one can generalize the inequality such that the measurement occurs at four different points in time, in which case we even only need one measurement C, i.e.

      [math](1)\,\,\langle C_1C_2\rangle+\langle C_2C_3\rangle+\langle C_3C_4\rangle-\langle C_1C_4\rangle\leq2[/math]

      Let's perhaps review why that inequality must hold if we have 'at any time a definite state'. If this is so, then at t1, the measurement C returns either the value +1 or -1. In order for the first term to be maximal, i.e. equal 1, the measurement of C at t2 must agree with this, i.e. either we get (-1)*(-1) = 1 or 1 * 1 = 1; any other sequence of outcomes would yield a value for that term of -1, and we could not hope to exceed the bound of 2 anymore. The same holds for terms two and three, i.e. the outcome of the measurement of C at t3 must agree with that at t2, and the outcome of the measurement at t4 again must agree with that at t3.

      Now let's look at the last term, the one with the minus sign. Clearly, if we want to maximize the value of the inequality, then this term must be negative, which together with the minus in front of it means that a positive quantity will be added to this value. In order to make this term negative, the measurement of C at time t4 must disagree with the measurement of C at t1, i.e. we either must have (+1)*(-1) = -1 or (-1)*(+1) = -1.

      But this requirement is in contradiction with the requirements imposed by the first three terms: if C1 must agree with C2, and C2 must agree with C3, then C1 must agree with C3; since the latter must also agree with C4, it follows that in order to maximize those three terms, C1 must agree with C4, in contradiction to the condition for the fourth term. So the maximum value of the inequality is indeed 2.

      So the setting is the following: I have a physical system, which traverses a sequence of states, or realities, or whatever you wish to call it. Then, I make the same measurement C at four different points in time, which are arbitrarily chosen. If there is truly only one definite state at any point in time, then the value of the quantity (1) can never exceed 2.

      But experimentally, one finds that there exist physical systems for which it does just that. Since the assumption that there is one definite state at any given time was all that went into the derivation of this inequality, therefore, this assumption must be false; I see no other conclusion to draw.

      • [deleted]

      • Post #49

      Jochen

      I was going to look at your previous post, and see if I can address it in the terms it was written. Although not an excuse, I am really getting exhausted with the amount of physical work I am currently having to do. So I said something which covered the key point. But first, then I'll get more coffee...

      "I only say 'one or another' because we don't know what outcome that'll be"

      The outcome of what? Each physically existent state is an 'outcome'. And you are measuring the previous ones, ie trying to identify what they were (let us ignore whether that is actually possible for the sake of this argument). So, in this particular sequence of realities (ie physically existent states), you would have an explanation of each discrete state and what differed each time between them, and what caused that.

      Now, you use the word "will", so I presume you are referring to the next state in the sequence (forget whether it has actually already occurred, we are considering this in terms of knowledge of realities). So, with the knowledge of previous states you could predict what the next in the sequence, would be. Indeed, if you knew everything about the succeeding states then your prediction will be correct. Physical existence does not occur in accord with strange rules. It is us who cannot identify everything that is occurring and therefore attribute it with 'strange' behaviour.

      If we do not know everything, then we can make a probability statement about what physically existent state will occur next. In doing so, one can average out other states, or deploy any other valid technique which helps.

      How can you measure too early or too late? You measure. That is, in doing so you are considering a physically existent state. By definition, you can only consider states that existed previously.

      "I have a physical system, which traverses a sequence of states, or realities"

      How, what is this that persists in the same physically existent state over time And if it does, then by definition you will always get the same result from any measurement at any time.

      More coffee, and I will try to get to the bottom of what you are saying.

      Paul

      • [deleted]

      • Post #50

      Jochen

      Re your posts. By the way, I do appreciate the effort you are putting in to responding.

      As I indicated previously, experiment is only valid if it is based on presumptions which correspond with physical reality. Your overt presumption is valid, ie "we can make measurements that tell us at least something about those states". You covert one is not. Because you then immediately assert a 'physical system' with a degree of persistence in one physically existent state over time. In reality you could not "perform two measurements on it" at different times, as at different times it is not necessarily the same existent reality, until proven otherwise, indeed it is highly unlikely to be. You can perform (with help!) as many measurements as you want, at the same time, ie on the same physically existent state.

