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Questioning the Foundations Essay Contest (2012)
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Toward an Informational Mechanics by Karl Coryat
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Author Karl H Coryat wrote on Jun. 19, 2012 @ 16:34 GMT
Essay AbstractThe assumption that classical information derives from underlying objects or things, with absolute properties, may be impeding progress in physics. Taking Wheeler's "it from bit" seriously requires placing informational systems in relational context to each other. Doing so raises questions about the nature of the early universe, and suggests a possible "simplest-case scenario" for ultimate reality. An informational mechanics could describe the world as a compact, evolutionary system of information, where objects, spacetime, and physical laws emerge in relation to topologically connected complex subsystems ("observers"). Under this program, interpretational concerns may be neglected, while recovering the full predictive power of 20th-century quantum mechanics.
Author BioA desire to hobnob with rock stars led me to a career in music journalism out of college, but I've always stayed close to my science roots. I've been making music under the name "Eddie Current" since taking freshman physics at University of California Berkeley (BA Biological Science), and while I lost my top form long ago, I can still recite π to 40 or 50 decimal places. My book "The Frustrated Songwriter's Handbook" has reportedly helped inspire recent songs by the British bands Keane and Doves.
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J. C. N. Smith wrote on Jun. 28, 2012 @ 23:09 GMT
Karl,
Thank you for an interesting, thought-provoking, well-written essay. I plan to read it again at least once, probably several times. Perhaps you can address one question from my first read? I thought I was following along with you pretty well until I came to the following statement on page 7: "Perhaps for an alien race with its own origin and history, the universe would reveal an entirely different set of constants and laws." I guess my first question relates to the sort of hypothetical interactions we might have with this alien race should our paths cross. How would our discussions of physics unfold? Would we be able to reach agreement? Whether yes or no, why or why not?
My own essay here (Rethinking a Key Assumption About the Nature of TIme) takes the more conventional views that there is a real universe and that the universe has one, and only one, real history. I'll be thinking about whether and/or how our views might find common ground.
Thanks again for a thought-provoking essay, and good luck in the competition!
jcns
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Author Karl H Coryat replied on Jun. 29, 2012 @ 03:14 GMT
Hi JCN, Thank you for the comments. May I say I enjoyed your essay as well. I've gone back and forth on this question. For a human and an alien to meet (under the scenario proposed in the essay), we're talking about two complex subsystems interacting which have no contextual history in common. I think it might be a bit like a man trying to mate with an oak tree, if you'll excuse the image. I don't think human and alien races would ever casually bump into each other, which is a possible resolution of the Fermi paradox, among many (here is
an old FQXi post on that topic).
However, just as we might be able to splice a human gene into the oak genome, it may be possible for information to be deliberately transmitted from one universe to another. Perhaps this information-in-common can accumulate into a kind of translation subsystem of increasing complexity, eventually leading to true communication and mutual learning as you described.
However, the problem remains -- how can you send the first bits of information to a universe you literally know nothing about? It may be that two disconnected observer groups simply can never learn facts about each other, the same way different worlds in MWI remain discrete branches of reality. (Of course this is all wild speculation.)
Paul Reed replied on Jun. 29, 2012 @ 06:04 GMT
Karl (JCN)
Should such occur then they are no longer 'alien' (just not-human) in the sense that they are of the same existence. Assuming communication could be established then we (or they) may become aware of another sensory system that did not occur when such evolved here (or there). And hence another aspect to reality which was previously unknown. It will only be intellectual awareness, not sensoritly experienceable. Though they (or we) might be able to give us some technolongy which converts the 'information' they can receive via this sensory system, to a form we can receive. All of which demonstrates what is actually going on.
Paul
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J. C. N. Smith wrote on Jun. 29, 2012 @ 14:50 GMT
Karl, Paul,
"For a human and an alien to meet (under the scenario proposed in the essay), we're talking about two complex subsystems interacting which have no contextual history in common."
I'm having a difficult time wrapping my head fully around this concept/scenario. If there is no underlying "it," then we (as well as any hypothetical alien race) are both totally free to create, from whole cloth as it were, a concept of objective reality. Correct so far? But if there is no underlying "it," then any hypothesized alien race would be part of *our* creation from whole cloth. So then if this were the case, couldn't we take this creation in any direction we choose, so long as our reality remains internally consistent?
Even if there *were* an underlying "it," then we and any previously unencountered alien race still would be totally free to develop our own separate descriptions and explanations for the nature of that "it," perhaps dramatically different due to having developed different sensory capabilities (perhaps they see only gamma radiation or something).
So I'm wondering, should we ever have an encounter with an alien race, would there be any way this encounter could help us answer the question of whether there is or is not an underlying "it"? Why or why not?
My concern here, I guess, is that we're treading very close to solipsism, aren't we? If not, why not?
jcns
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Anonymous replied on Jun. 29, 2012 @ 19:53 GMT
JCN and Paul, thank you for the challenging discussion. For head-wrapping purposes, the scenario in the essay can be thought of as a kind of "evolutionary collective solipsism." I hope I did not just step in it with that remark, as solipsism is something of a dirty word. However, the scenario differs from traditional solipsism in at least three respects: (1) Solipsism denies multiple minds and multiple observers, as there is no way to account for a single objective reality seen by many individuals. I try to close this gap on page 6 of my essay. (2) Solipsism denies an objective history of the biological world. A solipsist might say that dinosaurs did not live on Earth, the evidence for them being only when "I" dig up a fossil, which is just silly. In the essay's scenario, organisms and their minds evolve, in parallel with the physical world that they observe. (3) Solipsism/idealism seem to imply that we can create whatever reality we desire. This is falsified by observation (I realize it whenever I try to play the piano). Objective reality and one's imagination are clearly different.
As for discovering a true alien lineage with its own independent history, nothing would prevent that from happening -- provided their history is consistent with ours. But I think this would be terribly unlikely. It would be a bit like two authors showing up at the same publisher at the same time with identical manuscripts. If the aliens observe four spatial dimensions and three varieties of charge, and they have flotons instead of photons, and they have a map of smalaxies that's different from our map of galaxies, I don't believe any kind of interaction could occur. The gears just wouldn't mesh.
So, I believe that if we found a distant planet with signatures of complex metabolic activity and energetic local entropy-reduction, or any kind of complex informational signal (like in "Contact"), this would falsify the essay's scenario and point to a world underlain by "it."
