Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: HellsMascot on 14/05/2013 20:17:19
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This could be a poll, but there would be too many options. What interpretation of quantum mechanics do you subscribe to (e.g. many-worlds, consistent history, deBroglie-Bohm)? The most commonly taught and widely accepted interpretation is that of the Copenhagen Interpretation. Do you feel this is a just treatment of our universe? Do you believe that the wave function describing a system, when observed, collapses into a discrete eigenstate? Perhaps wave function collapse is a human construct?
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Copenhagen interpretation. Maybe (I really hope so) in the future an extension of it will be able to account even for the process of measurement.
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To me the Copenhagen defines us as parts of whatever experiment or system we observe, and that makes sense to me too. Many worlds theories exist in different versions as I've seen, but I don't see it excluding us as part of the experiment we do? What I don't subscribe too is the idea of a arrow as a illusion.
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Quantum mechanics gives us a probability function. When an experiment measures a quantum state, you obtain a value among all the probable values given by the function.
Copenhagen interprets it as the reality. There is a superposition of all states until there is an observation, which fixes the state. There is no moon unless someone observes it. There is no causal explanation. Which is unsatisfactory.
Many worlds adds a simple causal explanation: each possible state exists in a different world. The act of observation generates a new path and a new world.
Both interpretations stop at looking for other causal interpretations. This is not a good scientific practice.
Gravity and electromagnetism have an infinite range, therefore everything is connected in our universe. All the probable states are in this universe, there is no superposition for one particle taking alone, but there is a superposition of all states for all particles of the universe taking as a whole. The proof of that is the Pauli exclusion Principle.
The Bohm-De Broglie interpretation is the good one but it does not explains anything much further experimentally than other interpretions.
See my theory for an extension: http://www.thenakedscientists.com/forum/index.php?topic=34413.0
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I'm partial to the many worlds interpretation because I find it philosophically fascinating. But in practice, I'm very much an empiricist, so I stay out of fights over which interpretation is best: they all agree with observation.
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This could be a poll, but there would be too many options. What interpretation of quantum mechanics do you subscribe to (e.g. many-worlds, consistent history, deBroglie-Bohm)?
The Copenhagen Interpretation. There is an article online which will be of interest in this thread.
Quantum Mechanics and reality, Bryce S. Dewitt, Physics Today, September 1970 at http://www.projects.science.uu.nl/igg/jos/foundQM/qm_reality.pdf
It has a section the quantum theory of measurement.
Also Quantum Theory Needs No 'Interpretation', Christopher A. Fuchs and Asher Peres, Physics Today, March 2000 at http://www.google.com/url?sa=t&rct=j&q=Quantum+Theory+Needs+No+%27Interpretation%27+physics+today&source=web&cd=3&cad=rja&ved=0CD0QFjAC&url=http%3A%2F%2Fwww.phy.pku.edu.cn%2F~qhcao%2Fresources%2Fclass%2FQM%2FPTO000070.pdf&ei=gwmUUbrHGeHj4APshoGwCw&usg=AFQjCNHx2NQe38nPXbN6KXbU7Vo_3VlmMA&bvm=bv.46471029,d.dmg
The most commonly taught and widely accepted interpretation is that of the Copenhagen Interpretation. Do you feel this is a just treatment of our universe? Do you believe that the wave function describing a system, when observed, collapses into a discrete eigenstate?
Yes.
Perhaps wave function collapse is a human construct?
Of course it is. It's something created by humans to describe nature. Humans require descriptions. Nature doesn't give a hoot about what we wish to describe. :)
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All the interpretations tend to be a bit mystifying to non-scientists like me, but a distillation of the De Broglie-Bohm interpretation invites some comments.
The cosmos in which we live is infinite; not only is every part in contact with every other part, every part is the whole. It is not sufficient to say that everything that can happen happens, as this implies progression. In infinity there can be no change, no progression and no differentiation in time or space.
Change, movement and the passage of time that we observe is an illusion arising from our 3 + 1 dimensional perspective.
Quantum mechanics is a window into the infinite, through which we are just learning to look. A measurement is simply the translation of quantum reality into our limited perception of reality.
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All the interpretations tend to be a bit mystifying to non-scientists like me, ...
Me too. It's difficult enough to study orthodox quantum mechanics, never mind studying all other interpretations. Especially those which cannot be tested by experiement such as the many worlds interpretation.
...but a distillation of the De Broglie-Bohm interpretation invites some comments.
I take it that you're familiar with it then?
The cosmos in which we live is infinite;...
That is not known to date. It's possible that the universe is spatially closed, which means that it's finite in extent.
... not only is every part in contact with every other part, every part is the whole.
That is incorrect. We are not in contact with most of the universe. E.g. there is no way to communicate with certain region of the universe, especially those galaxies which are moving away from us at speeds greater than the speed of light.
It is not sufficient to say that everything that can happen happens, ...
There is no way to prove such a think like that.
In infinity there can be no change, no progression and no differentiation in time or space.
What does In infinity there can be no change,.. mean? If taken literally then I quite disagree. Almost everything is chaning with time so how can you say that there can be no change at all? Is that what you really mean to say?
Change, movement and the passage of time that we observe is an illusion arising from our 3 + 1 dimensional perspective.
I don't know where you got that idea but it's quite wrong. There are very few physicists who would agree with such a statement. Except, of course, Julian Barbour who thinks that time is an illusion. Most, if not all, physicists don't accept that view.
Quantum mechanics is a window into the infinite, ...
Huh? Why? You're really confusing me. Please explain where you're getting these notions from or at least justify them for us.
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I'm partial to the many worlds interpretation because I find it philosophically fascinating. But in practice, I'm very much an empiricist, so I stay out of fights over which interpretation is best: they all agree with observation.
Beautifully said my good man!
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Perhaps wave function collapse is a human construct?
Don't know exactly what you mean but if you mean that the collapse is *caused* by the human act of observation (the man who opens the box to see if the cat is dead or alive), then it's not. The collapse is caused by the act of measurement.
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Don't know exactly what you mean but if you mean that the collapse is *caused* by the human act of observation (the man who opens the box to see if the cat is dead or alive), then it's not. The collapse is caused by the act of measurement.
Quite. And an 'act of measurement' occurs for any interaction with the system (e.g. any particle interaction). Observers not necessary. The idea that an 'observation' or 'measurement' must involve a conscious observer seems as popular an error as the idea that the Uncertainty Principle is a consequence of the Observer Effect.
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Quite. And an 'act of measurement' occurs for any interaction with the system (e.g. any particle interaction). Observers not necessary. The idea that an 'observation' or 'measurement' must involve a conscious observer seems as popular an error as the idea that the Uncertainty Principle is a consequence of the Observer Effect.
M'man! You're awesome! That is precisely the way I see it. In fact the paper by Bryce de Witt that I posted a link to above addreses that exact thing. You should give it a read. I think that you'd like it.
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Pete, thanks for the thorough examination of my post. I will try to do justice to a response.
I take it that you're familiar with it then?
I struggled with "Wholeness and the Implicate Order" and have dipped into a few other things, eg "The Undivided Universe". I am quite willing to accept that my interpretations may have been wide of the mark, but surely testing one's understanding is a part of learning.
That is not known to date. It's possible that the universe is spatially closed, which means that it's finite in extent.