      The phrase: "depending on the state of the system" is ontologically incorrect. The state is the system. And if it is in a different state, then it is a different system. Or more precisely physical reality. This reveals the whole misconception of physical existence which we all have. Bits have just fallen off St Paul's Cathedral. So it is not the same physical entity as it was previously, is it? But no, we all, understandably, talk in terms of 'it has changed'. That is, it remains, but is different. Which is impossible. If it is different it is different. What we mean is, our conceptualisation of physical existence, which is effected at a higher level than what occurs, is unaffected. Indeed, we will continue to call something St Paul's until it is a pile of rubble. In other words, the superficial features whereby we differentiate physical entities no longer pertain. But in reality it was never St Paul's, but a sequence of physically existent states. This being the only way there could have been that existence.

      Anyway! Your two measurements depending on the state of the system, are two measurements of something different. Either you are measuring the same thing or you are not. And I presume from the language you are referring to these being measurements performed on something that has existed. Whether it is two or two thousand is irrelevant. What matters is that you must measure the same existent state. In which case I am at a loss as to why you do not have an outcome. The fact that this outcome may be a probability statement is irrelevant. If it is impossible to discern what occurred, then one invokes a probability on what did.

      Then: "and thus, if measurements can tell us anything about these states, both C and C' will produce one outcome or another". Assuming that these two measurements were different types of measurement, but performed at the same time, ie on the same physically existent state, then they will only be different in so far as they are measuring different aspects of that state. Which is a statement of the blindingly obvious. However, linking this to your second post: "because we don't know what outcome that'll be". You are, supposedly, measuring a state that existed, not the one that succeeds it. You can, of course, make a prediction as to what will happen next, based on an understanding of the succeeding sates in the sequence.

      The fact that you have had to make predictions about what happened before that is a function of the fact that you cannot accurately and comprehensively identify what did happen. Needless to say, given the dynamics of combination/permutation, this is not necessarily a problem, and may be a necessity. And you can relate the two measurements, or however many were effected on the same state, in order to achieve the optimal prediction as to what occurred. And at a different time you will, assuming the state has altered, incur a different set of measurements. And so on. You can average them, or invoke any other procedure, in the face of being unable to discern what actually happened, in order to derive the best depiction of what did in the sequence.

      "all theories in which there is 'one physical state at a time' must necessarily obey the above inequality"

      Not so, the "inequality" is a function of you deeming what are actually different states as being the same. "We would find that the majority of physical systems---cars, horses, bushes, shoes etc.---indeed fulfill the inequality". That is because thy do not exist in that form, ie they are not states, we conceive them to be. "But eventually, we would find physical systems that don't". No, it is a case of, if you decompose physical existence to its actual existential level, you will. And everything ultimately exists in this way. That being, one definitive physically existent state at a time.

      Paul

      I think we still need some clarifying of terminology. If I talk about a physical system, I don't mean something that is the same at different points in time (like you, I'm not sure the notion is coherent: if something is literally 'the same' across time, in what sense does any time have passed at all?). Rather, I mean a collection (or sequence) of distinct physical states: what we call, in everyday language, a 'chair', on this picture then is a sequence of states (chair1, chair2,...) at different times (t1, t2,...), and each of the chairi is (or may be, at least) different from any other.

      In the derivation of the inequality, that is indeed the picture of physical system that is being assumed. Let's consider the 'system' to be a collection of states s1, s2, ... Then, measurement of C at time ti probes the state si, yielding Ci.

      So operationally, what I do is: I take my 'system', i.e. sequence of physical states, and make measurement C1. The system at that point is in some state which we may call s1. Since for any state si, measurement of C yields either +1 or -1, I get one of those results, and write it down.

      Then, I wait some time, during which the system will probably traverse some states, and make measurement C2. Again, I will receive a value of either +1 or -1. Since now the system is in a (generally different) state s2, the value need not agree with the one I got for C1 (note that by s2, I don't necessarily mean the state of the system that comes after s1---there could be any number of states 'in between', but since I don't do any measurement on them, I'll never know).