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Author Karl H Coryat replied on Jun. 29, 2012 @ 19:55 GMT
Sorry, I got logged out. -Karl
J. C. N. Smith replied on Jun. 29, 2012 @ 21:59 GMT
Karl,
Thanks for the good reply. You've clearly done some thinking about this. You've encouraged me, by your essay and by your posts thereon, to do some additional thinking about it, too. It certainly is a fascinating topic. Moreover, your essay, unlike a few others I've read definitely is responsive to the theme of the competition.
Your observation about falsifiability is important, too, of course, being what keeps us within the realm of science.
No problem about the anonymous posting; it's happened to all of us at one time or another.
Keep up the good work!
jcns
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Paul Reed replied on Jun. 30, 2012 @ 06:28 GMT
JCN
For an alien and human to ‘meet’ means that one, or the other, or both, are able to sense the other through at least one of their sensory systems. There may be a million aliens in your room right now, but you have no ability to sense them. On the other hand, they may be able to ‘hear’ you, or whatever. The point is, so long as one validated (ie not some lone individual with a vision!) sensing is operational, then these two are of the same existence. They are only alien in so far as they are not human.
There must be an underlying ‘it’, because sensory systems have developed to take advantage of it. But we can only know of ‘it’ through these systems, ie we are trapped in a sensory closed loop. In this sense, one could say it has to be regarded as an abstraction. But that is all we have. And if you stand on a railway line you will most certainly appreciate the effects created by its existence, and have enough sense to move before interacting with it! The alien might have different sensing systems, and get squashed.
Paul
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J. C. N. Smith replied on Jun. 30, 2012 @ 21:08 GMT
Paul, Karl,
"There must be an underlying "it", because sensory systems have developed to take advantage of it. But we can only know of "it" through these systems, ie we are trapped in a sensory closed loop."
At first glance, this appears to be a rather convincing point in favor of an underlying "it." Karl, I'd be interested in your take on this.
It's clear that not all sentient beings (even among our garden variety fellow earth-bound creatures) share exactly the same sensory systems. As a rule of thumb, it would appear that the more sensory systems to which we have access, the better able we are to understand and cope with our environment. One of our "known unknowns" is the number and nature of types of sensory data which may exist but which we are not able to access (or even imagine). Can people who are born blind imagine what it is like to be sighted? I don't know, but it's possible that we could be in an analogous position with regard to some types of sensory data thus far unknown to us.
jcns
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Author Karl H Coryat replied on Jun. 30, 2012 @ 22:03 GMT
JCN and Paul, I think that's missing the main argument of the essay. The idea is that (informational) sensory systems evolve in parallel with the environment being sensed, all of the above becoming increasingly complex in the process. If our earliest ancestor(s) had sensed magnetic monopoles and six spatial dimensions, perhaps today we'd be saying that "magnons" must be fundamental features of our highly complex six-dimensional universe. After all, we would only know these features through our biological (and technological) magnon-sensors and intuitively knowing what it's like to move through six dimensions of space.
There's nothing that would prevent us from discovering novel properties of the world that manifest as new kinds sensory data, provided they are not inconsistent with our other discoveries. We might for instance discover, as we probe at higher energies, that quarks decompose into "even more fundamental" entities, and we might find compactified extra dimensions as proposed by string theory. However it's unlikely we'd uncover a third variety of charge or a fourth large spatial dimension -- again, the gears wouldn't mesh. As an analogy (you can see I like analogies), we can always discover more decimal places for pi as we calculate with greater and greater precision, these merely providing better approximations of pi. But we are unlikely to find a second version of pi that's 3.24 or 4.0. Not in this universe, anyway!
Paul Reed replied on Jul. 1, 2012 @ 07:21 GMT
JCN/Karl
(JCN) but all sentient organisms do share the same sensory processes, in that, generically, the functioning of any given type of sensory system is identical across all organisms. How that manifests between species type, and indeed between individuals of the same species, is another matter. Indeed, one can go further, in that all sensory systems are logically identical (independent physical effect, detected, processed). This is just a function of evolution.
Which brings us to the next point. That which is received by sensory systems is physically existent. It is only ‘information’ in the sense that those systems, should they be in the line of travel and therefore receive (ie interact with) that physically existent effect, can process it. Put another way, a brick, or an ear, cannot process light, an eye can. But this acquired functional role has no effect on their physical existence. They are not altered by it, and neither are they developing in reaction to it. Sensory systems evolved, and continue to do so, to make use of pre-existent physical phenomena. But they can only sense that which is there, independently of them. Or to be precise, we can infer that from what is received. On top of that, in order to progress, hypothecation is allowable, to overcome certain practical problems in the sensory process, or we can use technology to enhance it, but not to second guess an alternative existence.
Paul
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Author Karl H Coryat wrote on Jul. 1, 2012 @ 19:33 GMT
I'd like to clarify something I wrote in the previous thread. I mentioned that our earliest living ancestors sensed photons rather than, say, magnetic monopoles. It is more precise to say that they sensed something *which we humans would describe as photons*. The most you can say is that the behavior is consistent with what we'd call photon behavior. In relation to the rich contextual history which we complex self-aware substructures have accumulated, photons are seen to obey strict boundary conditions; but in relation to the crude contextual history of our earliest ancestors, the boundary conditions are looser. I think this is what Paul Davies means when he says (on page 7 of my essay) "the laws start out unfocused and fuzzy" but that observations "help sharpen those laws."
Paul Reed replied on Jul. 2, 2012 @ 07:07 GMT
Karl
What any given ancestors sensed, is a function of the capabilities of their sensory system at that point in time. What they received, ie what collided with them, was exactly the same in terms of generic physical constitution, as it is today.
Paul
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Author Karl H Coryat replied on Jul. 2, 2012 @ 20:02 GMT
Paul -- For photons to have an invariant "generic physical constitution," one must assume that they are objects with absolute properties. My essay questions that assumption.
Paul Reed replied on Jul. 3, 2012 @ 09:39 GMT
Karl
OK, so if that is the case, then photons do not constitute what we mean by the label elementary particle (see my post over in JCN's blog). The definition of that being a 'substance' (and there may be various types) which physically, in generic terms, persists in the same physically existent state over time. In other words, some innate properties which comprise the 'substance' are always existent. Their value, or what they manifest as, at any given point in time may well vary from circumstance to circumstance.
