Essay Abstract

Since Wheeler's 'it from bit'-proposal, informational ontologies have been considered a promising candidate for the description of the fundamental nature of the world. However, few concrete examples seem to be in wide discussion. I introduce a very general model ontology, based on the notion of information states in an information space, whose dynamics are supplied by computations acting on them. I point out that such approaches lead to hard problems, the paradigm example of which is the mind-body problem: the underdetermination of the mental by the physical. However, I argue that a relational or relative-facts account of information sheds new light on this and similar problems, such as for instance the problem of deciding which computation is realized by a given physical system, and the quantum measurement problem.

Author Bio

Jochen Szangolies studied Physics at the University of Siegen, and is currently a PhD student at Heinrich Heine-University Düsseldorf, where he works on quantum contextuality and quantum correlations.

Download Essay PDF File

Jochen

"anything can be either round or not round..."

Physically, anything can only be anything, it cannot be anything else. If the representation is effected at a higher level of differentiation than what physically exists, then 'things' can be categorised as the 'same'. But this does not create physical existence, it creates a categorisation thereof.

"perhaps the world, at its (very deep, to speak with Wheeler) bottom, is nothing

but an information space"

Yes, and perhaps it is a shoot em up game operated by green giants. My point being that it could be any possibility, but we can never know. We can, in science, only address what is potentially knowable, and that is driven by a simple physical process. Otherwise, we are involved in belief.

"But this difference structure is not enough to recover the physical state of an object".

Not so. You are presuming, see above, open-endedness. Whereas we are trapped in a existentially closed system. The only way to identify what that constitutes, from within, is to compare everything and identify difference. This being objective within that closed system. You are also presuming that we can effect some form of 'direct access' to reality (that is, even the possibly limited one we can know), ie "But this difference structure is not enough to recover the physical state of an object", which we cannot. We can only compile information/knowledge which at best we can deem to be the equivalent of reality. It may well be that we do not have the capability to compile a representation at the existential level, but that is a different, practical, matter.

"Hard Problem III: Consciousness"

What happens in the process of converting a received physical input to a perception thereof is irrelevant to the physical circumstance.

Paul

    Perhaps I did not do a very good job at explaining the project I am engaged in with my essay. Basically, when I say 'anything can either be round or not round', I do not mean that the same physical object can be one way or the other, but that among all possible objects, there are those that are round, and those that are not. Thus, the alternative 'round/not round' partitions this set in two. You can then proceed to further characterize any object by deciding more alternatives, like in the game 'twenty questions', until you have picked out a unique one. The answers to all the questions then give the 'information state' I associate with the object.

    You're right that this does not, prima facie, tell us anything about the object itself. But the fact that such a procedure is possible at all might tell us something about the kind of world we live in: it does not seem necessary that this kind of thing should be possible; for instance, we could live in a world of indistinct, boundariless 'things' that cannot be individuated in any way, but it doesn't seem that we do (never mind for the moment the problems quantum mechanics brings to individuating objects).

    So, one possible hypothesis that explains why we can associate an information state to every object is that, at bottom, there's really nothing but that information state; this is, of course, not a unique explanation, but it seems at least a plausible one. My project in the essay is then to take this hypothesis as given and investigate its consequences. In particular, I focus on what I call 'hard problems', after the hard problem of consciousness: problems that involve a sort of fundamental underdetermination of higher-level facts by lower-level facts. These, I argue, are simply analogous to the fact that from just some information---some string of bits, or other symbols---no unique meaning is derivable: information encodes only differences, not 'things in themselves'.

    So I'm not really saying that 'this is the way the world is'. I'm just attempting to investigate the consequences of a particular way it might be, and often is popularly supposed to be. Whether these investigations manage to uncover something of interest, I'll leave to the reader to decide!

    Jochen, this is a very thought provoking essay. I think your concept of relative facts has a pedigree going back to the Victorian relativity of knowledge, or perhaps further to Leibniz as you mention, but you have given it a modern informational perspective. I am convinced by your arguments that understanding hard problems such as consciousness require a good grasp of the interpretation of information.

    Let me pose you a question. Suppose I told you that I have a communication channel with some form of intelligent live in another universe. I don't know whet language they communicate with or anything about their laws of physics. I just know that the same rules of logic apply for them and they can read strings of bits that I send them. They can reply with similar strings. Could we establish a meaningful communication with them even though there is no common ground between us?