      Again, I wait some time, let the system traverse however many states it is inclined to, and measure C3, wait again, and measure C4, each time getting either +1 or -1, depending on the states of the system (remember, I'm using 'system' just as a shorthand for a set of physical realities existing, say, at a particular location across some amount of time; in the same sense, I could use the designator 'St. Paul's' to refer to 'the physically existing reality at 51°30′49″N 0°05′53″W between 1710 and today'; there's room to quibble with this, but since you'll probably understand the sentence 'meet me at St. Paul's tomorrow', such a definition is at least intelligible).

      Then, according to the arguments in the previous post, if I calculate the quantity

      [math](1)\,\,C_1C_2+C_2C_3+C_3C_4-C_1C_4[/math]

      I will never obtain a value exceeding 2, and thus, in particular, the average over all such procedures will never exceed 2, i.e. the inequality

      [math](2)\,\,\langle C_1C_2\rangle+\langle C_2C_3\rangle+\langle C_3C_4\rangle-\langle C_1C_4\rangle\leq2[/math]

      will always be fulfilled, *if* a physical system is a succession of different physical states---that is, if for any time ti, there exists a state si. I am not making the assumption that these states are the same, or in fact that there is any continuity between them---they are like states of St. Paul's, independently existing realities. I'm not deeming what are different states the same---that they are different (but definite), and hence, yield different values for C, is the foundational assumption in this derivation.

      So in the face of the experimental violation of (2), this is the assumption we need to throw out: there is not at all points in time a definite physical reality.

      Hi Jochen,fantastic essay, congratulations.

      You cite Leibnitz "take, for instance, ink splattered onto a page by shaking a quill. The distribution of ink blots on the page now will be effectively random. But nevertheless, one can always find a mathematical formula describing this distribution. Thus, the mere presence of such a mathematical formula does not indicate lawfulness. Rather, Leibniz argues, something should be considered lawful only if the complexity of the mathematical formula is less than the complexity of a simple description of, say, the ink blots on the paper."

      You can find a realization of that approach in my very short essay: http://fqxi.org/community/forum/topic/1609

      Do I risk a lot reducing your discussion with Deutch and your conclusion to the perception issue? Finally that issue is absolutely crucial point before we start any discussion in physics. Could we agree e.g. that hardware and software cannot be separated in Nature (that entity we call Reality)? So there is no underlying hardware but also no fixed reference point?

      Thanks

        Jochen

        So, we have a sequence of physically existent states.

        At any given time you effect a measurement, which yields an outcome reflecting some aspect of the state in existence at that time. At another time, repeat,...etc, etc. You are tracking some physical attribute, as manifest in the sequence of physically existent states when you happen to instigate a measurement.

        The fact that it is either +1 or -1 is irrelevant, that is just the valuation of that physical characteristic. It is different. If nothing else, it occurred at a different time. [And it is probably in a different spatial position, etc, etc- but forget that]. So it cannot be physically the same as something which occurred previously. Every occurrence is different, every physically existent state must, by definition, be unique. You are equating these different states via the valuation of this feature, which is incorrect, then finding an inequality which does not exist.

        Apart from which, I am left wondering where the difference between us is. You start the post by depicting what I have been saying all along. Then at the end of the post state: "there is not at all points in time a definite physical reality". Which directly contradicts your start point, because what is a physically existent state, if it is not a reality, and how can it be an existent state if it is not definite? How does something exist, but 'vaguely'? What happens, on some occasions is there no spatial position, or charge or whatever?

        On a minor point, it is not a case of time passing. It is about rate of change, irrespective of what is involved. One has to presume that alterations have different rates of occurrence. So, say we could harness the fastest rate of change in physical existence as the tick in a timing device. That would enable the differentiation of every state. Then, in some cases for any form of difference to occur in some types of alteration, there might be a duration of 3, 14, whatever, ticks. Time has passed, there has just not been any alteration.

        Paul

        I'm starting from your assumptions, in order to try and show that they are contradicted by observation. In deriving the inequality, I do not equate any different states, but explicitly consider them to be different from one another. The valuation of the physical characteristic corresponding to the measurement C is assumed to apply only at one given point in time, to one specific physical state. If C yields a different value than before, it's clear that the state has changed; but even if it doesn't, I don't consider this sufficient to indicate that the state is still the same, and thus, assume that it isn't. It's precisely from these assumptions that the inequality follows.