But another point here is that, in simple terms (and I really wish-like everybody else-that we knew what was actually happening), any given light can be characterised as a physical effect in a physical photon (or definitive number thereof). It is the physical result of a physical reaction with a physically existent state (ie the reality which can now be ultimately seen if that light, and others that follow, come into contact with an eye when travelling). Precisely how it works....but that is sufficient to reveal a key point, ie light, more or less, is able to retain its existent state over time (ie as it travels). Meanwhile, the reality which it 'represents' (ie from the perspective of the sensory system) has ceased to exist.
Paul
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Thomas Howard Ray wrote on Jul. 6, 2012 @ 15:53 GMT
Karl,
How about "evolutionary collective tautology?" Then you avoid the implications of a solipistic creature contemplating only itself. Wheeler's famous drawing of the "eye" includes an interval, suggesting the underlying objectivity of a world made of information.
Great reading! I hope you get around to reading my own essay which is also based on Wheeler's it from bit philosophy. Good luck in the contest.
Tom
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Paul Reed replied on Jul. 7, 2012 @ 07:23 GMT
Tom
Yep, what we can know is what we can know.
Paul
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James Putnam wrote on Jul. 8, 2012 @ 15:35 GMT
Karl H Coryat,
Could you please explain what is information in the context that you are applying the word? Does it have a physical explanation or representation in your context or is it pre-existing understanding without origin? By understanding, I mean does the information have meaning? If so, is it information that understands information? If it does have an origin and that origin is not physical, then is it its own cause? If it is its own cause, did it begin from something before the first bit. Was there a first bit that gave existence to other bits? Is there a state where information does not have meaning? If not, then what is the origin of meaning? Do you assume that information arrives with meaning intact. When it does arrive what receives it? Is it received by information? Would you use the words arrive and receive? If so, what do they mean in your context? If not, then what is the information doing?
James
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Author Karl H Coryat replied on Jul. 8, 2012 @ 20:40 GMT
Hi James, thank you for taking the time to read my essay. I'm afraid I can't answer all of your questions; no one can. It seems as if we are only now figuring out what information is. I refer you to "Information and the Nature of Reality," co-edited by Paul Davies, for the state of the art as to our understanding of what information is, from various perspectives. The book was largely the inspiration for my essay.
Does information have meaning? Only in relation to other information. Information accumulates, to use a crude analogy, somewhat like compound interest in the bank: as money accumulates in the presence of other money, so does information in the context of other information. (We don't really have the equivalent of a bank or a customer here.) Information is clearly physical in some sense -- but if by "physical" you seek an underlying object or objects from which this information derives, then my essay would suggest instead adjusting what is meant by the word "physical." It refers merely to the automorphism invariance of systems, not necessarily to objects obeying absolute laws.
"Interact" is perhaps a more general word than arrive or receive, which carry unnecessary connotations. Information has no meaning as a closed system, and therefore information does not appear with meaning intact. The very reason why we can even have a concept such as "meaning" is that we are sitting atop a mountain of information and have this rich contextual history to draw from. The spirit of my essay is that it may be difficult or impossible to understand ultimate reality if one assumes that objects or physical laws (or meaning for that matter) necessarily appear in the world intact and absolute, irrespective of anything else.
James Putnam replied on Jul. 8, 2012 @ 20:56 GMT
Thank you Karl,
I enjoyed reading your essay. It is just that in the transition to information, it appears that the theory borrows the attributes of a physical world while denying it. By the way, I do think that information is primary. It is what we use. Everything else we think exists or does results from our interpretation of information. However, so far as I understand it, empirical evidence consists of information about changes of distance and time. So, I accept distance and time as the two fundamental indefinable properties. Anyway, that is just what I think. I do have a question?
"The very reason why we can even have a concept such as "meaning" is that we are sitting atop a mountain of information and have this rich contextual history to draw from. The spirit of my essay is that it may be difficult or impossible to understand ultimate reality if one assumes that objects or physical laws (or meaning for that matter) necessarily appear in the world intact and absolute, irrespective of anything else."
What are we that we can assign meaning to information? Thats pretty much what I was getting at, by questioning where or when meaning comes into existence or use. Where does the intelligence reside?
James
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Author Karl H Coryat replied on Jul. 8, 2012 @ 21:57 GMT
I would say that meaning is evolutionary. A mouse or a flatworm do not contemplate meaning per se, but they are spectacular at processing information, and this is always done in the context of other information (which is laid out in detail in the Davies book I mentioned). For humans, a species that has fairly recently acquired language and high-level reasoning faculties, it may be tempting to think that what we call "meaning" has always existed in more or less the same form. I'd suggest that the ability to contemplate meaning -- and ultimately meaning itself! -- emerged as a very slow process. It's similar to something I call the comb-over effect: When we see someone with a bad comb-over, we wonder how he could go out thinking this makes him look like he has hair. But we forget that he's been parting his hair like that ever since it began to thin. It has been a long and gradual process but eventually leads to the person looking ridiculous.
The idea of "borrowing the attributes of a physical world while denying it" sounds a bit like the historical opponents to Copernican theory. That idea was considered absurd by many, because the Earth was, by its very definition, that which did not move. So it seemed as if Copernicus was invoking something that was both the Earth (clearly a non-moving object), and yet moves! Crazy talk, right? Only when you throw out the assumption that the Earth is stationary does the picture make any sense.
James Putnam replied on Jul. 8, 2012 @ 22:37 GMT
Thank you Karl,
I feel that I have gained some understanding of your reasons for the ideas you present in your essay. Like I said, I enjoyed your essay. Thank you for your responses to my questions.
James
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Paul Reed replied on Jul. 9, 2012 @ 08:54 GMT
Karl
Information is not physically existent. Some physically existent phenomena are only potentially 'information' in the context of sensory systems which have evolved to utilise them. So, if and when, any such given physical phenomenon interacts with (ie their lines of travel coalesce) the receiving organ of a sensory system which can process it, then it becomes information in respect of that functional role in that sensory system, and indeed ceases to exist. Its physically existent state is unaffected by the subsequent activity.
In other words, if light hits a brick wall, it is not utilised because brick walls are not the front end of a sensory system. However, if that light had hit an eye, then…..The physical existence is the same in both cases. What the mouse or the flatworm can then do is irrelevant. That is just a discussion about physiology, biology, sociology, etc. This all wraps in with my point about ‘aliens’ above.