Of course it is not known, but the concept of an undivided universe in which: "The velocity of any one particle depends on the value of the guiding equation, which depends on the whole configuration of the universe" lends itself to speculations along those lines.
I generally follow John Gribbin in distinguishing between the Universe and the cosmos. I do so in an attempt to avoid confusion, but, unfortunately, terminology is by no means standardised, so it doesn't always help.
That is incorrect. We are not in contact with most of the universe.
This is absolutely true, but can we be sure that just because we lack the ability to achieve this contact, contact is impossible?
There is no way to prove such a think like that.
I suspect that some confusion has crept in here because you quoted only part of my sentence. "Saying that everything that can happen happens" implies progression. Hopefully, we can agree on that without digressing into a definition of "happen".
Almost everything is chaning with time so how can you say that there can be no change at all? Is that what you really mean to say?
Precisely! "With time" everything is changing. Only if you regard infinity/eternity as a very long time can you draw a comparison between what happens in time and what is in eternity. That is a concept with which I would certainly take issue.
I don't know where you got that idea but it's quite wrong. There are very few physicists who would agree with such a statement. Except, of course, Julian Barbour who thinks that time is an illusion. Most, if not all, physicists don't accept that view.
I didn't get the idea from Barbour's "The End of Time", but to my surprise I found much in that book that provided a scientific route to ideas I had formulated from a more philosophical approach.
I take your point that most physicists don't agree with Barbour, but then, scientific veracity is not a matter for a democratic vote, is it?
Huh? Why? You're really confusing me. Please explain where you're getting these notions from or at least justify them for us
I will gladly try to explain how "these notions" come about, but it may take a while, so, it being 2am, I shall save that for another post.
Bohm and Barbour will, hopefully save me from having to take the ideas into "New Theories". :)
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M'man! You're awesome! That is precisely the way I see it. In fact the paper by Bryce de Witt that I posted a link to above addreses that exact thing. You should give it a read. I think that you'd like it.
To be fair, 'measurement as interaction' isn't an original idea ;-) When I first heard about Schrodinger's Cat and Wigner's anthropocentric ('the buck stops here') recourse to consciousness for the wavefunction collapse, I couldn't see where, in the process of reaching conscious awareness (whatever that means), collapse might occur, or why. In the limit, Wigner view seemed no different to the EWG (Many Worlds) universe before consciousness evolved, and substituting the mystery of consciousness for the mystery of wavefunction collapse seemed a lazy trick. The claim that consciousness (undefined) is somehow privileged is also Special Pleading (http://en.wikipedia.org/wiki/Special_pleading) on a par with the deist claim that, everything must have a cause - er, except God (also effectively undefined, except as the arbitrary 'first-cause-by-definition' and philosophical back-stop).
Naively, I also couldn't see why an observer must necessarily physically affect the observed system so as to collapse the wavefunction. IOW, surely the Observer Effect (http://en.wikipedia.org/wiki/Observer_effect_(physics)) applies only to a subset of observations? For example, imagine that Schrodinger's cat is one of those genetically engineered glow-in-the-dark cats. In this situation, the observer doesn't require any external interaction with the observed system, photons or otherwise; he/she can observe the light emitted directly by the cat to determine whether it is alive or dead. This makes observation a passive activity - so how can it physically affect the system? I later found the observer could be considered part of the system as soon as information from the system reached him/her, but this again seemed to mirror the EWG interpretation, as the observer would then become part of the system superposition, and as the superposed states were independent and non-interacting, each observer superposition would be aware of only a single history.
I don't have the maths, but I used to wonder whether one could use Feynman's 'Sum Over Histories' path-integral approach, where the contributions of histories to the probability amplitude are summed over all possible histories, and cancel each other by interference, leaving only the classical outcome. It would tidy up a lot of loose ends... ;)
The paper you linked to is interesting, with some elegant statistical arguments, and makes the point that, if nothing else, 'Many Worlds' has the advantages of being entirely causal, and raising interesting questions about measurement theory.
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The way I think of the Copenhagen definition is that you are part of a experiment. The experiment per se are not actively involved in setting parameters and limitations. You do that, before, also defining a outcome by it. And that's the 'part' I'm referring to there. The cat is a beautiful example on that you don't know a outcome, but your choice of parameters and limits will still make a difference. And where they end you still should, practically seen, find relations defining a outcome doing some forensic work on any 'real experiment'. You can't assume consciousness to define the 'mechanics' of a universe unless you define the universe itself to have that consciousness. Although it will still be correct to define it such as 'relations' defining a outcome, including you making your choice of experiment, as well as observation..
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The way I think of the Copenhagen definition is that you are part of a experiment. The experiment per se are not actively involved in setting parameters and limitations. You do that, before, also defining a outcome by it. And that's the 'part' I'm referring to there. The cat is a beautiful example on that you don't know a outcome, but your choice of parameters and limits will still make a difference. And where they end you still should, practically seen, find relations defining a outcome doing some forensic work on any 'real experiment'. You can't assume consciousness to define the 'mechanics' of a universe unless you define the universe itself to have that consciousness. Although it will still be correct to define it such as 'relations' defining a outcome, including you making your choice of experiment, as well as observation..
Sorry, I can't make head or tail of any of that...
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All of this reasoning though, builds on existent arrow. But so do statistics, and motion, relative or not. Show what does not build on a arrow and I will point out to you where it comes from historically, and that should involve a arrow somewhere in its buildup. To me it's the exact same as when I refer to 'c' as a 'clock', I need that 'speed' first, to be able to define it as a clock. Without that definition I don't have one.
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Heh.
unreasoning reasoning huh :)
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Relations define outcomes. Relations being all parameters, what we know and what we might infer from earlier experiments. Making that experiment we define and limit the relations as good as we can, to get a clear and consistent 'repeatable experiment'. And your choice of observation must also play part of what you see.
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This is not a definition set in stone, defining 'relations'. What we think cause a outcome may change with more knowledge and experimenting, but it will still be relations. And I don't expect consciousness to be excluded from those relations. It, as much as a arrow, define a experiment, testing some idea. But the universe should be able to exist without it, or else we have to assume that the universe is a consciousness. In the end that might be a question of personal taste, but experimentally seen I expect the one where a universe exist, even though no 'life' existing at all.
But you can't ignore it, after all, it's consciousness that defined what we think we know, so far.
And by 'personal taste' I refer to quantum logic and super-positions, defining something without a linear arrow microscopically. But the universe we exist in is defined by outcomes, and linear time.
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I have been wrestling with the concepts of time and eternity/infinity for several decades. It seems that the more I try to think about either, the more they seem to be entwined, scientifically and philosophically, with everything else, and the more complex they become. To some extent, QM and its possible interpretations do seem to provide opportunities to tie up a few loose ends. To offer the explanation which Pete asked for, and to make it meaningful, would involve going back over quite a long process of thought. I would not attempt to do that in a single post. Whilst I would welcome comments at any stage, and would point out that a conclusion might be some way off, I would also invite anyone to shout "stop!" at any point.
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The Infinite Cosmos Part 1.
