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Is Newtonian Mechanics an advantage or a limitation in astrophysics?


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What happens to the physical laws when chaos figures into the mix?
This is a great question! I think a lot of the answer has to do with one's view of how "chaotic systems" work: I get somewhat exasperated by the "they just look chaotic because we can't see enough detail to see the causes and effects" excuse.

 

I'd love to hear Hilton's view too (I think I'm going to have to buy your book...the excerpts are interesting!)...

PS. Yes I am obsessed with thermodynamical issuse...:P
Yes I am obsessed with chaotic systems! :(

 

My small difference in initial input, :P

Buffy

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Hi CC and Buffy,

 

Welcome back from the NYC wilderness, CC. Thank you both again for your comments and questions. The subtitle of the sequel to The Virtue of Heresy is “Agricultural Astrophysics”. Guess why?:hal_skeleton:

 

CC said:

On the one hand we have Newton's laws, deterministic, the past, present and future predictable with certitude, time-reversibility, time-symmetry, where the same role is played in either direction of time. And on the other, we acquired from the 19th century an evolving world view, where entropy is a nondecreasing property of nature that introduces the 'arrow of time,' irreversibility, according to the second law of thermodynamics, where past and future no longer play the same role.

How do you reconcile these divergent points of view?

 

Buffy said:

I think a lot of the answer has to do with one's view of how "chaotic systems" work: I get somewhat exasperated by the "they just look chaotic because we can't see enough detail to see the causes and effects" excuse.

 

I do not share this interpretation of Newtonian Mechanics. Newton assumed the unidirectional passage of time as an axiom, and did not, as far as I am aware, use time as a geometrical dimension (ie he used time in the Euclidean sense). Both time and space are used in classical physics as we experience them, not as we imagine they might be or become. No further abstract inflections are used or indeed permitted. Reversibility or symmetry in time is certainly not part of a mechanical, cause-and-effect description of the world, and are considered impossible in reality. The notions of entropy and chaos came later on, when physics had become intimately bound to mathematical freedom. Personally, my troubles begin at Maxwell’s equations of electromagnetism, which for me exist at the cusp of relativistic physics.

 

So, although I share your great interest in thermodynamics (and am about as interested in chaos theory as I would be in last week’s breakfast), I do not see a Universe that blatantly verifies the broader implications of thermodynamics. There is ubiquitous evidence of cycles involving reversibility, and none that I know of that indicates a universal linear decay to entropy and heat death. Are we not with these theories trying to explain something that does not really exist? Elsewhere, I used the analogy of the ball on a roulette wheel: If we knew enough about the initial and ambient circumstances, we could predict its destination with precision. In my view, the same would hold for galaxy rotation. It is a machine and an engine, so a galaxy’s behaviour is the effect of the sum of causes. We find the so-called “anomaly” in galaxy rotation because we are leaving something out of the calculation, that is, it is a function of our ignorance, as chaos is. Think carefully about the anomalous perihelion advance of Mercury. Doesn’t it make you uneasy that GR arrives at a fairly precise prediction by considering Mercury’s motion as a 2-body? Get my drift?

 

Oliver Manuel and I are setting up collaboration with Richard Mackey of the Australian National Astronomical Observatory to investigate the effects within the Solar System of solar inertial motion (SIM). The shifting barycentre of the Sun causes it to follow an epitrochoid orbit that is intrinsically random, and which in turn results from n-body interactions within the Solar System. This solar orbital shift will interactively affect the orbital dynamics of planets, and probably most noticeably of Mercury because of its proximity. However, as I recall, this mechanism was built into neither GR nor NM precession calculations. We can suggest that NM was wrong because it left something out, but can we explain why GR was right? For your interest, read Mackey’s excellent review of Rhodes Fairbridge’s work on the Sun (Journal of Coastal Research, Special Issue 50, 2007). See http://www.griffith.edu.au/conference/ics2007/pdf/ICS176.pdf . I have a PDF if you require it.

 

This is another paper on the subject published in Physics Letters A, vol. 367, pp. 276-280 (2007):

Gravimagnetic effect of the barycentric motion of the Sun and determination of the post-Newtonian parameter gamma in the Cassini experiment, by S.M. Kopeikin (University of Missouri-Columbia, USA), A.G. Polnarev (Queen Mary University of London, England), G. Schaefer (Friedrich Schiller University of Jena, Germany), I.Yu. Vlasov (University of Guelph, Canada).