        As for how something could 'vaguely' exist, well, I'm probably as clueless as anyone else. I can model it mathematically, but what it means, I can't tell you. At the moment, my working hypothesis, so to speak, is that the quantum realm is such that there is more than one consistent story that can be told about what 'exists' or happens, and these stories don't necessarily have to agree. This doesn't make intuitive sense, but our intuitions are derivative of classical concepts and thus, misleading when those concepts no longer apply.

        A possibility I'm thinking about at the moment is that classical information really is synonymous with communicable information---that it's simply that information that I can freely share with you. But that's not the only kind of information: quantum information, in general, can't be shared. But to the extent that the concepts used in our thinking are derived from the concepts which we communicate---to the extent that thinking is shaped by communication, is maybe a form of communication ('talking to yourself' internally) itself---, to this extent it becomes at least comprehensible why we have such troubles coming to grips with a quantum world: the concepts which shape our thinking, since arrived at through communication, are fundamentally classical, and thus, incapable of fully capturing a quantum world.

        Hi Jacek, thanks for your comment; I'll have a look at your essay. I'm not sure what you mean with 'the perception issue', but as for hardware and software, my thinking is really that the same hardware can be seen as implementing different computations; and likewise, the same computations can be implemented on different hardware. So there's no one-to-one correspondence between both. In this sense, if we're the computation, as Deutsch says, there is no fact of the matter regarding the hardware---one would have to fix a reference in order to make the relative facts definite. So, since I believe that just the relative facts are enough, in this sense, yes, I think there's no underlying 'hardware'.

        Jochen

        "It's precisely from these assumptions that the inequality follows"

        Assuming this is so, ie you are saying what I am saying, then I am lost. Can you please explain this notion of 'inequality' and why it proves there is no definitive state in plain english. Sorry about this, but there is something wrong here and I cannot spot it, mainly because of the way it is expressed (in numbers).

        Put simply: some aspect of a physically existent state, when depicted in terms of 'one/the other' gives a different result when measured for each state in a sequence thereof, which therefore proves that the states do not exist in a definitive form (or perhaps even in a sequence of 'one at a time'-that was inserted later having noticed a previous post, commented on below).

        I am assuming at present it has something to do with expressing a particular aspect of the physically existent state in terms of -1 +1.

        Or it is something to do with the presumption of correlation, spins and two particles, ie a depiction of at least an aspect of a physical existent state which is flawed in some way (your post 25/4 05.36). Indeed, as I write this I realise that this form of operation was not included in your stated presumptions. Whereas I do not know what actually constitutes a physically existent state, because I am expressing things generically. All(!) I am saying is there must be one, only one can occur at a time, and it must be definitive, otherwise what we know as physical existence cannot occur. And I then notice in that post: "About your assertion that there is 'one state at a time', this too is something in direct conflict in quantum mechanics. Leggett and Garg have shown that it leads to inequalities..." Now hang on a minute, I thought from subsequent posts you were agreeing with 'one at a time', your 'proof' was about the lack of definiteness??

        Or it is something to do with: "If now the measurement did not depend on that state, this function would not depend on s at all---but then, this so-called physical state would be completely without consequences on the physical world, at least as we perceive it". The word 'depend' worries me. As I said, observation/measurement cannot have any effect on the physical circumstance, because of the sequence order of that interaction and the fact that the subsequent processing is not a physical process. And incidentally, that is then followed up by a false differentiation of physical systems, in that one type referred to does not exist.

        [When I switched to Cox & Forshaw as a reference to go through Einstein, I spent several frustrating days trying to derive a set of equations only to then realise that the vertical (distance between mirrors) of the triangle they used to represent the circumstance cannot be given a value (they deemed it to be 1 for simplicity) because that renders the other sides to have a specific value given the presumptions as to what they represented. Such a simple, apparently innocuous act, ie 'let this be 1 for the sake of simplicity because we are not interested in it'. It was obvious once one realised it, but I only kept going because I knew the end conclusion must be wrong].

        Now, given your language I am assuming you mean that any given physically existent state might not (or is always not) definite. Not that an aspect of any given physically existent state (or the whole of that state) is not definite in the sense that we cannot identify it, but it does occur in a definite form. I just say this to double-check.

        The interesting point here being (and of course you are not alone) encapsulated in: "As for how something could 'vaguely' exist, well, I'm probably as clueless as anyone else. I can model it mathematically, but what it means, I can't tell you".