Theories as to the origin of the cosmos should not be confused with theories about the origin of the Universe. The two things might be the same, or they might not. The evidence we will be considering suggests that they are not the same in our frame of reference, but the same evidence may also indicate that there could be other frames of reference in which the situation might be different. The best current theory of the origin of the Universe is the Big Bang theory. This indicates, quite plainly that our Universe is finite; at least it had a beginning. Of course, it is often argued that although our Universe had a beginning, it might not have an end, therefore it could be said to be infinite. However, this is really a case of confusing “infinite” with “unbounded”. Scientists and philosophers may speculate about its future, but the “moment of creation” is clearly pinpointed. This invites the question, “what came before the Big Bang”. The frequently offered answer, “nothing”, turns out to be quite unsatisfactory. In fact, the more closely one looks at this "nothing" the more "somethingy" it becomes. The bouncing universe theory, which postulates that the universe expands from a big bang, then contracts to a big crunch, followed by another big bang, over and over again, seems to provide a possible way out, but this runs into the problem of infinite regression, and with it, the equally problematic “infinite series”.
On closer inspection, the bouncing universe turns out to be just one of a multitude of multiverse theories, all of which, in some way or another, involve the idea that our Universe is not all that there is. A convenient term for this greater concept is the “cosmos”. We then have to ask if the cosmos might be finite or infinite. At first sight, we may seem only to have transferred this question, and all the difficulties that accompany it, from the Universe to the cosmos. However, such may not necessarily be the case. For a start, we have almost certainly parted with the Big Bang as the start of it all. I suppose we could suggest a big bang as the origin of the cosmos, but as far as I am aware there is no evidence to indicate that this might be the situation, so we would simply be plucking a theory out of the air, which is not the best foundation for anything resembling serious enquiry.
What about an infinite cosmos in which our Universe is temporarily embedded? This feels as though it could have quite a lot going for it; but does it stand up to closer scrutiny? As we have seen, cosmological evidence points to the probability that we live in a finite Universe; this accords with our perception, and may lead to the further “perception” that we live in a finite cosmos; one which, like the observable Universe, has three dimensions of space and one of time; a cosmos that, somewhere, had a beginning. If this were the case, it would mean that the cosmos must have had a precursor. This seems to leave us once again with the problem of infinite regression. The only way to avoid infinite regression is to postulate an infinite precursor. We have now exchanged one question for another: “How can a finite cosmos arise out of an infinite precursor without dividing that eternal precursor, and in so doing, dividing infinity?
At this point we have to ask ourselves what we mean by dividing infinity. In mathematics we can have more than one infinity, but in reality, infinity must be everything; if it does not include everything, it is not infinite. If we postulate more than one infinity, then neither is infinite, because the contents of one infinity must always be excluded from the other; which is nonsense.
Before we can make any real progress we have to answer one very pertinent question: Can there ever have been a time when there was absolutely nothing? Some scientists talk of the Universe having been created from nothing by a quantum fluctuation. Unfortunately, this does not solve the problem, because a fluctuation must, by definition, be a fluctuation of something. In some scientific descriptions the apparent creation from nothing of virtual particles in the vacuum requires that there be particles already there; presumably, these are necessary catalysts. Even if we accept that virtual particles simply appear as a result of quantum fluctuations in the vacuum, then we must regard the vacuum as something in which fluctuations can occur. All we have done is push the problem further into the past. It seems impossible to escape the conclusion that there can never have been a time when there was absolutely nothing; otherwise there would be nothing now, but this leads us to ask the question: If we live in a finite Universe, how can the change from infinite to finite have been accomplished without changing the infinite? We might reason that the nature of infinity is such that whatever we do to it, it remains unchanged. Like zero, which can be multiplied or divided by any number, but remain unchanged, infinity might be unchanged by any action to which it might be subjected. I suspect that this might be the most promising line of enquiry, but I also suspect that it will lead to more complications than might at first be evident.
Whatever it might be that has always existed must exist in eternity. This statement might seem so self-evident as to be tautologous, but it is a point that is worth making, and keeping in mind. Another point, one that is perhaps less obvious, that is worth stressing, is that an object that is finite can never become infinite. Although, in theory, it can increase “for ever”, it can never reach a point where it is infinite; it would always be moving towards infinity, but would require “infinite time” in order to arrive at infinity; which, quite obviously, it could never do. It is said of an “open universe” that it would go on expanding for ever. Scientists seem to have no problem with that idea, and, perhaps chalk it up as yet another type of infinity, along the lines of Georg Cantor’s “infinity of infinities”. However, we must not forget that Cantor’s infinities were mathematical infinities. In the “real” world, even the assertion that a finite object could “increase for ever” is misleading, because it assumes the possibility of an infinite progression. Although this is a mathematical possibility, it involves some serious complications in the physical world. Mathematical “truths” do not always equate to physical realities. It seems that there must be a distinction between “endless” and “eternal”. In our finite frame of reference we cannot see an end for something that apparently goes on for ever; yet we still have to distinguish between that and something that is eternal. The seemingly endless may have an end somewhere. We may not be able to see it, or even imagine it, but there is no way we can be sure it is not there. On the other hand, that which is eternal has, by definition, no end. We may not be able to imagine this, either, but the definition is there, and if we change it in any way it is no longer eternal.
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Don't know exactly what you mean but if you mean that the collapse is *caused* by the human act of observation (the man who opens the box to see if the cat is dead or alive), then it's not. The collapse is caused by the act of measurement.
Quite. And an 'act of measurement' occurs for any interaction with the system (e.g. any particle interaction).
Yes, but you have only changed the name of the process.
What's an "interaction"?
For example, when a beam of light is bent, without absorption, by a glass prism, does the beam "interact" with the glass or not? Explain why yes or why not.
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What do you say about this Lightarrow? :)
to get a interaction you need a 'system'. Even when you define it as being something interacting with itself. That makes it one of two possible definitions, either something 'unitary' can interact with itself, or it can't be 'unitary'. If it isn't unitary, then you either have a hidden parameter, or what you call unitary can't be. If it can interact with itself then it must consist of more parts than one.
All from my own way of thinking of it, that everything 'observes' and adapts to everything (aka relations) :)
And welcome to the philosophy forum :)
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Btw, maybe the arrow could be seen as a 'hidden parameter' here, as we have probability?
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... you have only changed the name of the process.
Just emphasising the semantic difference between the term in physics and the term in common usage.
What's an "interaction"?
It is the mutual effect of two objects on each other, involving the transfer of energy between objects and/or fields.
For example, when a beam of light is bent, without absorption, by a glass prism, does the beam "interact" with the glass or not? Explain why yes or why not.
Yes, it interacts. Considered as a wave, the frequency remains constant but the phase velocity is changed entering the glass; the refractive index of glass varies with frequency, so the change in phase velocity of the different light frequencies varies, resulting in the frequency dependent refraction & splitting of the beam. Considered as particles, the photons interact with the electrons in the glass, by scattering, absorption, and re-emission (see Feynman's 'QED', ch.3, p.107).
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Some minor quibbles:
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I suppose we could suggest a big bang as the origin of the cosmos, but as far as I am aware there is no evidence to indicate that this might be the situation, so we would simply be plucking a theory out of the air, which is not the best foundation for anything resembling serious enquiry.
There may be no evidence for the big bang as origin of the cosmos, but neither is there any evidence for the greater cosmos itself, so the idea of a greater cosmos is open to the same objection. If you accept the possibility of the greater cosmos without evidence, is it reasonable to reject ideas concerning that cosmos for lack of evidence? The big bang as origin of the cosmos must surely remain a possibility unless there is contrary evidence. An analogy that comes to mind is a death that might be murder, but there is no evidence of it. You would not then exclude the possibility that the murderer was a woman because of the lack of evidence that a woman did it - there is no evidence that anyone did it.