Paul
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Georgina Parry wrote on Jul. 15, 2012 @ 00:15 GMT
Hi Karl,
I really enjoyed your essay. Its very clearly written and very interesting and thought provoking.
It seems to me there has to be a source of the information rather than it just existing. It makes sense, for me, for there to be a source, information/potential sensory data, and output. Having the objects as well does not prevent string like histories, or EM information in the environment being on spherical surfaces. Just data and output doesn't work for me. Which may be my inability to get my head around the alternative. Does the orphan information just exist in the way that we might think of an object just existing? If a blind cave shrimp feels its environment it is obtaining information from the objects,it seems to me. Were is the information in the environment if there is no object to feel?
Really liked your explanation of chipping away at many histories to make the reality that is known. It is an interesting idea though again one that I cannot say I think is correct. I think there is a lot more information in the environment than is selected and a fabricated reality is formed from that. So from the Multiverse of possibilities originating at the source events that occurred, not many (different)histories.
Also really liked what you were saying about Aliens having a very different perception of the universe because of their different sensory capabilities.They might also process information differently, being adapted to their environment and way of life.
Well done. Good luck in the competition.
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Author Karl H Coryat replied on Jul. 15, 2012 @ 03:26 GMT
Hi Georgina, thank you for the kind words. I realize that the ideas in this essay are a bit out there, and I understand why people may not be getting their heads around it on first look.
The orphan information exists in the same way we think of objects as existing -- the primary difference is its history. A fundamental object is generally assumed to have existed in that absolute state since...
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Hi Georgina, thank you for the kind words. I realize that the ideas in this essay are a bit out there, and I understand why people may not be getting their heads around it on first look.
The orphan information exists in the same way we think of objects as existing -- the primary difference is its history. A fundamental object is generally assumed to have existed in that absolute state since shortly after the Big Bang. In the informational scenario, data is constantly being generated in relation to existing data, and it accumulates in this way as an ongoing evolutionary process. There is one and only one fundamental entity, the bit; the particles of the Standard Model emerge to us complex systems and our technology. Wheeler described this emergence process: we hear our detector make a click, and we say, "Aha! A particle did that." But perhaps it was just the appearance of a new bit in the world, and in the very rich context of the situation and everything we've learned about the laws of physics, we interpret the bit as a fundamental object speeding through space and hitting the detector.
Think of the blind cave shrimp as a bundle of bits, like us, only simpler. However, we don't see bits when we interact with this bundle. A cave shrimp, an object, is what appears to us. It is like a projection that we are biologically equipped to create in our mind (which itself consists of numerous interacting bit-bundles), and these projections are what make up much of the human experience. This is a phenomenon that has evolved over billions of years. As for the cave shrimp itself, it's hard to imagine what experience is like for such an animal, but in some sense, objects emerge in its mind as well. Though more crude, the shrimp is similarly equipped to create these internal projections so as to enhance its fitness.
The hardest part of this scenario, as you've noticed, is trying to envision the environment as information rather than objects. Space-with-objects seems "normal" to us, and standalone information seems very abstract. But, I claim we should be skeptical of a reality consisting of objects, because it might be unnecessarily extravagant. While it's true the universe may contain ~10^90 objects carrying their accompanying bits of information, perhaps we could have the same identical experience in a universe containing only ~10^60 bits and zero fundamental objects. If that's the case, I claim that Occam's razor points toward the latter. The universe might be much simpler and more elegant than we intuitively assume, and having just one type of fundamental entity, in a compact, self-driven system of mutually relational subsystems, may be all that's needed to account for what we see. To me, an unknowably vast landscape of spacetime with various bosons and fermions obeying seemingly disjointed physical laws of mysterious provenance -- let alone multiple universes out there! -- seems ugly in comparison, like the hilarious building in Vladimir Tamari's essay.
I've been struggling with various approaches to explaining these ideas. Thank you for the opportunity, and for your time.
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Georgina Parry replied on Jul. 15, 2012 @ 09:36 GMT
Hi Karl,
thank you for explaining.
I can't help thinking that the process of perception is being taken for the foundational physics in the "it from bit" view. Yes images or ideas of objects, that we name, are formed from received data, after learning has occurred.
The learning component is important. I watched an interesting video, I think it was TED one, about people who were born blind learning to see.For most people this learning would occur in very early infancy. Interestingly it was the relative motion that allowed them to identify separate objects rather than just object outlines. The vital information for differentiation was not concerning the objects themselves but the relationships.
Without motion overlapping portions of shapes could be misinterpreted as separate objects, and separate objects could be considered the same object.So it seems to me, the "it" comes from knowing it is a separate "it". ie.requires the pattern of neuron connections in the brain of the observer and the received "bits". You seem, to me, to be implying the same in your reply.
That processing is all on the internal side of a reality interface by my way of thinking.It is producing the observer's Image reality. A fabrication of what exists externally. On the other side of the interface is the existence and change of material objects and sensory data; and ongoing production of data; the observer independent Object reality. To deny the source of the information seems to be doing away with a large part of reality.
I do appreciate the challenge to think about things differently and the time you have taken explaining your viewpoint to me.
Good luck.
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Dirk Pons wrote on Aug. 8, 2012 @ 10:01 GMT
Karl
As I understand you, 'If the world is informational...' then we don't have to focus so much on the physical theories, and may instead hope to find an information-first theory based on observation context.
It is an intriguing thought. How does one approach a problem like that...are the existing informational methods up to it yet?
Thank you
Dirk
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Author Karl H Coryat replied on Aug. 8, 2012 @ 19:09 GMT
Dirk: That's exactly right. The physical theories and their mathematical representations are excellent approximations, but their inadequacy to fully describe nature was revealed by quantum mechanics and its multitude of interpretations. By assuming that mysteriously created objects obey mysteriously created laws, QM and GR appear to be incompatible, for example. We need to develop a theory based on what actually is fundamental, which I argue is information and not objects/laws.
Alas, our current informational theories are only skeletal at this point, but that's what intrigued me about Bob Coecke's graphical system. By graphing the flow of information, we can at least get an idea of how informational context can gradually evolve from simpler to more complex. KC
Member George F. R. Ellis wrote on Aug. 12, 2012 @ 20:32 GMT
Karl
I really like the focus of this essay on informational mechanics as a ``generalization of quantum mechanics that embeds contextual data into descriptions of subsystem interactions''. This is completely in line with the idea of top-down causation associated with contextual effects. You suggest that "all a theory really needs to address is the beautiful world of automorphism-invariant information." That is rather similar to the emphasis
Auletta, Jaeger, and I have put on how equivalence classes as characterising top-down action. I also applaud your sensible take on quantum measurement.