    I think the answer is yes. We could start by sending them strings of prime numbers as in Sagan's "Contact", then we could move on to mathematical expressions that defined prime numbers and other sequences. Prime numbers do not come from our laws of physics. They are based on pure logic so they would start to learn our mathematical language in the same way as Friday learnt English from Robinson Crusoe. Once we had some understanding we could start to tell them about the laws of physics in our universe and they could tell us about theirs.

    We would never know for sure that we were understanding them, or them us, because we would be relying on what could be coincidences in the patterns of bits, but the more we communicated the more sure we would become that useful information was being sent.

    The interesting thing here is that to establish understanding you need redundancy in the message. If it was compressed to a string as short as its Kolmogorov complexity they could not understand anything from it. They would just see a random string of bits. What I am trying to say is that the relative part of the information is contained in redundancy and without it the actual string of bits carries no useful information. I think this is an interesting paradox which might explain why redundancy in the form of symmetry appears in physics. Without it there would be no meaning in any information and therefore no sense of consciousness.

      Jochen

      Yes, but my point was you need a definition of 'round'. And, by definition, if you define that precisely, you will only relate it to one existent state, ie that unique one you refer to. That is, anything can only be anything, it cannot be something else.

      And, OK, but you do not have to take this approach to get to that answer. 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 alteration. Which means that the physical existence we can know is 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.

      The issue being that I doubt if we can investigate/identify reality at that level, even though this is how it occurs. Think of a film, but an incredibly fast film. Ultimately, there is a discrete frame.

      "information encodes only differences, not 'things in themselves'"

      You derive knowledge about things, by differentiation from other things. This is the only way to proceed in a closed system. Here is a fun way of putting it(!):Every statement has the same logical form, ie a comparison to establish difference, which necessitates a reference. But, an absolute extrinsic reference is never available, because that can only ever be the possibility of an alternative, ie if it 'is', then by definition it is not extrinsic. That is, given A (where A is 'is'), there is always the logical possibility of not-A, however, this cannot be defined from within A, as a reference from within not-A is required for that. So all that can be defined is A, from within A, and that that is not not-A. But not what not-A is.

      Paul

      Thanks for your thoughtful comments, Philip. Yes, I believe you can trace the concept of relative facts quite far through history, but I'd like to stress that I don't consider myself a relativist when it comes to knowledge. I think probably relative facts should best be understood in analogy to temporal facts: think of the sentence 'it is raining here'. This is false now, but was true a few days ago. So in a sense it is relative to the present moment. But there's also an absolute truth to it in the sense that there's an absolute matter of fact to what moment is 'now'. In the same way, there's nothing relative to my saying 'the electron's spin is up', after having performed the appropriate measurement. The electron's spin is up relative to my experiencing it as such, but given my experience, there's an unambiguous fact regarding its spin; the electron's spin is not both up and down in the same sense that it's not both raining and not raining here at this moment. Not all relative facts are actual, just as not all moments are 'now'.

      Regarding your thought experiment, I think that's a very interesting question. In a way, it's asking: 'can you get semantics from syntax?', and is thus perhaps also a distant cousin to the Chinese room experiment I mention in my essay (the discussion of which mostly landed on the cutting room floor, though). I believe that the most widespread opinion would be that this is not in general possible: a string of bits does not tell you what mapping to use to convert it to something you find intelligible.

      But thankfully, we need not be encumbered by what the most widespread opinion on the matter is! Indeed, in a way my argument rests on the stipulation that while syntax does not determine semantics, it can at least constrain it. If this reply considered of just a single letter, you would probably not consider it interesting regardless of what language it might be in!

      Nevertheless, I think the issue is not quite clear-cut. However we wish to communicate, we have to assume some common ground, and I'm not sure mathematics and logic (whose truths we can take to be a priori) really suffice. Suppose for instance I were not speaking English, but another language with the peculiar property that any valid utterance constructed therein is also a valid utterance in English, but with a different meaning. The scenario is staggeringly unlikely, but does not seem impossible: we could be having two entirely different conversations.