In mathematics we can have more than one infinity, but in reality, infinity must be everything; if it does not include everything, it is not infinite. If we postulate more than one infinity, then neither is infinite, because the contents of one infinity must always be excluded from the other; which is nonsense.
It's not clear to me that this is the case, because it is not clear to what you are referring with 'infinity'; spatial extent? temporal extent? Some explanation/clarification required.
With regard to your later musings on temporal infinity, it's worth bearing in mind temporally closed models of the universe, such as Hawking's model, which describes time itself starting at the big bang. His model had the universe and time ending at a big crunch, which we now believe to be unlikely, but as he put it, if time starts at the big bang, it makes no more sense to ask what was before the big bang than it does to ask what's north of the North Pole.
Of course, there is now a variety of hypotheses about pre-big bang physics, so Hawkings model has lost its original appeal, but it seems to me that the concept of an origin of time (and possibly an end of time) could be applied to the greater cosmos (metaverse?) just as to our universe.
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Dlorde, your quibbles are certainly not trivial. Gratifying as it might be when someone agrees with you; in terms of the development of ideas, it is much more valuable when they disagree.
I take your point about there being no direct evidence for the "greater cosmos". One of the ideas that is central to my thinking is that if there had ever been a time when there was nothing, there would still be nothing now. There will probably be plenty of people who will disagree with that, and I look forward to reading their arguments, however, "given that there can never have been a time when there was nothing, it follows that there must be more to our Universe than meets the eye", because our Universe seems to be finite.
..... it is not clear to what you are referring with 'infinity'; spatial extent? temporal extent? Some explanation/clarification required.
A large part of the thrust of my reasoning is that infinity is not a very large space, and eternity is not a very long time. In fact, making any distinction between infinity and eternity has relevance only in our limited understanding. Time and space may have no part in the definition of infinity, but we lack the vocabulary to define it in any other way.
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The Infinite Cosmos Part 2
Assuming, as, apparently, most scientists do, that the Big Bang theory is correct, our Universe must be finite; in so far as it had a beginning, but even if it could continue to expand “for ever” it could never become infinite. Even claiming that it is expanding towards infinity is misleading. If it can never reach infinity, in what sense can it be said to be approaching infinity; in fact it never gets any nearer. It cannot even be likened to a hyperbolic curve which, although it never becomes a straight line, at least in theory, does, quite obviously, become increasingly straight, and therefore more like a straight line. On the other hand a finite object that seems to go on for ever must always be an infinite distance away from infinity.
Consider the following. Those intrepid adventurers Alice and Bob, in their respective space craft, are the only occupants of an infinite void. Each perceives her/him self to be at the centre of the void. There is an infinity of void extending in every direction. This sounds like being at the centre of a sphere, but it makes no sense to describe infinity as spherical. A sphere has, by the very fact of being a sphere, a boundary surface; infinity does not. However far you travel in infinity, you will never reach a boundary. A sphere has one fixed centre, but as the perceptions of Alice and Bob have already shown, there is no fixed centre to infinity, nor would it matter how far apart they were, they would still have infinity in every direction.
What changes when Alice or Bob moves? Any movement that either, or both, may make is movement relative to the other, not movement relative to the infinite void. Inevitably this leads to the conclusion that whatever movement you make in infinity makes no actual difference to your position in infinity. Before you move you are at the central point; after you move you are at the central point, but you were already there, so have you moved?
Before we leave Alice and Bob in the void, consider one more strange thing. Suppose that Alice and Bob are an infinite distance apart; if each moves, say, a billion light years towards the other, are they still an infinite distance apart? Intuitively, it might feel as though they should be two billion light years closer to each other, but that would mean that the distance between them was less than infinite. If the distance between them is less than infinite, it must be finite. This means that something that was infinite has become finite. Reverse their journey and something finite suddenly becomes infinite. This is impossible. It seems that if they are an infinite distance apart, however they may move, the distance between them will always remain infinite. Perhaps the logical conclusion to draw from this is that in infinity, distance has no relevance at all. This must lead us to ask if, in fact, Alice and Bob in infinity can actually move relative to each other, or if the concept of movement is simply transferred from our dimensions.
Whatever relevance distance might or might not have in infinity, it does seem that the occupants of infinity may be able to move relative to one another, but not relative to the infinite background. There is a ring of familiarity here; this scenario has a distinctly Einstienian feel. In 4-dimensional spacetime we can move relative to one another, but can identify no static background against which we can establish absolute motion or rest. Given that relative movement is, at least in theory, possible in infinity, we must look at the question of time, because movement is change and change requires time in which to happen. However, we should not lose sight of the fact that there is a major difference between spacetime, as we experience it, and infinity. In spacetime we can identify no static background against which we can measure motion, but having said that, there is no way we can know with certainty that it is not actually possible to move relative to spacetime. In infinity, it is infinity itself that seems to provide that background, but here we can establish that movement relative to that background is not possible.
It seems that all we are doing is asking ever more questions, without answering any. To attempt to answer at least some of these questions, we must look even more closely at the idea of infinity. For the sake of simplicity the term “infinity” will be used to cover “eternity” as well; in fact, the terms are interchangeable. In general usage eternity is simply “infinite” time, but there is a serious caveat here that must not be overlooked. Eternity is not an infinite expanse of time; in fact, it does not involve time, the two concepts are, incompatible. The concept of eternity as being in any way “temporal” arises from our limited ability to comprehend the nature of infinity. Like “Flatlanders” who cannot even imagine a third spatial dimension, we try to examine infinity using only our four dimensions of spacetime. Thus we find ourselves constantly falling back on temporal analogies and terminology, which are, at best, only of limited value, and, at worst, are downright unhelpful.
We might wonder if we could devise some sort of thought experiment to probe eternity and time as we have just done with infinity and space. Here, the task becomes much more difficult because as far as time is concerned we have only one observable dimension. What is worse, we seem constrained to move in that dimension only in one direction, and, for all practical purposes, at one speed. Of course, in a thought experiment, we can use perfect instruments and ideal conditions, so there is no bar to our travelling at sufficiently close to the speed of light for relativistic effects to become important. In other words, we can construct our experiment in such a way as to make time dilation significant.
If Alice could travel along the time dimension at close to the speed of light, relative to Bob, and then return, she would find she had aged less than Bob. Unfortunately, even in our idealised experimental conditions, there seems to be no way of reversing Alice’s travel along the time dimension. We might invoke closed time-like loops, but these come with problems of their own that might only serve to distract us from our consideration of infinity/eternity.
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One of the ideas that is central to my thinking is that if there had ever been a time when there was nothing, there would still be nothing now.
This is where closed or one-ended temporal systems can be considered. If time itself 'begins' when the universe or cosmos initiates (our language is inadequate for this kind of thing!), then there is never a time when there is nothing, yet it's debatable whether time is infinite in the past direction...
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Accepted! If time began with the Universe, then there was no time (unbounded or otherwise) before that. However, that is little more than a semantic device. It is no more a scientific proposition than saying "God created the Universe", so we can ask no more questions about its origin.