George Ellis
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Author Karl H Coryat replied on Aug. 14, 2012 @ 00:37 GMT
George, thank you for your comment. I'm seeing several threads common among many of the essays this time, for example arguments against reductionism and against particles-as-ontological-objects -- ideas that are clearly related.
I checked out your paper with Auletta and Jaeger. I especially appreciated the sentence, "Mechanical devices, like a thermostat, are able to implement information control without any intervention of biological elements and in purely mechanical terms. This is however an erroneous point of view, since such devices have been built by humans to act in a certain way. Therefore, the functional element (and the goal) is already inbuilt." This rather obvious aspect of technology tends to be overlooked when discussing quantum measurement and measuring devices. Similarly, even the simplest living organism seems not to be just a bottom-up collection of matter particles doing complex things, but rather is an informational entity whose complex interactions derive out of a legacy of evolving context. No wonder it's so hard to create a living system by putting together a bunch of inorganic molecules.
Daniel wrote on Aug. 24, 2012 @ 02:58 GMT
Dear Karl
You might find very interesting that there is an
informational derivation of quantum theory . The assumption of a relational and informational reality seems very interesting and it would be nice to study all its consequences.
I have also thought of ways of conceiving the world without the primitive notion of ''an object''. It is a natural extension of Machian thoughts on the foundations of dynamics, and I have developed this in my essay
Absolute or Relative Motion...Or Something Else?, which you might find interesting.
Good luck in the competition,
Daniel
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Author Karl H Coryat replied on Aug. 24, 2012 @ 23:46 GMT
Daniel -- I am interested in reading both the paper and your essay; thank you for the links. As my essay argues, I suspect that the world is only as complex as the system or systems (within) that measure its complexity; thus, complexity in the world becomes a relational function of biological and technological evolution. Superficially that may sound like a facile "philosophical" statement, one that escapes falsifiability. However, if the world is in fact fundamentally informational and relational, then there would be no other accurate way to describe complexity except in those purely relational, informational terms. And it should be possible to demonstrate this in experiments of sufficient sophistication.
S Halayka wrote on Sep. 12, 2012 @ 16:25 GMT
Hi Karl,
I really liked your essay a lot. Your writing covers a lot of ideas and possibilities in a way that is interesting to read.
In particular, I like how your essay draws the distinction between data and information. That string of randomly-generated bits is taken to exist as a single datum, and without the existence of other such datum there is no chance for information to be...
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Hi Karl,
I really liked your essay a lot. Your writing covers a lot of ideas and possibilities in a way that is interesting to read.
In particular, I like how your essay draws the distinction between data and information. That string of randomly-generated bits is taken to exist as a single datum, and without the existence of other such datum there is no chance for information to be had -- information is an entirely relational concept that involves multiple data. This is no ephemeral concept devoid of realism -- the vast majority of us rely on fully-implemented relational databases every day when we interact with banks, websites, etc.
I do wonder though if the distinction between data and information is still tripping other people up. For instance, I see phrases like "information compression" in some of the literature. Of course, in actuality, it is the data that is compressed/decompressed, and the information content of the data is what governs the compression limit -- information-poor data is very compressible, information-rich data not so much. Again, this is no ephemeral concept -- it is the core principle used in relational databases and really any lossless dictionary-based compression algorithm (ZIP, RAR).
From this perspective, if a blue photon always imparts X momentum upon absorption, and we repeat that observation over and over, we will never gain information. In fact, the law that we derive (constancy of momentum per photon frequency) springs directly from exactly how information-poor the data is. I take this to mean that even if such an observation leads to one type of information-rich chain of events (ie. saying "It's blue!" via sound waves and then giving each other a high five) versus another (ie. aliens making some equivalent statement via pheromones, etc), the root event is still the same no matter what. To me, it is the root data that really counts here, and that leaves little or no room for mysticism. I think this is what you are trying to say too?
- Shawn
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S Halayka replied on Sep. 12, 2012 @ 17:04 GMT
To be fair, not all is lawful, and perhaps this is precisely why quantum mechanics is information-based. I am considering the probability cloud that represents an electron's possible position. If one repeatedly performs an experiment in which they make a measurement to gain a datum about an electron's actual position, then one will come to find that over time the data are non-repetitious -- the angular components of the position are fully random, and the radial component partially random (still random, but the probability drops off based on radial distance). I suppose that one could call this the "law of lawlessness" (since it is a constant kind of random), but no one seems to explicitly say it and stick to it, which is why I think that we get varied opinions on what information really is (and isn't) -- none of this gives credence to the phrase "information compression", and I doubt that anything would.
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Author Karl H Coryat replied on Sep. 12, 2012 @ 23:36 GMT
Shawn -- My reply is below.
Author Karl H Coryat wrote on Sep. 12, 2012 @ 22:39 GMT
Shawn -- Thank you for reading my essay, and for the comment. I hadn't explicitly considered the distinction between data and information, but I think you've hit the nail on the head. Data is an absolute thing, but information is necessarily relational. This was the message I took away from the book "Information and the Nature of Reality," co-edited by Paul Davies. One chapter stuck out for me as problematic -- Seth Lloyd's -- perhaps because he did not make this distinction. He writes, "Quantum mechanics, via decoherence, is always injecting new bits into the world." Those would be bits of data, but are they information? If so, relative to what? The same can be asked regarding a string of random numbers, or highly compressible (low-information) data as you insightfully mentioned. I feel these questions are not only relevant, but perhaps even fundamental.
S Halayka replied on Sep. 14, 2012 @ 20:11 GMT
Hi Karl,
I believe that you're correct in that information is a kind of emergent property that arises only in the context of multiple data. From what I can gather, it's primarily the distinctness of the data that gives for high information content per datum. I'm not sure how deep you have gotten into the math of information theory, so please forgive me for going into further detail: there...
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Hi Karl,
I believe that you're correct in that information is a kind of emergent property that arises only in the context of multiple data. From what I can gather, it's primarily the distinctness of the data that gives for high information content per datum. I'm not sure how deep you have gotten into the math of information theory, so please forgive me for going into further detail: there will be a point.