 

arxiv.org/abs/gr-qc/0604060v6

 

Best

Hilton

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...

So, although I share your great interest in thermodynamics (and am about as interested in chaos theory as I would be in last week’s breakfast),...

 

...There is ubiquitous evidence of cycles involving reversibility,...

 

 

 

Hello all,

 

Hilton, although there are no gross instabilities associated with the motion of planets within the solar system, numerical data integration has shown that orbits are in fact chaotic. Deterministic equations used in celestial mechanics to forecast future positions of planets are obligatorily limited. For example, if we know the orbital position of the Earth to within one meter today, the exponential proliferation in the margin of errors— characteristic of chaotic systems—entails that we can acquire no knowledge of the Earth’s orbital position 200 million years into the future (Murray, 1999).

 

1. Does this bother you in any way?

 

 

It has been known since the work of Henri Poincaré (1854-1912), the French mathematician famed as the founder of algebraic topology and the use of his theory on differential equations in celestial mechanics, that dynamical systems are characterized in terms of the kinetic energy of its constituents plus the potential energy due to their interactions.

 

Furthermore, as Ilya Prigogine (yes, he again), 1996, informs us, interacting fields lead to resonance patterns that “wander erratically” through regions of space, increasing entropy, leading to long-range correlations, chaos and the breaking of time symmetry—irreversibly—that acts over very long time-scales and that profoundly alter the macroscopic state of the system.

 

“…phase transitions are ultimately defined by the thermodynamic limit…Phase transitions correspond to emerging properties. They are meaningful only at the level of populations' date=' and not single particles. This contention is somewhat analogous to that which is based on Poincaré resonances. Persistent interactions mean that we cannot take a part of the system and consider it in isolation. It is at this global level, at the level of populations, that the symmetry between past and future is broken, and science can recognize the flow of time. This solves a long-standing puzzle. It is indeed in macroscopic physics that irreversibility and probability are the most conspicuous.” (Prigogine, 1996, p. 45). [/quote']

 

So it seems that interacting fields cause chaotic perturbations and/or coherent patterns, along with irreversibility.

 

 

2. Does this view concord or grate against classical mechanics in your opinion?

 

 

 

Finally, the laws of nature are both simple and complex. They deal with both certitudes and probabilities, with reversible and irreversible phenomena. They describe a universe of asymmetrical and chaotic motions, of equilibrium configurations and stable orbits. This order and disorder represents both microscopic and macroscopic systems.

 

3. Do these apparently contradictory codes not open the way to an evolutionary description of things* present in the observable universe*?

 

 

 

 

Edited to add:

 

*By things it is meant constituents such as, elementary building blocks and nest of forces defined as elementary particles, atomic nuclei, atoms, molecules, ions, and field, such as gravity fields and electromagnetic fields: along with the properties energy, entropy. Let's include, too, planets, stars, galaxies, clusters, superclusters and people.

 

*By observable universe it is meant the universe out to the visual horizon (so as not to speculate as to what resides beyond).

 

 

 

 

Thanks,

CC

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Can you contribute your definition of "observe"?

an evolutionary description of things present in the observable universe
Mine is anything from an electron "absorbing" a photon, to a phototropic single cell organism having a "photon-receiving" protein (which contains the above), to us, and all the other life forms (and the matter they are made up, or compartmentalised out of) in between, including, of course, the ones that don't interact with photons directly (like fish in deep caves, or other things in the deep ocean, say), but still interact with their interaction with the planet -type of thing.
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Hi CC

I shall attempt to answer your questions in terms of the essential theme of this thread, although I find them difficult to understand, and sometimes double-edged, because a question itself can depend upon contained assumptions that are either obscure or in my judgement, off-track. However, I will do my best.

Hilton, although there are no gross instabilities associated with the motion of planets within the solar system, numerical data integration has shown that orbits are in fact chaotic. Deterministic equations used in celestial mechanics to forecast future positions of planets are obligatorily limited. For example, if we know the orbital position of the Earth to within one meter today, the exponential proliferation in the margin of errors— characteristic of chaotic systems—entails that we can acquire no knowledge of the Earth’s orbital position 200 million years into the future (Murray, 1999).