In spite of such eye-catching book titles as "A Universe From Nothing"; the "nothing" always seems to turn out to be something.
If there was nothing before the Universe; was there an infinity of nothing?
If not: what came before the nothing?
Why might it be preferable to imagine that there was nothing, rather than that there was a timeless cosmos?
Which of those is, logically, more likely to give rise to a universe with space and time?
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It is no more a scientific proposition than saying "God created the Universe", so we can ask no more questions about its origin.
True enough, although it has advantages over the God proposition, not least Occam's Razor.
But the same objection applies to your subsequent questions, none of which are scientific, yet you still ask them:If there was nothing before the Universe; was there an infinity of nothing?
If not: what came before the nothing?
Why might it be preferable to imagine that there was nothing, rather than that there was a timeless cosmos?
Which of those is, logically, more likely to give rise to a universe with space and time?
Incidentally, I'm not sure the idea of a 'timeless cosmos' has any useful meaning, but maybe there's a place for it in the maths...
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What's an "interaction"?
It is the mutual effect of two objects on each other, involving the transfer of energy between objects and/or fields.
It's not enough, if you want the term "interaction" to be synonimous of the term "measure" I have used in this context. See down.For example, when a beam of light is bent, without absorption, by a glass prism, does the beam "interact" with the glass or not? Explain why yes or why not.
Yes, it interacts. Considered as a wave, the frequency remains constant but the phase velocity is changed entering the glass; the refractive index of glass varies with frequency, so the change in phase velocity of the different light frequencies varies, resulting in the frequency dependent refraction & splitting of the beam. Considered as particles, the photons interact with the electrons in the glass, by scattering, absorption, and re-emission (see Feynman's 'QED', ch.3, p.107).
But if the light beam is made of single photons, a photon's wavefunction doesn't collapse after having passed through the glass prism, so that "interaction" is not a "measure" of the quantum state of the photon (in particular, of its frequency).
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What do you say about this Lightarrow? :)
See my answer to dlorde. What do you think about it?
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Incidentally, I'm not sure the idea of a 'timeless cosmos' has any useful meaning, but maybe there's a place for it in the maths...
Before addressing the question of any useful meaning for a timeless cosmos, or any place for it in the maths, consider Cantor's mathematical infinities. His countable and uncountable infinities found their places in the maths of the time, and have remained there. What is rarely considered is that he discovered that there existed an infinity of these infinities. Where does that fit into "the maths"?
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The Infinite Cosmos Part 3
We should return to the question as to how a finite cosmos might arise out of an infinite precursor without dividing that eternal precursor, and in so doing, dividing infinity. Three possible answers come readily to mind: (i) there is an infinite, intelligent, entity that can create a cosmos without undergoing any change in itself; (ii) the infinite precursor, although not an intelligent entity, was able to spawn a finite cosmos without undergoing change. One must then assume that this finite cosmos gave rise to our finite Universe; and, (iii) we live in an infinite cosmos, which somehow was unchanged by the “creation” of our finite Universe. Any of the above possible answers appears to involve a leap of faith, but we have to ask if any answer might be more logically likely than either of the others.
Theologians would undoubtedly opt for the first choice. An entity such as this lends itself to the concept of God, the creator, whose eternal realm is outside our cosmos, but who, with a little imaginative thought, could be woven into our world view. It is still tempting to argue that the act of creation would divide eternity into two “halves”, but this temptation may arise from our restricted view of the proposed spiritual realm. Because our only experience is of linear time, it is very difficult for us to achieve any real feeling for the nature of eternity. There may be some mystics who can do this; Fred Alan Wolfe would have us believe there are yogis who can; and he justifies this call to belief on the basis of quantum mechanics. Michael Talbot cites numerous examples of mystics and psychics who can experience a timeless, spaceless realm. Julian Barbour’s “Platonia”, without resorting to mystics and psychics, paints a picture of a timeless “world” in which all movement and change are illusions, establishing that scientists as well as mystics can have profound thoughts about eternity. In addition, as we reasoned above, movement and change seem to be possible in infinity, provided they are restricted to the frames of reference of individual occupants.
If one were formulating an argument regarding the possible existence and nature of any causal entity that might have been responsible for the “creation” of the cosmos, it could be tempting to start the argument from a consideration of the nature of the observable Universe. Indeed, many theist arguments have this starting point. It is easy to think that, because progressively fewer scientists seem to believe in God, any argument that relied on observations of the Universe would become progressively less powerful as time passed. Such a position would, undoubtedly, spring from a belief that arguments from the nature of the Universe would necessarily be based on the apparent order of the Universe, which many interpret as indicating intelligent design. However, order and apparent design are not the only arguments that might be made from observations of the Universe, and some of the other lines of reasoning could actually become stronger with time. For example; if one argued that there is life in the Universe, therefore its creative force must also include life; this could, a few decades ago, have been countered be the assertion that the Universe contained matter; therefore the creative entity must also have included matter. This would necessarily have cast considerable doubt on any spiritual or supernatural claims made about the “creator”, who, at least from the theist perspective, needed to be spiritual. To some extent, Einstein wrecked this counter argument by pointing out that matter and energy were interchangeable, and it seems easier to equate a spiritual being with pure energy than with any kind of matter. However, things have progressed even further since Einstein, and the counter argument is by no means as secure as it would have seemed before quantum physics raised serious questions about the reality of what we perceive as matter. Even more doubt is cast on the concrete nature of matter by the theory of the “holographic” Universe, a concept which, although it is still very much a minority view, cannot necessarily be ruled out.
If one acknowledges that the material world is not necessarily as solid as it seems to us, and if one accepts that an effect is unlikely to be greater than its cause, then it seems reasonable to argue that whatever is responsible for the existence of our Universe must contain the essential elements of our Universe in order to be able to “create” them. For example; the Universe contains life and intelligence, therefore the “cause” of the Universe must also contain life and intelligence. Naturally, there are many who would disagree with this line of reasoning and maintain that it is unscientific. However, that pillar of scientific reason, Richard Dawkins, appears to support this view.
One of the arguments that Dawkins uses to refute the belief that the Universe had a creator is that any being who could create something as complex as the Universe would, itself, have to be very complex, and would therefore have had to evolve. Apart from the fact that he seems to be ignoring the scientific concept of the “Boltzmann Brain”, he is, at worst, sidestepping the whole idea of eternity, and, at best, trying to apply Darwinian evolution to eternity. It is worth repeating that eternity is not an endless succession of time, it must be something quite different, and if there is one thing that evolution needs it is time.
Evolution provides us with the best, possibly the only, scientific explanation, not only for life on Earth, but for the existence of our Universe, as we perceive it. Unless and until science can provide us with a better explanation than the Big Bang for the origin of our Universe, then, within this Universe, evolution rules – OK. I am not suggesting that the evolution of the Universe from the initial singularity, or whatever the starting point might have been, was Darwinian “survival of the fittest”, but an evolutionary process can, nevertheless, be traced from the Big Bang to the present day. The important thing to remember about evolution is that it involves change, and change, as noted above, requires time.
Asserting that there is a creator who created the cosmos might seem to lead to an infinite regression situation in which the next question must be: Who created the creator? – and so on, ad infinitum. However, this regression can be brought to a halt at any point by invoking an eternal creator. In the spirit of William of Ockham, we would have to concede that nothing was achieved by protracting the list of creators beyond one.