      Also, your concrete suggestion of using prime numbers in fact assumes something more as a common ground than just math and logic. The first difficulty is making something intelligible as a message in the first place: electromagnetic waves, for instance, may be wholly meaningless to beings without sense organs to detect them (more so in the case of entirely different physical laws). But if we can solve this---as you said, we have a communications channel capable of transferring strings of bits---, then still we need, I believe, hope that some contingent facts of their intelligence have worked out approximately similar to ours. One example might be that they'd also have to use a place-value system to represent numbers, perhaps. Another that they need to have a concept of prime numbers, which is perhaps not a priori: even if math is the same everywhere, not all of its elements may be equally useful. Among all possible mathematical systems, we use a vanishingly small set, and a case can be made that the set we use is dictated by the universe we find ourselves in (natural numbers, for instance, might not be useful in a universe that's in some sense 'fluid', with no clearly individuated objects).

      Also, even if they are a scientifically advanced species, I'm not sure we can accept it as a given that they use mathematics in the way we do at all: Wigner wondered about the 'unreasonable effectiveness' of mathematics in describing our universe; perhaps, if their universe is structured differently, mathematics doesn't really apply as well (though I struggle to see how this could be possible). Another suggestion might be to follow the arguments of the fictionalists, which hold that mathematics is wholly conventional; an interesting effort in this direction is Hartry Field's 'Science without Numbers', containing a construction of Newtonian gravity without reference to mathematics.

      But I think your observation that if this can work at all, it depends critically on the structure of the message we send is a very good one: it's similar to what I've heard called the distinction between the 'outer' and 'inner' message (I think maybe I got this from Douglas Hofstadter?). Basically, anything that is to serve as a message first needs to identify itself as such: this is the content of the outer message, which ideally says: 'hi, I'm a message, decode me; here's how:...' in such a way that it can be understood by every conceivable intelligence. This means that the outer message must be phrased in a kind of 'universal' language; but it's IMO not really clear whether such a language can exist. It always needs the assumption of some commonality between the intelligences attempting to communicate; but I think a plausible case can be made that each such commonality (except perhaps for trivial ones, such as 'existence') constrains the set of intelligences in such a way that one can at least conceive of one not meeting the relevant criteria.

      Paul, I'm not sure I need a definition of 'round': what I really need is a concept of difference, so that, if I had a round object, I could classify all other objects according to 'different from that object'---i.e. not round---, or 'not different from that object'---i.e. round. Note the counterfactual language: I do not need an actual round object, I just need the possibility of classifying objects in this way.

      Regarding your statement: 'what occurs, does so, independently of the processes which detect it'---this, I think, is not true in general. In fact, in quantum mechanics, the Kochen-Specker theorem states that if you assume this, you run into conflicts with experimental observations (i.e. the assumption above enables you to derive a certain inequality which must hold for all theories obeying this assumption, but which is violated by experiment, indicating thus that quantum mechanics does not obey the assumption). This is generally called the 'contextuality' of quantum theory: what you observe depends on how you look, and in particular, on what else you look at. You might say that facts in quantum mechanics can only be considered definite relative to a given experimental arrangement (which is a bit of what Bohr was all about).

      Jochen

      If you do not have a definition of 'round', or indeed any other such factor, then how do you deploy it as a reference? This underpins the fact that any judgement involves comparison to identify difference, which necessitates a reference (this is what you are referring to as relative facts). In identifying anything, we can, as you say, only do so by difference. But this is because we are trapped in an existentially closed system. In other words, we cannot externalise ourselves from it, and against an external reference (which is still a reference) invoke a judgement as to what 'is' within our system. Furthermore, if your reference is not in accord with our physical existence then your categorisation will not be either.

      "'what occurs, does so, independently of the processes which detect it'---this, I think, is not true in general"

      Leaving aside the somewhat strange caveat of "in general", can you then please tell me how the detection systems and any subsequent processing has a physical affect on physical existence? Given that, to be able to detect something, that something has to be in existence first. I am not aware of any physical process which operates contrary to the sequence order of existence. Leaving aside the fact that what you observe (ie receive) is light, which is a representation of an occurrence, not the occurrence. Although the same sequence order point applies to light anyway, in that it is physically existent in its own right.

      Paul

      Thanks for your comments, Joe. Perhaps I should clarify that when I talk about everything being expressible through the symbols 1 and 0, I only intend that to mean anything that's expressible, say, in natural language, or mathematics. There the claim is near trivial: whatever you can say with an alphabet of fixed symbols, you can say with an alphabet consisting only of 0 and 1. All you have to do is assign to every symbol of your original alphabet a unique string of 0s and 1s, and then re-write what you want to say in these terms. This may seem like a triviality, but it's deep in the sense that it leads to the realization that the information in a message does not depend on the symbolics it's clad in.