 

1. Does this bother you in any way?

No, not at all. Chaos is a term from mathematics that I don’t like to use. I prefer “randomness” or tidal effects in mechanics. I am puzzled why my ability to predict 200 million years ahead should be a factor in science. Surely this is trivial? It cannot be a test of the verity of NM in astrophysics, and in any case, how could we check our answer if we dared to suggest one? Predicting even 100 years ahead is not greatly important because if Homo sapiens survives with technology and sanity intact, in 50 years time it will be far better equipped to predict 50 ahead with far greater resolution than now, so although we can and do make almanacs that model centuries into the future, they are trivial with remoteness. As I have mentioned earlier, it seems that uncertainty in our measurements increases exponentially with scalar remoteness in space or time, and that is why, on the cosmic scale, I prefer to work nearby. However, other astrophysicists are more daring than I am, and I accept that astrophysics and astronomy do explore territory where randomness is a big factor. We can have no knowledge of the future; we can however make predictions and calculate statistical probabilities, but they are generally more reliable the closer they are to home.

It has been known since the work of Henri Poincaré (1854-1912), the French mathematician famed as the founder of algebraic topology and the use of his theory on differential equations in celestial mechanics, that dynamical systems are characterized in terms of the kinetic energy of its constituents plus the potential energy due to their interactions. Furthermore, as Ilya Prigogine (yes, he again), 1996, informs us, interacting fields lead to resonance patterns that “wander erratically” through regions of space, increasing entropy, leading to long-range correlations, chaos and the breaking of time symmetry—irreversibly—that acts over very long time-scales and that profoundly alter the macroscopic state of the system. So it seems that interacting fields cause chaotic perturbations and/or coherent patterns, along with irreversibility.

 

2. Does this view concord or grate against classical mechanics in your opinion?

 

I see no conflict at all with classical mechanics, just a chasm of difference in the way things are described. I find the philosophy expressed above excessively complicated, convoluted, and abstract. It seems to me that psychologically, such expressions (eg Prigogine’s) are symptomatic of a desire to be smart beyond the facts, to downplay one’s own ignorance. Even before this thread started, I argued that reality physics can achieve essentially the same results in measurement as mathematical physics, but is just so much easier to use. That is the advantage of NM (classical physics) in applied astrophysics, in my opinion.

 

Finally, the laws of nature are both simple and complex. They deal with both certitudes and probabilities, with reversible and irreversible phenomena. They describe a universe of asymmetrical and chaotic motions, of equilibrium configurations and stable orbits. This order and disorder represents both microscopic and macroscopic systems.

 

3. Do these apparently contradictory codes not open the way to an evolutionary description of things present in the observable universe? By things it is meant constituents such as, elementary building blocks and nest of forces defined as elementary particles, atomic nuclei, atoms, molecules, ions, and field, such as gravity fields and electromagnetic fields: along with the properties energy, entropy. Let's include, too, planets, stars, galaxies, clusters, superclusters and people. By observable universe it is meant the universe out to the visual horizon (so as not to speculate as to what resides beyond).

 

Hoo boy! What can I say? There are no “apparently contradictory codes” in nature, only in our thoughts about them. From my standpoint, all such a philosophy can achieve is to somehow legitimise abstract conjecture, giving it a measure of scientific foundation that results in nothing physically useful (although it may have some intellectual, artistic, or spiritual value). We just get way ahead of ourselves. However, I do not pretend to be qualified to advance such an argument, and by putting discretion ahead of valour, gently nudge this thread back on course.

Best

Hilton

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not to speculate as to what resides beyond

See that here a speculation is already being presented.

You are speculating that what (if anything) we presume to be beyond the visible horizon is already 'imponderable', which it isn't, obviously (since how could you or anyone ponder it if it were "unknown"?). This by itself illustrates the "problem" with the "unknown" --like the infinite, we can only conceive of it, but not see it (sound familiar?).

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Chaos is a term from mathematics that I don’t like to use. I prefer “randomness” or tidal effects in mechanics. I am puzzled why my ability to predict 200 million years ahead should be a factor in science. Surely this is trivial? It cannot be a test of the verity of NM in astrophysics, and in any case, how could we check our answer if we dared to suggest one?
Ignoring for the moment the notion of "verifying" results, I'm interested in your ideas of the interaction of quantum effects on such predictions.

 

I've wasted much time in discussions here on the misaprehension that *any* sort of randomness means that *no* results are predictable--a "conundrum that is explained easily by the fact that most systems are *convergent*, cancelling out quantum randomness for many--but not all!--predictions of future outcomes. This is in fact quite similar to your approach of using NM as "close enough" for most predictions (i.e. we actually use a similar argument to justify two somewhat unrelated issues in physics! Imagine that!).