Theists choose to call the ultimate cause of the Universe “God”, and God is interpreted in a multitude of ways by various religions, sects and philosophical schools. Atheists, on the other hand, look for explanations that do not involve God, or any intelligent “designer”, and tend to produce their own array of explanations. In spite of claims and counter-claims by both sides, it seems that either position is ultimately a question of “faith”. Perhaps this is why everyone hates an agnostic. The agnostic has seen through both sides, and, like the true scientist, is keeping all options open in case some hard evidence should be produced in favour of one side or the other.
Returning to our list of “possible answers”; Ockham’s razor might usefully be invoked in order to dispose of the second option, which contains an “entity” for which there is no apparent “necessity”. In reality, the second answer differs from the third only in form, and in the fact that it lets in the possibility of infinite regression.
Scientists might be more inclined to go for the third choice, particularly if they happen to be atheists. On the face of it, the third choice seems to involve less “faith” than does the first. However, an exploration of the concept of infinity, and how our apparently finite cosmos might be dovetailed into it, raises some interesting physical and philosophical questions.
The alternative to a created cosmos seems to be an infinite cosmos. Understanding this may require some thought. An infinite cosmos must contain everything that exists, or ever can exist. There can be nothing that is outside an infinite cosmos: no matter, energy, space or time; no potential and no uncertainty. Outside an infinite cosmos there can be no creation, because there can be no creator. Nothing can be added to an infinite cosmos, because all that is, or can be, is already included in it. Nothing can be taken away, because there is nowhere for it to go.
I guess this is a good place to make a break as there will probably be quite a few objections to this latest installment, so I'll need time to consider some possible answers. :)
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I go by the interpretation put forth by Feynman... there is a way to calculate the probability of something happening, (the path integral), but all else is asking questions that have no meaning, and any imagined answers will probably look as silly to future generations as the flat earth theory. By the way, 'QED, the Strange Theory of Light and Matter' is available as an ebook, with a new introduction that is relative to this discussion.
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But if the light beam is made of single photons, a photon's wavefunction doesn't collapse after having passed through the glass prism, so that "interaction" is not a "measure" of the quantum state of the photon (in particular, of its frequency).
If you read Feynman (QED pp.101 & 107), you'll see he explicitly describes the scattering interaction as the photon being absorbed by an electron and a new photon being emitted. If you prefer the wavefunction collapse interpretations, the absorbed photon's wavefunction clearly must collapse. There is a probability amplitude for photons to pass through the glass without interacting, but for the observed refraction, the scattered photons are also required.
As already mentioned, the frequency of the light doesn't change, but its phase velocity does (depending on frequency). The use of 'measure' in physics generally refers to an observation (collapsing the wavefunction if you like), but in QM, any interaction has this effect, so 'measure' is the subset of interaction that involves observation. That's all I was saying.
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... consider Cantor's mathematical infinities. His countable and uncountable infinities found their places in the maths of the time, and have remained there. What is rarely considered is that he discovered that there existed an infinity of these infinities. Where does that fit into "the maths"?
Cantor's transfinite numbers are esoteric and interesting, but I'm not sure what you mean by "Where does that fit into 'the maths'?" - the maths is the maths of transfinite numbers (wikipedia has a reasonable stab at it). When I said there may be a place in the maths for a timeless cosmos, I was referring to the maths used in the physics of cosmology; it was just a rhetorical statement.
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Yes Lightarrow :) To me it's a question of what 'reality' should be seen as. The first thing I would like to measure is if the photon would differ for passing that glass. If it won't, then that need a explanation, as I would from my first definition expect anything meeting another object to interact, especially if passing through it.
You can argue it a lot of ways, naively lift up the issue about a 'size' and ask yourself what the probability is for something without size to interact at all? That one you can use neutrons for too. But light is presumed to interact with matter. If it didn't we wouldn't be here.
And it will change angle passing that glass, depending on incoming direction. So it must interact to me unless I assume that light not to interact depending on angles, meaning a light beam 'hitting' a perfect transparent glass straight on then gets excluded from interaction. So, the question to me would be if it really can be proven that this photon in no way change momentum energy frequency passing that glass?
And that is in a way the exact same question as the one about a 'ideal elastic collision', aka light reflected from a perfect mirror.
You could also argue this way, presuming no measurable change, assuming all light quanta to be identical, then there was no glass represented from the definition of a photon interacting with matter. And that one might fit a idea of 'relations' and light non propagating. Because in such a definition we just look for a logic, we do not discuss what the logic implies when when it comes to questions about if the universe is closed or not. Neither do we ask ourselves what 'energy' really, really, is :) We just look at causality, experimenting on it, to then define relations.
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... I would from my first definition expect anything meeting another object to interact, especially if passing through it.
As already mentioned, there is an amplitude for photons to pass through the glass without interacting.
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Want to expand on that one dlorde?
Using my definition it either interact, meaning that it will differ, or it doesn't. If there is a measurable change then that should be a result from a interaction, and that includes everything that differ from how we define it to have behaved before, including its propagation.
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Using Feynman's many paths then I would say that that is a result from statistics, defining probabilities. That to me is the same concept as I described when discussing light as non propagating. We use logic to define it, and if that logic won't fit a macroscopic definition then it doesn't matter, as long as the logic makes sense and give us a prediction. From it you can, or you can't, 'prove' how it really should be, also depending on what experiments you can imagine up to define your hypothesis. And to me the most important things there should be the experiments and the logic, if that works then the theory can wait a little :) Because a theory is just as good as our preconceptions, and we all have such.
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But if the light beam is made of single photons, a photon's wavefunction doesn't collapse after having passed through the glass prism, so that "interaction" is not a "measure" of the quantum state of the photon (in particular, of its frequency).
If you read Feynman (QED pp.101 & 107), you'll see he explicitly describes the scattering interaction as the photon being absorbed by an electron and a new photon being emitted. If you prefer the wavefunction collapse interpretations, the absorbed photon's wavefunction clearly must collapse. There is a probability amplitude for photons to pass through the glass without interacting, but for the observed refraction, the scattered photons are also required.
As already mentioned, the frequency of the light doesn't change, but its phase velocity does (depending on frequency). The use of 'measure' in physics generally refers to an observation (collapsing the wavefunction if you like), but in QM, any interaction has this effect, so 'measure' is the subset of interaction that involves observation. That's all I was saying.
I don't have Feynmann's QED available in this moment, so don't know what he means, but let me contest your interpretation. If that were a qm measure, why you can't say which is the photon's energy after coming out of the prism?
(As you know, infact, the photon's wavefunction is still in the same superposition of frequencies which had the photon before entering the prism).
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lightarrow
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Yes Lightarrow :) To me it's a question of what 'reality' should be seen as. The first thing I would like to measure is if the photon would differ for passing that glass. If it won't, then that need a explanation, as I would from my first definition expect anything meeting another object to interact, especially if passing through it.
The photons which enters the prism is not the same as the one who comes out, but in a very subtle way; the photon which comes out has now frequency entangled with its direction: if the detector detects the photon at a specific angle, then its energy is specific (larger angles = larger frequencies) but *you can't say the photon's energy before detecting the angle of arrival of the photon*, so its wavefunction hasn't collapsed after coming out of the prism.