      Thanks for the interesting pointers, I was not familiar with most of these things.

      My opinion is that logic and mathematics is sufficiently universal that intelligent minds could establish communication by sending messages, even if they have no common grounds, I realize that this would be controversial among philosophers of mathematics. It depends on the old debate about whether mathematics is discovered or invented. I think that basic mathematics concepts such as numbers are discovered but I am not sure I can really make a totally convincing argument. Perhaps some day they will be able to test these things by generating intelligent minds in computers that have no idea of the outside physical world. They could allow two such minds to communicate with just strings if bits and see if they can manage to understand each other

      This does not necessarily mean that your idea of relative facts is wrong, just that you have to take care about what it means. It requires messages written in a very special way and a high level of intelligence to get over the lack of context. That is not representative of information in physics.

      Paul, if I understand you correctly, I think that in a sense I want to do away with any reference, and leave as a foundation to an ontology only those kinds of facts I have termed 'relative'---those which, if there were a reference, would be definite. Consider a statement like 'it is raining here'. Without any further specification, you won't be able to decide its truth. This specification comes in the form of the present moment---'it is raining here now' has an absolute truth value (it's false). So temporal facts can be considered as an archetype for relative facts: time provides an indexical, and its value decides what is true at a given moment. Or consider the example of a rainbow I gave in the essay: the position of the rainbow is only definite relative to the position of an observer (which could be a camera, for instance). Different observer positions yield different rainbow positions, and this circumstance is expressed in the form of relative facts.

      As for your other point, it is well accepted that in quantum mechanics, speaking about the definite value of some observable quantity absent an experimental context leads to contradictions, as exemplified by the Kochen-Specker theorem. Consider an observable A, which may, upon observation, yield two different results---say, '' and '-'. This may be something like an electron spin that can be either up or down, or you may imagine something more prosaic, like an urn that may contain either a red or a green ball. The observation now is just pulling out the ball, and noting its color.

      Now consider a second observable of this type, call it B (sorry for not being terribly creative with naming here). A and B have the further property that they are co-measurable: that is, if you measure A, then B, and then A again, the second measurement of A reproduces the first result. Thus, measurement of B does not interfere with measurement of A: if you had the value A = , then measure B, you'll know that you'll again get if you measure A again. The same goes for measurements of B. So, A and B may be considered properties of the system, some intrinsic characteristic that is merely revealed by measurement.

      But now consider a third measurement, again of the two-outcome type, which I'll call C. Again, C is co-measurable with A, in the same sense that B is. Now we have a system that appears to have three properties, corresponding to the values of A, B, and C. It is natural to suppose that these are intrinsic characteristics of the system, revealed by measurement.

      However, quantum theory tells us (and experiment confirms) that this is not the case: the assumption that, if A and C are measured together, and A and B are measured together, the value of A is independent of whether it is measured together with B or C leads to a contradiction with experimentally observed results. But then, we cannot attach to the system an intrinsic value for A after all; rather, this value has to depend on the context in which it is measured (on whether B or C is measured simultaneously, that is). This is the contextuality of quantum mechanics. If this seems similar to Bell's theorem, it should: essentially, Bell merely guarantees the impossibility of one measurement influencing the other (according to assumptions of local causality) by assuming spatial separation between measurements; but the contradiction comes from the same properties of quantum mechanics as in the Kochen-Specker case.

      One possibility I have been thinking about to facilitate such communication may be to collapse the outer and the inner message into one, taking advantage of the possibility of self-reference: one could encode a message saying 'this message contains n bits of information' in a string of Kolmogorov complexity n. That way, any alien intelligence could at least approximate the content of the message by merely estimating its Kolmogorov complexity. (The obvious problem here is that Kolmogorov complexity is only definite up to an additive constant, though.)