        There are two choices: either physical existence occurs in a definite form, or it does not. The precise detail of what does not is irrelevant, it does or it does not. Now, logic dictates that within the terms of the existentially closed system which constitutes physical existence for us, something which is existent must be so in a definite form. Whether we can detect that or not is irrelevant. Otherwise, what at any given time is 'there', how does 'it', which is not definite become some other 'it' which is also not definite (or probably not). The whole system just collapses. Yet people are (apparently) happy to start from this peculiar base. A similar sort of argument applies to the 'one state at a time' point. People know there is difference, but whilst the only way to explain this is by sequence (ie one state at a time), they are again (apparently) quite happy to ignore this and invoke some form of muddled compromise.

        I do not understand your concept of information. Within our existentially closed system, 'stuff' 'exists' independently of the mechanism which enables awareness of it (in those entities which possess the relevant capabilities). You seem to be equating information with existence.

        Paul

        OK, then let's back up a little. I'm not sure I know how to state things 'in plain English', but I'll give it a shot.

        First, let's define some terms. A physical system S is for now assumed to be a sequence of physical states si, occurent at times ti. The system traverses or evolves through its states---this is the origin of any observable change. A measurement can be regarded as a 'question' asked of the system; there are certain such questions that the system will always answer with either 'yes' or 'no', i.e. for any given state s[sun]i, the answer to C can only be either +1 (yes) or -1 (no). We can call these states then yes-states or no-states.

        The rest is essentially simple algebra, and no further assumptions will be introduced. All that operationally is done, is to measure the system at four different points in time, and note whether it answers 'yes' or 'no'. This will give us a list of four numbers, such as, for example

        [math]

        C_1 = +1\\ C_2=-1 \\ C_3 =-1\\ C_4=+1[/math]

        From these four numbers, we calculate the quantity

        [math]X=C_1C_2+C_2C_3+C_3C_4-C_1C_4[/math]

        As you can see for yourself, in the given example, X = -2. For another example, consider

        [math]

        C_1 = +1\\ C_2=+1 \\ C_3= +1\\ C_4=+1[/math]

        Here, X = 2. If you want to, you may play around with these numbers; you'll find that there is no case in which X exceeds 2. This is then the inequality

        [math]\langle C_1C_2\rangle+\langle C_2C_3\rangle+\langle C_3C_4\rangle-\langle C_1C_4\rangle \leq 2[/math]

        which in words just means '(the average of) X is always smaller or equal to 2'. (Here, we take the average for technical reasons that need not bother us; it's clear that if the inequality is valid in every individual case, it's valid on average, as well.)

        But now consider what it would mean if X were to exceed 2. A possibility would be

        [math]

        C_1C_2 = +1\\ C_2C_3=+1 \\ C_3C_4= +1\\ C_1C_4=-1[/math]

        where I've now tabulated the products of results, not the resulty themselves. As you can see, this amounts to X = 4. How could this come about?

        Well, in order for each term to be +1, the factors must be equal either both +1 or both -1. In order for a term to be -1, they must be different, one +1, and the other -1. So the above table implies that

        [math]C_1=C_2,\,\,C_2=C_3,\,\,C_3=C_4,\,\,C_1\neq C_4,\,\,[/math]

        But this, as I've already argued, is impossible to satisfy under the assumption that there is one definite state at any given time: this assumption necessitates that C_1=C4 if the first three of the conditions are to be satisfied, in contradiction to C_1=/=C4, the fourth condition.

        But how can this be? Well, one possibility is provided by quantum mechanics: that of linear superposition. There, you find that whenever there are two states |s_1> and |s_2> possible, there is also a state like |s_1> + |s_2>. Then, it becomes possible to violate the inequality; and this violation is what we actually observe.

        This is contrary to everyday logic, certainly. But you can find logics with which it is perfectly consistent; Birkhoff and von Neumann have proposed a logic in which the distributive law is not a theorem that is modelled by quantum mechanical propositions, and Putnam has (somewhat (in-)famously) argued that this is the 'correct' logic to reason about the world. On this logic, for instance, the conclusions you draw would by and large simply not follow. The logic we use everyday is, on this view, merely a classical approximation to the logic the universe actually observes.