 

The issue I bring up here is that there are systems that are *divergent*: while you can predict within a small error range the exact position of Mercury 217,238,071 years, 2 months, 5 days, 3 hours, 12 minutes and 27 seconds from *right now* to probably within 10 percent accuracy with NM, I can pretty much assure you that you can't tell me whether it is going to be raining or not here in San Francisco in just 30 days from now.

 

Chaos does indeed have a role in divergent/convergent systems, and only a few seem to disagree with this. So the question I have for you is:

 

Does your worldview in which NM dominates accommodate quantum randomness, or does it require that quantum randomness to be completely predictable albeit with information that is unobtainable ?

 

This is slightly parallel to my earlier arguments insofar as I believe that while you are absolutely right that many astronomical effects are "close enough for Country music," that macroscopic quantum effects pose another potential problem--albeit in only a small fraction of instances--to relying on NM for astrophysical studies (trying here to get us back on topic!).

 

That stray photon really is the difference between 5.999999999.... and 6, :phones:

Buffy

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I shall attempt to answer your questions in terms of the essential theme of this thread, although I find them difficult to understand, and sometimes double-edged, because a question itself can depend upon contained assumptions that are either obscure or in my judgement, off-track...

 

I was under the impression that the questions were straight forward, a follow up to the previous post regarding the physical laws when chaos figures into the mix. I hope they are not double edged. I don't see how they could be off-track since it would seem that there are systems observed (empirically) to exhibit erratic motion, and other that exhibit mean motion resonance patterns.

 

 

No, not at all. Chaos is a term from mathematics that I don’t like to use. I prefer “randomness” or tidal effects in mechanics. ...

 

Ok, so let's use randomness and tidal effects instead of chaos.

 

 

 

It has been known since the work of Henri Poincaré (1854-1912), the French mathematician famed as the founder of algebraic topology and the use of his theory on differential equations in celestial mechanics, that dynamical systems are characterized in terms of the kinetic energy of its constituents plus the potential energy due to their interactions. Furthermore, as Ilya Prigogine (yes, he again), 1996, informs us, interacting fields lead to resonance patterns that “wander erratically” through regions of space, increasing entropy, leading to long-range correlations, chaos and the breaking of time symmetry—irreversibly—that acts over very long time-scales and that profoundly alter the macroscopic state of the system. So it seems that interacting fields cause chaotic perturbations and/or coherent patterns, along with irreversibility.

 

I see no conflict at all with classical mechanics, just a chasm of difference in the way things are described. I find the philosophy expressed above excessively complicated, convoluted, and abstract. It seems to me that psychologically, such expressions (eg Prigogine’s) are symptomatic of a desire to be smart beyond the facts, to downplay one’s own ignorance...

 

I don't see how the above could be described as complicated, convoluted, and abstract philosophy, again because the arguments are based on observations. In fact, it even seems that the above discussion is a classical one. I see no relativity, chaos theory of quantum mechanics above. I don't see, either, how Prigogine's comments have anything to do with psychologically or a desire to be smart. I think he had simply come across (not by chance) a way of describing nature that was perhaps more general, more coherent than the traditional academic stance, where the large and the small are governed by different principles (GR and QM).

 

 

 

 

Finally, the laws of nature are both simple and complex. They deal with both certitudes and probabilities, with reversible and irreversible phenomena. They describe a universe of asymmetrical and chaotic motions [“randomness”], of equilibrium configurations and stable orbits. This order and disorder represents both microscopic and macroscopic systems.

 

3. Do these apparently contradictory codes not open the way to an evolutionary description of things present in the observable universe?

 

 

Hoo boy! What can I say? There are no “apparently contradictory codes” in nature, only in our thoughts about them. From my standpoint, all such a philosophy can achieve is to somehow legitimise abstract conjecture, giving it a measure of scientific foundation that results in nothing physically useful (although it may have some intellectual, artistic, or spiritual value)...

 

Are you saying there is no such thing as irreversible phenomena, that our perception of “apparently contradictory codes” is based somehow on the anthropomorphic or anthropic principle? I don't see where you get philosophy and abstract conjecture out of something that is observed in nature (e.g., mean motion resonance patterns ensuring the repetition of specific geometrical configurations, perturbations, random motions intrinsic in globular clusters, a direction in time, irreversibility).