      As for the philosophy of mathematics, the problem I have with Platonism is the problem I have with all kinds of dualism: the lack of a causal nexus to connect the two substances. How does an abstract mathematical concept, which has no mass, energy, no location in space and time, and can't exert any forces, produce the relevant influence on a physical meat brain in order to make the latter aware of it? If the number 3 exists in some abstract realm, how does it reach down from there and make itself known to me? (Just as an aside, this is a problem Leibniz also struggled with: in order to get his primitive substances, the 'monads', to act in synchrony, he found he had to postulate a kind of 'pre-established harmony', such that each monad could evolve for itself, and nevertheless lead to the appearance of interaction.)

      I think that anything that has causal contact to the physical ought to be considered physical itself: after all, we only know the physical world through chains of causality (photons hitting surfaces, being reflected by them, hitting our retina, causing certain reactions there, eventually producing patterns of neural activity). So it's hard to see how something could exert causal influence on the physical, without itself being physical (at least to me).

      Personally, I think that the philosophy of mathematics that fares best is rooted in structuralist ideas: mathematics captures the relations between objects, the structures build from them (I consider this to be very well in line with my ideas about relative facts, but I think a detailed argument would take us too far afield). Natural numbers, for instance, embody the relations between individuable entities: stones used for counting, say.

      But that doesn't mean that we can't 'discover' new mathematical ideas that are not suggested to us by the physical world: just as we can think about things regardless of whether they exist, we can consider their relations. And if we have made no mistake in our imaginings, then what we end up will not be arbitrary, but rather, fixed by simple consistency requirements. In this sense, mathematics, even if not enjoying an independent, Platonic existence, nevertheless isn't just some arbitrarily made up stuff.

      Another point of view is that there are many sets of axioms, and which ones you consider is essentially arbitrary; but what then follows from them, is not. So for a theorem t and a set of axioms A, that say A -> t is a priori, and true independently of human considerations (it's basically a relative fact: t is not absolutely determined, but A -> t is); but out of all the infinitely many possible sets of axioms, which ones we consider, that is I think ultimately contingent (on our being humans, on our living in a universe of 3+1 dimensions, etc.).

      Jochen

      "I think that in a sense I want to do away with any reference, and leave as a foundation to an ontology only those kinds of facts I have termed 'relative'---those which, if there were a reference, would be definite"

      First you cannot do this, there is always a reference. Second, you are maintaining a reference anyway, and re-labelling it as facts which have a certain attribute and are therefore definite (ie facts). But that is the whole point of science. So the real question is, in the context of our physical existence what constitutes a fact?

      Judging the validity of that statement has nothing particularly to do with time. Although none of the statement is that specific, it involves the concepts of rain, in a spatial position, at a time. The validity of that statement therefore depends on that event having occurred as depicted, ie what constitutes 'rain', where was 'here', and when was 'is'.

      Your point in respect of the rainbow is incorrect. The actual spatial location of the rainbow does not alter. Its relative spatial location does, obviously, because its actual spatial location is being expressed with respect to a different spatial location.

      The point is that everything can only be identified by comparison and the identification of difference. Which necessitates a reference. And in order to ensure comparability of outcomes, consistency of reference must be maintained.

      Probably the simplest way to respond to your next paragraphs is to point out that A, B, C are not A, B, C subsequently, they are something else. Because you said "then". The only way A, B, C could still be A, B, C, then, ie at a different time, is if a) nothing about them ever alters, b) nothing has altered in the duration being considered, c) the physical attribute being considered has not altered (although this is not really a condition because the entity is therefore different).

      To illuminate this point here are three introductory paras to another paper of mine:

      1 Distance is an artefact of physically existent entities, it being a difference between them in terms of spatial position. Existence necessitates physical space, but that can only be assigned via entities. So distance can only involve entities which exist at the same time. And they can only exist in one physically existent state at a time.

      2 Therefore, any given distance is always unique, since it reflects a definitive physically existent circumstance at a given time. The notion which presumes there could be varied results when quantifying it, either in terms of space or duration, is a fallacy. Whatever the measuring methodology, there can only be one outcome.

      3 Unless this is understood, a problem arises when distance is expressed conceptually in terms of duration. The concept being that it can be measured as the duration which would have been incurred had any given entity been able to travel along it, either way. But this is not possible, because there is no duration available during which that can actually happen, so it must be understood that there is no duration, as such. That is, the result is just an alternative expression to, and the equivalent of, a specific spatial measure. Misunderstanding this leads to the flawed application of the equation x = vt.