Consider receiving a string of symbols (data) consisting of "0123". This string contains 4 distinct symbols that all occur with the same frequency (1/4th of the time), which ends up producing a Shannon entropy (average information content per symbol) of S = ln(4). Converting that to binary entropy, it's S_b = S / ln(2) = 2. The same entropy is produced by the semi-randomized string "0312", which still contains 4 distinct symbols that occur with the same frequency -- the order of the symbols doesn't affect the result, only the distinctness. What the binary entropy effectively means is that if we wish to represent 4 distinct symbols that occur with the same frequency, then we're going to need at least ln(4)/ln(2) = 2 bits per symbol. This notion becomes very clear if we're already familiar with integer data types on the computer, where we know that a 2-bit integer type can represent 2^2 = 4 distinct symbols (0 through 4). Likewise, a 3-bit integer can represent 2^3 = 8 symbols (0 through 7), etc, etc.
A symbol by itself does not have information content -- the extremely short string "0" produces an entropy of S = ln(1) = 0. There is no distinctness to be had because there is nothing else in the string to compare our single symbol to.
To consider randomization further, consider the string "10100101010101101010101010101101011000101001110100010011010
11010", which is 64 symbols long. It's apparent that the string cannot be considered to be totally random just at the root level -- there are only two root symbols ("0" and "1"); any repetition of any symbol is a pattern; patterns are anathema to randomness. Secondly, we notice that there are patterns in the adjacency of the symbols as well; the composite symbols "00", "01", "10", "11" appear quite often. Since we clearly cannot avoid patterns, we must then look for balance (equal frequency) of the patterns at all possible levels in order to test for randomness.
As for actual figures regarding the frequencies of our string, we get...
For 0th level adjacency (singles), we go through the string one step at a time from the left to the right. This gives us the singles "1", "0", "1", "0", "0", etc:
The root symbol "0" occurred 32/64th of the time, as did the root symbol "1".
Balance in frequency occurred.
As for 1st level adjaceny (pairs), we go through the string one step at a time from the left to the right. This gives us the pairs "10", "01", "10", "00", "01", etc:
The composite symbol "00" occurred 7/63th of the time;
The composite symbol "01" occurred 24/63th of the time.
The composite symbol "10" occurred 25/63th of the time.
The composite symbol "11" occurred 7/63th of the time.
Imbalance in frequency occurred.
As for 2nd level adjacency (triplets), we likewise go through the string one step at a time from the left to the right. This gives us the triplets "101", "010", "100",
"001", "010", etc:
The composite symbol "000" occurred 2/62th of the time.
The composite symbol "001" occurred 5/62th of the time.
The composite symbol "010" occurred 18/62th of the time.
The composite symbol "011" occurred 6/62th of the time.
The composite symbol "100" occurred 5/62th of the time.
The composite symbol "101" occurred 19/62th of the time.
The composite symbol "110" occurred 6/62th of the time.
The composite symbol "111" occurred 1/62th of the time.
Imbalance in frequency occurred.
We will skip detailing the 3rd, 4th, etc level adjacency. The point is that at the n-th level of adjacency, there are 2^(n + 1) distinct symbols (root symbols for the 0th level, composite symbols for 1st level and higher), and we can consider the whole string to be random only if there is perfect balance in the frequency of the symbols at *all* levels of adjacency. Since there are imbalances in the frequency at the 1st and 2nd level, the string cannot be considered to be random.
Of course, if our string was indeed generated by a random process, then we would simply need to keep making the string longer and eventually a balance in the frequency at all levels of adjacency will naturally arise as time goes on. This is interesting to note because it means that we simply cannot know if a binary number generating process is truly random until we can analyze the symbols it puts out at an infinite level of adjacency, which requires that our string be infinite in length. It's literally impossible to tell with perfect certainty that a string of "0"s and "1"s is randomly-generated if the string is finite in length.
In essence, when someone says "quantum physics is random" and they show some data as evidence, it's still only an assumption because we do not have an infinite number of measurements to verify that statement. We can become more and more confident as we make more and more measurements (and find that the frequency in the patterns are balanced at higher and higher levels of adjacency), but we can never ever be absolutely certain until we take an infinite number of measurements. Consequently, if we were to make billions and billions of measurement and we found out that there was an unexpectedly large imbalance in frequency at say, the 23324324th level of adjancency, and this imbalance simply does not go away when we make billions and billions of more measurements, then we could say with a high degree of confidence that something deterministic is likely going on there, deep at the heart of things.
In other words, your essay indirectly puts a spotlight on one of our basic physical assumptions: “quantum physics is random” when you make mention of randomized strings. I think that's a pretty important observation, and some of your readers might pick up on it right away. This is another reason why I liked your essay a lot.
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S Halayka replied on Sep. 14, 2012 @ 20:17 GMT
(I'm not necessarily saying that quantum physics is deterministic at some deep level, just that we cannot know for sure either way until we make a ridiculous number of measurements)
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Author Karl H Coryat replied on Sep. 16, 2012 @ 05:53 GMT
Thank you Shawn, that's very intriguing. As I look at some of the other essays I realize I may have bit off too much, and perhaps I should have gone somewhere like the idea you suggested. I will probably have about 18 months to think about it before the next essay contest, so thanks for the ideas!
Karl
Shawn Halayka replied on Sep. 20, 2012 @ 22:50 GMT
I dunno... I think that your essay is pretty awesome as it is, and I gave it a solid community score accordingly.
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Member Benjamin F. Dribus wrote on Sep. 13, 2012 @ 05:16 GMT
Dear Karl,
I read your essay with great interest. I share a lot of the same general philosophy toward fundamental physics. A few questions and remarks.
1. Have you thought much about causal sets in this context? I don’t completely agree with all the hypotheses of causal set theory, but I admire its viewpoint, and it seems very relevant to your viewpoint as well. You reference...
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Dear Karl,
I read your essay with great interest. I share a lot of the same general philosophy toward fundamental physics. A few questions and remarks.
1. Have you thought much about causal sets in this context? I don’t completely agree with all the hypotheses of causal set theory, but I admire its viewpoint, and it seems very relevant to your viewpoint as well. You reference Fotini Markopoulou, who has written papers on this, and you also illustrate quantum picturalism and information trees, which are very close to the causal sets formalism. If you have not already read it, you might find Sorkin and Rideout’s paper on sequential growth dynamics interesting.