 

Maybe I'm not understanding what you are trying to say.

 

It seems to me, and I could be wrong (here is the double edged assumption), that the overabundance of resonant configurations observed in systems such as the solar system (far-from-chaotic or random distributions) leads us to an important clue as to the laws of nature responsible for the development and progression of a long-term selection process, of evolution.

 

Here is the problem then: Is classical mechanics responsible? If so, how can that be without the concept of field, without the problem of artificially fine-tunning parameters?

 

 

CC

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The simple fact about observation and events is that they are both things that require change. This requirement, however, is also a projection from the observer (along with the process of measurement).

We like to think we can 'freeze' time, or take a "snapshot", but clearly this simply isn't possible (we therefore "imagine" it). Observation and the learning that comes with it (to any sentient critter), are because of change. At least that's how it seems...

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I am new to this thread so forgive me if this has been asked and I missed it in reviewing previous posts.

 

Hilton,

 

Working in astrophysics you must use Lorentz transformations. Any distance calculation would be based on a relativistic Doppler effect - Galilean relativity wouldn’t work. With redshifts as high as 2 or 3 the error in distance calculation could end up being in orders of magnitude. Also, if you didn’t consider GR and expansion the uncertainty would be even higher. I don’t see how considering distance, speed, and redshift without considering expansion would give results consistent with the newest data. Have you derived a scale factor for your work that is different than the standard model? If so, what is it based on?

 

This said, would you say that a redshift 3 galaxy’s light has a travel time of between 9.4 and 11.5 Gyr? And, between 10.2 and 12.5 for z=5? If you agree or disagree I’d like to know your reasoning. If you come up with different numbers (or the same numbers for that matter) without using relativity that is very interesting to me and I’m eager to learn.

 

Again I apologize if this is a rehash Thanks in advance.

 

-modest

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The Universe I try to explain is that mainfestation I am sentiently aware of (what I observe and experience), and I therefore use observation and experience as the basis of my method. I also explicitly exclude mathematical arguments from my explanation of that infinitesimal fraction of the Universe that I claim to understand. As an astronomer, I use classical physics (Newtonian mechanics) to figure out what it is I am looking at. That may be considered by some to be a self-imposed limitation; I view it as an advantage. The questions this thread asks include whether or not a mechanical explanation of the observed universe is deficient or inferior to a (Einstein) relativistic explanation; if pure mathematical modelling assists or detracts from astrophysics; if one can draw realistic conclusions from gedanken experiments; whether mathematically-derived hypotheses like Black Holes and Big Bang theory enhance or detract from our quest for real-world answers to observational curiosities; and finally, if cosmology should even be part of the physical sciences rather than a subset of theology.

 

Essentially, this thread seeks to discuss the arguments put forward in my book "The Virtue of Heresy - Confessions of a Dissident Astronomer" (available from the Hypography bookstore).

 

Hilton Ratcliffe

Astronomical Society of Southern Africa

Alternative Cosmology Group

 

I posted an article on the Cosmology thread (Stability of the Hydrogen Atom)that supports Newtonian Math.

I consider him to be the greatest of the modern era scientists with Kepler as the first of the great mathmaticians.

I consider Einsteins math to be faulty and that includes his GR.

 

The popularity of the bible is what gives Einstein the popularity he has IMHO.

The bible is also faulty.

 

Mike C

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Hi everyone,

 

First CC:

I don't see how they could be off-track since it would seem that there are systems observed (empirically) to exhibit erratic motion, and other that exhibit mean motion resonance patterns.

 

The principle I am trying follow is that NM looks at randomness in observation and assumes that it is a sum of effects from physical causes, some of which are unknown. In applied astrophysics, we should not take answers from chaos theory in any form. Trying to find some formal mathematical description for randomness to the exclusion of digging up the missing causes defeats the object in my opinion, and I don’t accept it as part of applied astrophysics. Unfortunately we now tread perilously close to losing this thread to a debate about definitions. Put yourself in the position of an astronomer who makes an observation of something on the sky, and who then seeks to take it into the realm of astrophysics by seeking to quantify and describe it using physical science. Do you think Prigogine’s philosophy would help you to do that? I don’t, and that is why I suggest it is off track. Perhaps when we really get down to the nitty gritty of this thread, it is the practical implications of NM vs GR (and SR) in applied astrophysics excluding cosmology.