      Another general response to those paragraphs would be to point out that a) what we observe (ie receive) is a physically existent representation of the reality (eg light), not the reality, b) observation/measurement can have no effect on physical existence, because it occurs after that.

      Paul

      I have little time right now, so I'll just respond to two points briefly:

      First, how is the absolute position of the rainbow defined? There is no thing in the world that corresponds to the rainbow that has any definite location, as far as I can see. It's not the case that if observer 1 sees the rainbow as point x, and observer 2 at point y, each one is more right than the other: their perception is both equally valid.

      And as for the Kochen-Specker example, we can get rid of the contentious 'then': rather than measuring first one, then the other observable, you can measure them simultaneously.

      We have different opinions but also some common ground. My position is basically platonic. Many structures in mathematics seem to be interesting because they have physical counterparts, especially euclidean geometry and combinatorial problems, but I think mathematicians would have been interested in most such things even if they lived in a non-geometric universe. One observation that supports my view is that many mathematical concepts were invented by mathematicians before their importance in physics or practical applications was known, e.g. complex numbers, non-euclidean geometry, group theory, prime numbers even. Other concepts originally inspired by physics have been used by mathematicians to prove theorems that were originally stated without any reference to applied maths.

      However, I do not agree with the assertion by Tegmark that "in those [worlds] complex enough to contain self-aware substructures [they] will subjectively perceive themselves as existing in a physically 'real' world" You mention the novel "Permutation City" which as you know has a relationship to my earlier ideas about permutation symmetry over spacetime events. In that book in the end the simulation into which they copy themselves becomes real even though the computer stops running. I don't agree with this and if I understand you correctly I dont think you agree either because the simulation loses contact with the physical world. Tegmark would presumably disagree with us and agree with Egan.

      My reasoning is that this was just one version of the simulation and it was being simulated everywhere, in the rock or the wall as you cited from Putnam and Searle, or in the dust as Egan put it. Some simulations would have small differences so which ones would the copies experience? There is nothing to hold their experience to the original algorithm. How then can any copy be aware, and isn't our own consciousness just a program running on wetware?

      My solution is that what counts is universality. The features of mathematics that have most intrinsic interest are those that come from universal behavior of complex systems. They appear in many different places and that makes them useful and interesting. I think if you take the grand ensemble of all possible mathematical universes then there is a universality of behaviour that in some sense dominates the ensemble, like the laws of thermodynamics but more general. This is what determines the actual laws of physics and awareness is only possible through contact with that physical system. It seems to me that you and I therefore arrive at a very similar conclusion about awareness, even though we start from almost opposite philosophical positions. Do you agree?

      Yes, I think there's definite common ground. I'm also very interested in what I call 'ensemble theories of everything', one example of which is Tegmark's, another Schmidhuber's ('Algorithmic Theories of Everything), and perhaps also the recent and intriguing work by Lloyd and Dreyer ('The Universal Path Integral'). But one worry I think these proposals must address is the issue of triviality: if your explanation is consistent with everything, it explains nothing. This is why I can't really get on board with the extreme form of Tegmark's mathematical universe: one could postulate such a hypothesis basically no matter what. I'm not even talking about falsifiability; it's just that simply saying 'everything exists' is not a good explanation for our observations.

      This is essentially what leads me to search for a nontrivial implementation relation, i.e. for a way, given a physical system, to constrain possible computations that can be attributed to that system, for instance. The scenario in which the simulation shuts down and is nevertheless carried on in the dust essentially corresponds to a trivial implementation relation: everything computes everything, so every computation occurs *somewhere*. Like you, I find this dissatisfying. It would also lead to a complete dissociation between the computational and the physical: your dripping faucet could be seen as computing universes, the square root of pi, or the precise number of licks it takes to get to the center of a tootsie pop---absolutely anything at all.

      By universality, do you mean computational universality, or universality in the sense of a system's behaviour becoming independent of the detailed microphysical details? Because I have been looking at computational universality specifically, but I think the notion is still too broad: very simple systems are universal, and thus, can in principle simulate any given Turing machine, again leading to a version of the above worry. So I've begun trying to find a criterion (a kind of 'informational equivalence principle') that sort of uses a quantification of the information content in a computation in order to determine what computations can usefully be attributed to a physical system, the idea here being that if the information content of the computation greatly exceeds that of the system, in order to get the computation out, you sort of have to put in more work than is done by the system itself---so that the computation is not really performed by the system after all (like in my example where a code maps the symbol @ to the complete works of Shakespeare---you need something equal in information content to the complete works of Shakespeare in order to extract them from @, so the symbol itself plays really only a nominal role in the production of all this information).