2. You reference Hermann Weyl’s definition of objectivity in terms of group symmetry. Weyl, of course, is the father of group representation theory in physics. In particular, you mention that causal networks survive automorphism due to topological constraints. Since you are advocating a minimalist relationist viewpoint, I thought you might be interested in look at this from the other direction, in which one redefines covariance to mean preservation of causal structure. For instance, Lorentz transformations in ordinary SR change the order of spacelike-separated events (relativity of simultaneity), but preserve the causal order. If you define a “generalized frame of reference” to mean a refinement of the causal order, then you can redefine the covariance principle in these terms and completely do without the background geometry. I discuss this further in my essay here:
On the Foundational Assumptions of Modern Physics.
3. Have you thought much about quantum computing in this context? I ask this for a couple of reasons. First, it seems to me that a lot of the same formalism can be applied immediately to practical problems in quantum computing itself. Second, if the fundamental structure of spacetime is really informational, it might be possible that we could model fundamental-scale physics at much larger scales using quantum computers. I mention this briefly near the end of my essay.
Thanks for the great read! Take care,
Ben Dribus
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Author Karl H Coryat replied on Sep. 13, 2012 @ 09:04 GMT
Ben, Thank you for the thoughtful comment. It got me very interested in learning more about causal sets (I confess ignorance on the topic). I think there are a number of equivalently valid formalisms to look at this problem, some easier to grasp than others. I remember checking out your essay when it was first posted and taking a few notes; I'll be sure to look over it thoroughly and post comments over there. And, you make a great point about quantum computing. It's a good argument for why developing new kinds of information theories should be taken seriously. In fact, it's probably the only way to maximize the potential of quantum computing, and the potential, as you point out, is enormous. Thanks again and best of luck in the competition.
Conrad Dale Johnson wrote on Sep. 15, 2012 @ 13:58 GMT
Hi Karl,
I was very glad to find your essay and enjoyed it very much, since my own ( An Observable World ) also deals with the assumption that the informational environment we can observe must be based on some kind of underlying reality. I was especially happy with your thoughts on the history of the universe, since I'd wanted to include something similar in the last part of my essay. I...
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Hi Karl,
I was very glad to find your essay and enjoyed it very much, since my own (
An Observable World ) also deals with the assumption that the informational environment we can observe must be based on some kind of underlying reality. I was especially happy with your thoughts on the history of the universe, since I'd wanted to include something similar in the last part of my essay. I had to cut it, since the piece was already too long and too condensed.
The main difference between our approaches is that you're envisioning a physics based on a new kind of fundamental entity, the bit -- per your response to Georgina Parry above. You told her information exists in the same way we think of objects as existing. To me, the key point about information in physics is that it's observable, i.e. definable in the context of other observable information. Some kinds of physical information do resemble binary bits, but I don't think it's helpful to think of all such information as reducible to simple interchangeable bits. Instead, the question is how all the different kinds of physical information contribute to an environment that's able to define them all, in terms of each other.
If we open up the question of how information gets physically defined, we get a picture of the informational environment that's anything but elegant or unified . So I don't envision replacing our theories of fields and particles with an all-embracing theory of information. To the extent that it works to describe our world in terms of real, objective entities, I think we should certainly do so. My point is that we shouldn't be surprised that this kind of theory doesn't work at a fundamental level. It's only when we try to understand how our theories fit together, how they constitute a basis for the observable world, that we need to switch over to an informational viewpoint.
Anyhow, it will be some time before it's clear just how to make this conceptual shift, from the world as a structure of real things-in-themselves -- inferred from phenomena -- to the structure of communications that constitute the phenomena. I appreciate your brave incursion into this unexplored territory.
Thanks -- Conrad
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Author Karl H Coryat replied on Sep. 16, 2012 @ 05:44 GMT
Conrad, thank you for the kind worlds. You raise a good point about being too rigid about the definition of informational entities. I'd just add that if we reduce everything to simple interchangeable bits, we do so with the understanding that those bits are defined relative to something -- in our case, relative to the context in which they have been discovered and observed. Perhaps that has something to do with why information often manifests to us observers as binary bits. I'll certainly check out your essay to get a better idea of your view; thank you for pointing me to it.
Karl
Steve Dufourny Jedi wrote on Sep. 19, 2012 @ 13:36 GMT
Hello Mr Coryat,
I beleive strongly that the electromagnetism is so complex.I have several relevances considering the polarizations of informations. With a diffrent sense of rotation than m for the hv. The spherical volumes for the serie of uniqueness becomes very relevant. But in fact the complexity of this electomagnetic scale is very important considering the volumes of entangled spheres. In fact the volumes are the key !!! it permits to class the informations and the synchro and sortings.
See that the gravitation is for the most important volume, the electromaginetism is the number x very important, the gravitation the 1 more the volumes , it is very relevant.
The entanglement is like our universal sphere with the central sphere the most important volume. It permits to class better the forces, foundamental between the mass and the light in a pure evolution point of vue.The spherization is fascinating.
Regards
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Hoang cao Hai wrote on Sep. 19, 2012 @ 15:44 GMT
Dear
Very interesting to see your essay.
Perhaps all of us are convinced that: the choice of yourself is right!That of course is reasonable.
So may be we should work together to let's the consider clearly defined for the basis foundations theoretical as the most challenging with intellectual of all of us.
Why we do not try to start with a real challenge is very close and are the focus of interest of the human science: it is a matter of mass and grain Higg boson of the standard model.
Knowledge and belief reasoning of you will to express an opinion on this matter:
You have think that: the Mass is the expression of the impact force to material - so no impact force, we do not feel the Higg boson - similar to the case of no weight outside the Earth's atmosphere.
Does there need to be a particle with mass for everything have volume? If so, then why the mass of everything change when moving from the Earth to the Moon? Higg boson is lighter by the Moon's gravity is weaker than of Earth?
The LHC particle accelerator used to "Smashed" until "Ejected" Higg boson, but why only when the "Smashed" can see it,and when off then not see it ?
Can be "locked" Higg particles? so when "released" if we do not force to it by any the Force, how to know that it is "out" or not?
You are should be boldly to give a definition of weight that you think is right for us to enjoy, or oppose my opinion.