CC said:

I don't see, either, how Prigogine's comments have anything to do with psychologically or a desire to be smart. I think he had simply come across (not by chance) a way of describing nature that was perhaps more general, more coherent than the traditional academic stance, where the large and the small are governed by different principles (GR and QM).

I said, “…desire to be smart beyond the facts.” My position is that we actually know far less about our cosmic environment than our mathematical-theorist brethren would have us believe. Prigogine’s desire to marry GR and QM is admirable, but I do feel it is not pertinent to this thread. Having said that, I wholeheartedly support his contention that there is a direct causal link from small to big. (See chapter 12 “The Virtue of Heresy”). Without wishing to myself go off beam, it seems obvious to me that we could (if we applied our minds) find a NM model of atoms and sub atoms even that is less illogical and invokes fewer physical impossibilities than the Bohr and QM descriptions. Another place, another time…

Regarding the last para of your post CC, it seems we are not coherent and are missing each other. I say that there is no reversibility in time, no matter that we have equations in physics that are symmetrical in time. In reality, it doesn’t work. Regarding randomness: I take a mechanical view of physical systems (sorry Buffy, I know that makes you mad!) and if we can’t describe the orbit of Mercury properly, it’s because we have left something out of the equation, not because the theory is necessarily wrong. I have given references to a significant influence on the orbit of Mercury that has been ignored (as far as I know) by both NM and GR. How come GR gets the right answer then?

 

Modest said:

Working in astrophysics you must use Lorentz transformations. Any distance calculation would be based on a relativistic Doppler effect - Galilean relativity wouldn’t work. With redshifts as high as 2 or 3 the error in distance calculation could end up being in orders of magnitude. Also, if you didn’t consider GR and expansion the uncertainty would be even higher. I don’t see how considering distance, speed, and redshift without considering expansion would give results consistent with the newest data. Have you derived a scale factor for your work that is different than the standard model? If so, what is it based on?

Welcome to this thread Modest. I would suggest that you go over the posts from the beginning (homework!) because I’m pretty sure that these questions have been answered already, with supporting arguments. I will give a précis answer to your questions here:

I do not use Lorentz transformations. How can you show that Galilean relativity would give the wrong answer? Compared with what or tested how? Relativistic distances are based on assumptions that I do not make, and in any case come into effect only at great remoteness or extremes. I find no convincing evidence that the Universe is expanding, although I must mention that I consider these to be a matters of cosmology and therefore not within the spirit of this thread. I maintain that the redshift-based distance scale is unreliable and misleading. I do not know what the distances to cosmological objects are, and do not want to mask my ignorance with theory. I have not derived a scale factor, and use standard triangulation with high certainty for close objects, and other methods like period-brightness with less certainty for objects a bit further away, and beyond that make no claims about remoteness in space or time. I repeat what I said to CC above, I would say we know far less about the cosmos than most investigators would have us believe, and furthermore, I do not think that a prerequisite for understanding, quantifying, and describing astrophysical objects and systems is fluency in differential geometry.

I appreciate and share your desire to learn. Like you, I am fully open to being proven wrong.

Best

Hilton

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Hi Buffy,

 

Sorry, I missed your post. I'm not ignoring you! :)

 

Does your worldview in which NM dominates accommodate quantum randomness, or does it require that quantum randomness to be completely predictable albeit with information that is unobtainable?

 

My worldview (as it is being discussed here) is restricted to observational astrophysics, and does not include QM. Please refer to my opening post (quoted above by Mike C for easy reference).

 

Best

Hilton

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My worldview (as it is being discussed here) is restricted to observational astrophysics, and does not include QM. Please refer to my opening post (quoted above by Mike C for easy reference).

Fair enough. Just be sure to add "predictions may be wildly off in cases where systems are at least potentially divergent" to your list of limitations! :(

 

That which is static and repetitive is boring. That which is dynamic and random is confusing. In between lies art, :)

Buffy

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Hi Buffy,

 

Fair enough. Just be sure to add "predictions may be wildly off in cases where systems are at least potentially divergent" to your list of limitations!

 

Once again, I think you come close to showing me where I may actually be going wrong. I suppose it may boil down to whether or not I would make such a prediction. Do you have some examples that would illustrate this? I don't mean my own predictions from my own publications, just in principle, what kind of prediction would an astrophysicist need to make that would be wildly off in practice if she used NM?

 

Best

Hilton

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