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Bill S,
Thanks for the arguments. It seems that until we can understand infinity (and I mean understand, not describe), we'll always be like ants thinking that our bit of the nest is everything,
As a scientist myself, I can understand the frustration this may bring. At the age of eight, the annoyance of being told by schoolmates that the highest number in the world can be beaten by the highest number plus one is still a vivid memory. Cantor's work was impressive but it reminds me of the school playground again.
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Bill S,
Oh, and another thing.
If the cosmos or multiverses are infinite, surely there must be a infinite amount of information/knowledge. So our partial knowledge will always be zero. Yes, I know you can't divide by infinity but you know what I mean. So I know as much now as I did when I was eight.
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So our partial knowledge will always be zero.
Mathematically our partial knowledge would be zero, but there must be another way to look at it, because if you apply the same reasoning to matter in an infinite cosmos, then our "share" of that matter must be zero - yet we are here.
My protracted ramblings should - I hope - reach that alternative, I just hope others have the patience to stay with the animadversions of an old codger long enough. :)
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Want to expand on that one dlorde?
Sure, you can read it in Feynman's own words HERE (http://www.scribd.com/doc/7119268/2physics-Qed-Feynman-Qed-the-Strange-Theory-of-Light-and-Matter-Princeton-University-Pre) (page 107). If you need to get up to speed on his summing of probability amplitude arrows approach, start from the beginning of Chapter 3, 'Electrons and their Interactions' (p.87). He explains it more clearly and precisely than I ever could.
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If that were a qm measure, why you can't say which is the photon's energy after coming out of the prism?
If it was a measurement, you could know. Like I said, measurements are the subset of interactions where an observer is involved. All measurements are interactions, but not vice-versa. If an interaction occurs unobserved, you're not going to know the energy. If you arranged things so that you captured the scattered photons, you could measure their energy.
As you know, infact, the photon's wavefunction is still in the same superposition of frequencies which had the photon before entering the prism).
The way I read it, the wavefunction describes the quantum state of a particle. Feynman explicitly says (with italicised emphasis) that the scattered photon is a new photon, which means a new particle wavefunction (with the same frequency probability amplitudes as the incoming photon, but different in some other respects). From the point of view of the system of including incoming photon, electron and scattered photon, it's all part of the same evolving complex wavefunction that describes that system. Whether the wavefunction of a particle that persists through the interaction, e.g. the electron, collapses at the interaction would depend on your interpretation - go with Wigner and it doesn't collapse until a conscious observer (e.g. 'Wigner's Friend') 'observes' it; go with Objective Collapse interpretations, and it collapses when the system superpositions reach a certain complexity or size, etc.; go with Many Worlds and it never collapses, it just looks that way to an observer.
It's worth noting that Feynman doesn't mention the collapse of the wavefunction in his discussion of refraction, he deals only with the probability amplitudes of various actions. He'd probably say the interpretation doesn't matter if you can work out what happens without it (and you can).
I'm happy to take corrections and adjustments to my description if they correspond to my understanding of Feynman's description, or explain where my understanding of Feynman's description falls short.
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If it was a measurement, you could know. Like I said, measurements are the subset of interactions where an observer is involved.
I have some problems with the notion of an observer being involved. One has to define "observer" and I'm sure that we can all agree that the universe existed before observers where here and that life existed before it knew how to make an observation.
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I had the same problems Pete, in the end I found the best way to be to think of it as relations, and define them as 'observing' each other. That way you can add whatever you like to it, as long as it introduce something new, be it a measurable change or just something changing the relations I thought I knew before finding it. And a consciousness it just one more relation as I think, slightly differing in that it discuss 'free will', so becoming 'indeterministic' to me. Although that one is discuss-able :) depending on how you define chaos, probabilities, and randomness.
As well as what we find to be statistics naturally. Without statistics existing, and provable, to give us the logic, binding a past to a present, enabling us to predict a future, I wouldn't expect us to find a logic order (causality) to anything. Although maybe there is some other way to define it? But I still expect statistics to be the ground for defining it otherwise. What I mean is that we might not have thought about it this way in Newtonian society, as that was a 'clock work' universe presumed to be 'finite', but behind that we now find statistics, as I think :)
But everything is parameters, and it will be you that define them, depending on what you find yourself knowing at that time.
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I have some problems with the notion of an observer being involved. One has to define "observer" and I'm sure that we can all agree that the universe existed before observers where here and that life existed before it knew how to make an observation.
That's my point; interactions happen regardless of the existence of observers (which usually refers to consciousness, even if there is an intermediary device). There's no reason for special pleading for consciousness. So a measurement is just an interaction that has meaning for an observer. If interactions collapse the wavefunction, then a measurement will do so; if interactions don't collapse the wavefunction, then neither will a measurement. Or so it seems to me.
All assuming 'measurement' doesn't have some other meaning I'm unaware of.
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I would like to throw a thought into the ring to see if it is in any way new. To many, the idea of a cat being both alive and dead at the same time until it's observed by someone is a step too far, and this is because if a human observer is able to force a collapse of the wavefunction, a cat should really be able to do likewise. But what is it about us (and cats) that could drive this collapse? I reckon the answer is that both contain information systems, and trying to maintain highly complex information in multiple states may be more difficult than maintaining mountains of material in multiple states, so if the model in the brain is forced to simplify and take up a specific form, that would force the external reality to simplify too to remain compatible with the data. So, it isn't measurement that forces a collapse, but the integration of the resulting data into an information system which will then apply complex processing to it.
It may really be that when we look out into the universe through a telescope, we can potentially force whole uninhabited galaxies to throw off most of their possible states so that they can appear to us in a particular, specific form rather than a fuzzy mess of multiple possibilities. This would not result in any causality travelling back billions of years through time though, because it would only force that galaxy to take up a specific form now, while it's entire past history up to that point would remain fuzzy. The first complex observer to look at it would force a collapse, and that collapse would be transmitted throughout the universe in an instant such that no other observer could force an incompatible collapse of the wavefunction of the same object.
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The first complex observer to look at it would force a collapse, and that collapse would be transmitted throughout the universe in an instant.....
That's going to need some serious thought.
The first thing that comes to mind is that you have linked the two parts of this thread. Your instantaneous transmission could happen only if every part of the Universe were in contact with every other part.
Come back Bohm, all is forgiven! :)
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... what is it about us (and cats) that could drive this collapse?[/quote I reckon the answer is that both contain information systems, and trying to maintain highly complex information in multiple states may be more difficult than maintaining mountains of material in multiple states, so if the model in the brain is forced to simplify and take up a specific form, that would force the external reality to simplify too to remain compatible with the data. So, it isn't measurement that forces a collapse, but the integration of the resulting data into an information system which will then apply complex processing to it.
This sounds like an Objective Collapse interpretation, using complexity as the trigger. The problem I have with these interpretations is their arbitrariness. At one extreme, it reduces to Wigner's interpretation, i.e. only the complexity of a human consciousness will collapse it, and at the other extreme, it reduces to interaction collapse, i.e. any particle interaction is sufficiently complex to collapse it.
What's missing is some explanation of why (information) complexity is relevant (why not mass, or particle count, or number of interactions, ...?), and why it becomes critical at some arbitrary level. Could a mouse collapse it? a pidgeon? frog? ant? amoeba? and what about a non-biological information processing system, a PC?, IBMs 'Watson'? the internet? Where do you draw the line, and why?