Because in the process of research, the value of "failure" or "success" is the similar with science. The purpose of a correct theory be must is without any a wrong point ?
Glad to see from you comments soon,because still have too many of the same problems.
Regards !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
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Amanda Gefter wrote on Sep. 22, 2012 @ 20:17 GMT
Hi Karl,
As I mentioned on my thread, I enjoyed your essay immensely - it is well written, well argued and extremely interesting.
I am in full agreement with you that information, rather than objects, is ontologically fundamental as Wheeler first suggested and as the philosophy of ontic structural realism has fleshed out (Dean Rickles has a superb essay in this competition on this matter). To my mind, theoretical discoveries such as the holographic principle, as you noted, and the various dualities in string theory make an object-based ontology pretty impossible to uphold. (In case you're interested, I wrote a short
essay about this last year on the Edge website.) I also totally agree that information is fundamentally relational, as Rovelli boldly expressed.
Your idea that observations made by one observer places constraints on another is really fascinating - I have to spend some more time thinking about it to fully wrap my head around it. I'm also intrigued by your claim that multiple individual observers connected by contextual information correlations are topologically equivalent to a single super-observer. As I mentioned in my essay and on my thread, the idea of a single super-observer (an observer who effectively stands outside the universe independent of a coordinate frame) seems to violate certain laws of physics, as demonstrated in Susskind's horizon complementarity. However, if you can model the superobserver as many individual local observers, that raises some interesting new questions.
Thanks again for a wonderful read.
Best regards,
Amanda
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Author Karl H Coryat replied on Sep. 26, 2012 @ 23:08 GMT
Amanda, thank you for submitting your fine essay. I sincerely hope you win a prize. And by the way, please have my children....
Member Hector Zenil wrote on Sep. 25, 2012 @ 05:32 GMT
Dear Karl,
My own ideas on an algorithmic world are somehow consonant with your information mechanics, at least in some aspects. I also think that the role of the observer is key in modern theories of physics (beyond the current state given to the observer in quantum mechanics). A la Wheeler you place information at a lower level underlying the rest of reality. I would have liked you to contrast your ideas with that of the so-called digital physics where this assumption is the founding stone (although some may think I am making a conflation between information and computation, which might be the case). I found interesting your attempt of formalisation of objectivity by means of invariance with respect to the group of automorphisms, I would have liked to see it further formally developed.
In biology evolution of life is seen as indirected, in your view you seem to suggest that there is an increase of complexity, I think it is risky to do so because the notion of complexity would need to be much more precise and I also think it is not easy to justify such an increase of complexity from the evolutionary point of view (it is evident that some sort of "complexification" had to happen to come from, for example, unicellular to multicellular organisms).
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Author Karl H Coryat replied on Sep. 26, 2012 @ 23:05 GMT
Hector, I probably tried to cover too much ground and do too much in my essay. Perhaps it would have been better to take a core concept of informational physics, such as complexity or objectivity, and exhaustively explore the assumptions involved to reach a specific conclusion. Thank you for your comments -- you make very good points about complexity, and the problems with treating complexification as a trivial or autonomic process.
Hoang cao Hai wrote on Sep. 26, 2012 @ 03:01 GMT
Dear Karl H Coryat
Sorry for not having the same opinion with you.
You "absolutely" be yourself, do not rely on any "the reference"?
Hope you do not therefore ignore the essay and my new theory.
Kind Regards !
Hải.Caohoàng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY
August 23, 2012 - 11:51 GMT on this essay contest.
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Member Bob Coecke wrote on Oct. 3, 2012 @ 18:54 GMT
Hi Karl, I think that you are very right in identifying the diagrammatic language and relationalism. I always find most relational approaches to vague on what they mean by "relation", while the compact closed categories underpinning graphical reasoning are a natural formal way expressing what relations when interacting, and hence avoiding the need to give an explicit description of the underlying objects. Classicality (of ...) is then an extra ability to manipulate "whatever these relations carry", resulting in additional structure, the dots in the diagrams. These relations aren't really the mathematical relations, but something more flexible and general, closer to what we mean by relation in natural language.
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Author Karl H Coryat replied on Oct. 6, 2012 @ 00:08 GMT
Thanks Bob, I greatly appreciate that you took the time to read and respond. Rock on!
Sergey G Fedosin wrote on Oct. 4, 2012 @ 09:38 GMT
If you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is
and
was the quantity of people which gave you ratings. Then you have
of points. After it anyone give you
of points so you have
of points and
is the common quantity of the people which gave you ratings. At the same time you will have
of points. From here, if you want to be R2 > R1 there must be:
or
or
In other words if you want to increase rating of anyone you must give him more points
then the participant`s rating
was at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process. I hope the FQXI community will change the rating process.
Sergey Fedosin
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Sara Imari Walker wrote on Oct. 5, 2012 @ 17:00 GMT
Hi Karl,
Thanks for the great read! You present a lot of interesting ideas. I am curious, even with looking at living systems, how useful the concept of "contextual information" will turn out to be in the long run. Right now I agree that is is a very constructive way of thinking about these issues (which is why I like to use it as well!), but it is very difficult to formalize. This is one reason I have been leaning toward causal descriptions which are more rigorously defined. I'd be very interested to hear your thoughts on this.
Again, thanks for the engaging read!
Best,
Sara
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Cristinel Stoica wrote on Oct. 5, 2012 @ 21:56 GMT
Dear Karl,
Once again you wrote a very insightful and beautiful essay. It was interesting to me to see how far one can go without assuming that information is "about something". I think that what we can know are relations, and they can provide a complete description of reality (at least the accessible reality). You may want to take a look at some slides I prepared for a talk named
"Global and local aspects of causality", where I take the wavefunction and unitary evolution seriously, and try to explain quantum correlations by global consistency effects (this is not related to my essay, named
"Did God Divide by Zero?").
Best wishes,
Cristi Stoica
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Author Karl H Coryat wrote on Oct. 6, 2012 @ 00:17 GMT
Sara and Cristi, thank you for your comments. As I think I've mentioned on both of your pages, I found both of your essays very interesting. I've found so many enlightening ideas in this year's essays, and have a winter's worth of papers to read from the reference lists. I'll certainly go through both of your essays again. Thanks again and best of luck.
Eckard Blumschein wrote on Dec. 14, 2012 @ 06:42 GMT
Any complex square matrix is triangularisable.
Eckard
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