      Jochen

      "First, how is the absolute position of the rainbow defined?"

      It cannot be. We cannot know the absolute position of anything. Only its relatyive spatial position with respect to anything within our existentially closed system The observer's are each valid in respect of observer vis a vis rainbow. But if you want to know the position of the rainbow with respect to both observers then you need another reference.

      Re your second point. Ohysical existence only occurs in one definitive physically existent state at a time. So if state A & state B exist at the same time that is fine. I am leaving aside the practical difficulyies of measuring such. But one should start with how reality must occur, then try and resolve any problems we get as a result. Not construe a constitution for reality which fits our abilities.

      You need to read my essay

      Paul

      Computability is one form of universality where there are lots of ways to define "computable" and none that is the best, but they are all equivalent. Actually that is not quite true. If I define computable to mean something I can compute on my PC then that is limited by the memory capacity, but taking all possible finite computers together there is a limiting case where memory is not an issue.

      A similar thing happens in critical phenomena where you approach a critical point and renormalise as correlation lengths grow to get a theory which does not depend on microscopic details. This is another form of universality, as is chaos theory.

      In the "ensemble theory of everything" perhaps something happens where one particular form of universality is more common than anything else in the ensemble so it simply dominates.

      This might happen in the Lloyd-Dreyer universal path integral for example. If something dominates in this way it solves the issues about very specific crazy models because they dont really contribute anything. It is the much bigger collection of complex models in the ensemble that matter. Anything that is not close to the universal behavior has measure zero contribution relative to models that cluster around the universal point.

      Now the Lloyd-Dreyer path integral is a quantisation and the path integral itself could be a computable structure, so it should have been in the original ensemble and in some sense gets quantised again under the path integral. I conjecture that this structure actually dominates the path integral in which case it is a structure which when quantised leads to itself. This is also what you get when you use multiple quantisation ad infinitum, you get the picture.

      Of course that is ridiculously speculative but it is not inconceivable that some such argument could be made concrete. This would require defining what it means for two models to be close or equivalent which is one reason why category theory might be relevant.

      I agree with you that there's no well-defined 'actual spatial position of the rainbow'; however, 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...

      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, and these observed positions are all we can meaningfully talk about. I consider, in a way, the set of all possible observed positions of the rainbow: these are my relative facts (or rather, this is one relative fact: the position of the rainbow relative to the observer). Any given observer then picks out one particular 'actual' fact.

      About the Kochen-Specker theorem, I think you don't quite appreciate its force: it is simply an investigation regarding what follows from statements such as 'physical existence only occurs in one definite physically existent state at a time', and finding these consequences---the independence of certain observations from the context of these observations---to be inconsistent with observation.

      The connection you mention between computational and critical universality is something I've thought about a bit, but never was able to carry it beyond some vague feeling of 'there might be some relation'. I guess you're right to connect it to chaotic phenomena: there's the notion of 'edge of chaos', where systems show nontrivial emergent behaviour. Hmm, maybe one could connect the two via Wolfram's class 4 cellular automata? The conjecture then would be that every class 4 system---every system on the 'edge of chaos'---is a system showing both critical and computational universality, thus linking the two.

      As for your other speculations, I think that maybe I'm thinking in a broadly similar direction: my own idea is that what's invariant across all such phenomena should be the way information travels from one system to another, or the laws that govern this information exchange. Because when you think about it, no matter what the details of the microphysical implementation are, you'll always have to somehow inform the behaviour of a system about the behaviour of other systems. And (a natural conjecture for somebody with a background in quantum information, perhaps) this information exchange is described by quantum mechanics (which is where, if that's right, one could take advantage of Weizsäcker's or more recent ideas linking quantum information to 3-space, and things like that).

      So, matching your ridiculously speculative ideas with my own, there might emerge a broadly similar picture: the information exchange between two systems, if governed by quantum mechanics, is intrinsically probabilistic, and in some 'large' limit, maybe the crazy possibilities have a small enough meassure to be ignored...

      But I hear the ice creaking beneath me, so I'll better return to more serious work for the day.