        Now I have some troubles with that explanation, resting mostly on what's sometimes called 'the principle of tolerance' (by Carnap, I think?): that different logics, in a much as they are consistent, really say the same things, and if they appear to disagree, then you simply got your interpretations confused. But consequently, I also think that classical logic does not have an a priori privileged place.

        Ultimately, it's really similar to how you can't imagine higher-dimensional geometries: though some people claim to be able to, I never could visualize four orthogonal directions in space, let alone five or more. But that doesn't make them impossible---it's just a function of my own limitations, but those don't impose limitations on nature.

        Jochen

        Sorry, bit rushed, off to Bruch's violin concerto a Festival Hall.

        Ok, I got all that before, so I now know I am not missing anything. And the issues, as alluded to previously, are:

        1 This physical characteristic which only has values of 'yes/no' is but one aspect of any physically existent state. It does not represent the entire state.

        2 The repetition of one or the other of these values does not mean it is in the same state again, because if nothing else, the time is different.

        3 This is a somewhat dubious comment, but I am not convinced that 'yes/no' really differentiates the existent possibilities of whatever this particular physical characteristic is. This is just a simplified conceptualisation.

        4 Why are the outcomes represented in this way? This represents some form of presumed relationship. My reading of this scenario, as defined, is that you have 4 readings from 4 different times in the sequence. Another way of expressing this is what is X and why is it being worked out this way? The logic of what is being said just seems to relate to the representation of a particular aspect of a physically existent state, and some form of presumed relationship. As you say, it is "simple algebra", but what in physical existence does it relate to?

        "there is one definite state at any given time: this assumption necessitates that C_1=C4 if the first three of the conditions are to be satisfied, in contradiction to C_1=/=C4, the fourth condition"

        I do not understand this. There seems to be some hidden assumptions here, and this does not relate to the attribute of definitiveness anyway. It relates to the permutation of this physical characteristic in sequence, because it purportedly must occur in some sort of relationship. That is fair enough, if proven, then a certain attribute if it has been y in the previous state, must be x or z in the succeeding one, or whatever. What has this got to do with definitiveness, and/or one state of the sequence at a time? Obviously, if these values have to be calibrated by probability, that is irrelevant, out inability to discern the actual value does not alter physical existence.

        Paul

        OK, responding to your points in order:

        1. I'm not assuming that the physical characteristic represents the physical state; I'm merely assuming that each state has a definite value of this characteristic.

        2. I explicitly don't assume that it's the same state again just because I get the same measurement result for C.

        3. There's no 'maybe' answer: we can do measurements that, whenever we execute them, yield either +1 or -1. An instance is spin: it's been measured millions of times, and it never comes out, say, 'pink'. (And one can regardless extend the discussion to cover such cases, say account for measurement errors etc.; but things get more complicated then.)

        4. X is being worked out that way because I can. It's got no a priori physical significance, other than of course in any theory which obeys the 'one definite state at a time'-principle, it's smaller than or equal to 2.

        What this has got to do with definiteness is of course that in any theory with definiteness, there's at all times a definite value for C; and if there's at all times a definite value for C, then X is bounded by 2.

        Hope you enjoy the concert!

        Jochen

        Yep, did not stay for Berlioz, left on a high.

        Re 3. "we can do measurements that, whenever we execute them, yield either +1 or -1" Yes, but is the physical characteristic 'on/off' is what I asked, not what measurements we are capable of doing. For example, if spin means literally spin, or indeed some other form of movement, then why is it being represented by 'on/off. I can understand an answer that says because that is the most we can identify. But what is an existent state of something spinning/moving? Answer: each degree of alteration in spatial position/orientation, which is somewhat more than 'on/off'.

        Re 4 I asked what is the logic which justifies relating the readings this way? My understanding is that they are just 4 measurements of a particular physical aspect of 4 physically existent states in sequence. If I wanted the average, I would add them up and divide by 4, or if there is some sort of rule (can only have 2 -1's consecutively)...etc, etc. But what is the relationship why justifies this processing of the results? And what is X, indeed, I think I had best ask what is C?

        Sorry for all these questions, but there is something happening here I cannot pinpoint. A definite value of some feature which is only expressable (or apparently only exists) in terms of +1 -1 means that some expression of a sequence of these can only result in a limited number of outcomes. Which sounds like a simple expression of maths, until I understand its relationship to physical existence. But apparently it is enough to substantiate a very fundamental fact about existence

        Paul