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... Your instantaneous transmission could happen only if every part of the Universe were in contact with every other part.
That would be OK if it was a case of primordial entanglement (e.g. originating at the big bang). The problem I have with it is that it smells of special pleading for consciousness. I'm wondering quite how it would work in practice; would the first creature of sufficient information processing capability cause collapse as soon as the first photon from a distant galaxy hit its retina, or would there be a delay until the resulting signal had been through the visual cortex? Would it take multiple photons? how many, how much processing? would some early hominin look up at the sky at night, see a distant galaxy as a barely visible dot and collapse its wavefunction without even knowing what it was? And why, in David's example, if the universe was superposed this way, would only the one galaxy wavefunction collapse, wouldn't it be entangled with the rest of the universe? and what about intelligent life elsewhere in the universe - had the first arrivals already collapsed the universal wavefunction millions of years before we arrived on the scene?
It just doesn't smell right to me.
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What's missing is some explanation of why (information) complexity is relevant (why not mass, or particle count, or number of interactions, ...?), and why it becomes critical at some arbitrary level. Could a mouse collapse it? a pidgeon? frog? ant? amoeba? and what about a non-biological information processing system, a PC?, IBMs 'Watson'? the internet? Where do you draw the line, and why?
Maybe, as I wrote, it's not exactly a matter of "complexity" but of irreversibility / loss of coherence (which is related to complexity but not the same thing).
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Maybe, as I wrote, it's not exactly a matter of "complexity" but of irreversibility / loss of coherence (which is related to complexity but not the same thing).
Well yes, but that's just restating wavefunction collapse. Loss of coherence == decoherence (http://en.wikipedia.org/wiki/Decoherence). Decoherence is what the observer sees as wavefunction collapse.
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Well, I wasn't really suggesting that a galaxy could maintain a form that never simplifies through any collapses of wavefunctions until an intelligence finally gets round to looking at it after several billion years, but what I actually have in mind is that things can maybe maintain a certain amount of superpositions until it reaches a point where it's too hard to maintain them all, at which point some kind of simplification must occur. The galaxy will therefore repeatedly simplify itself as it becomes too hard to maintain all the superpositions, but every time it does so it will immediately start to generate new superpositions again which will in turn collapse when they become too complicated to maintain. Having a complex data system analyse the situation would merely hasten a point of collapse by increasing the complexity of the system as a whole.
It's easy enough for a single bit of data to be both a zero and a one at the same time, but to try to maintain that for billions of bits and with a program which must simultaneously run along trillions of different paths to process tham is not going to be at all easy. What we'd need to test this idea though is a way to measure the total amount of complexity involved in order to see if there is some consistent level where a collapse of the wavefunction becomes more likely than not.
I envisage real material as being outside the universe and merely contacting with it at a multiplicity of points, a bit like a spider with many legs hanging onto a web. Outside of the universe where the spiders reside there is no speed limit of c, but the movement of all the points of contact with the web are limited by c. Each leg continually multiplies into many new legs, following the waves in the web and maintaining an external, instant communication system between all these points. When the wavefunction has to collapse due to complexity, the spider simply lets go of the web with many of its legs and absorbs them back into itself.
This means that when we send a photon through a double slit, the photon starts out as a single leg of a spider and immediately multiplies into many legs as the wave spreads out. Some of the legs go through one slit, while some go through the other, and a lot of waves hit the gap in between or the surrounds. Those that continue on through keep multiplying and radiate out from the slits, interfering with each other and ending up hitting the screen in an interference pattern, but for a photon to land on the screen, all the energy has to be sent to a single point. The spider determines which leg the full energy of the photon will be transferred to and the rest of the legs simply let go of the web. Alternatively, several of the legs remain attached to the web and each of them transfers the photon provisionally so that a superposition of different possible realities is to be maintained until some later complexity forces a further simplification.
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Now that the two themes in this thread seem to have come together, I shall have to go back and do some re-reading, but first; a bit more of the ongoing saga.
The Infinite Cosmos Part 4
John Wheeler said that “Time is nature's way to keep everything from happening all at once”. This may sound like a flippant comment, but it is in fact quite a profound observation. We might say that eternity is the absence of time, and that in eternity everything must happen at once. However, even that statement is misleading: in order for something to happen there must be some passage of time. In eternity, everything just is.
Whatever one can do with mathematical infinities, it seems inescapable that any physical infinity must be immutable. The corollary of this is so important it is worth repeating. An infinite cosmos cannot be multiplied nor divided. It can have nothing added to it, because there is nothing outside it that could be added. It can have nothing taken away, because to take something away would either make it less than infinite, or it would mean that there was something other than the all-embracing infinity, which would constitute a contradiction in terms.
Even Cantor recognised that the absolutely infinite differed from his other "infinities". He is said to have equated the it with God.
The observable Universe, as we have seen, appears to have started its existence at a specific point, and must therefore be finite. It is important not to think of the Big Bang as having happened at a particular point in time, or at a particular location in space. Many cosmologists assure us that time and space were created with the Universe. However, the Big Bang has to be seen as a pivotal point in the history of the Universe. Given that there can never have been a time when there was nothing, it follows that there must be more to our Universe than meets the eye. For convenience we will call this extra something the “cosmos”, and will, at this point, not be diverted into considering whether that might be a “multiverse” or simply some sort of vacuum energy state from which the Universe appeared in accordance with the “rules” of quantum uncertainty.
If we were able to divide infinity, for example, by two, what would we be left with? One possibility seems to be that we would have two halves of infinity. Each half would be less than infinite, thus it would be measurable. Measure this quantity and multiply it by two and we have a measure of infinity, which is nonsense. The second possibility must be that each “half” somehow becomes infinite. Mathematically this seems reasonable; after all we can multiply or divide zero by any number we choose, and the outcome will be zero. Perhaps we also could do this, mathematically, with infinity. Consider Cantor's infinities: the whole numbers constitute an infinite series, so do the even numbers and the odd numbers. Thus, Cantor demonstrated that, not only were there numerous infinities, but they were not all the same size. It is evident that the infinity containing the even, or odd, numbers must be half the size of the infinity containing the whole numbers. Could it be that question is answered, that we can divide infinity and that any parts into which we divide it will be infinite? There seem to be at least two reasons why this cannot be the case. The first is that even Cantor does not seem to have performed mathematical calculations with the infinite set of all infinities; this appears to be the only one of his infinities that is not actually a mathematical infinity. The other is that, practically there is the complication that anything that is truly infinite must contain everything; there cannot be two infinities, because each would have to contain the other.
Applying the Reflection Principle to the infinite set of all infinities would lead to the following contradiction: The reflection principle holds that within a universal set, containing all sets, it must be possible to find a set that contains any property found in the universal set. The obvious contradiction is that the universal set contains all other sets (that is one of its properties), but this property cannot be found in any of the other sets.
Wikipedia says: " In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite."
Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers, whereas an uncountable infinity cannot. Useful as these concepts may be to the mathematician, none is an "absolute" infinity, and cannot therefore be considered as more than "unbounded".
Wikipedia says: "Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite."
Perhaps "transfinite" would be a less confusing term to use for mathematical infinities; then "infinite" could be reserved for what Cantor referred to as "absolutely infinite". This latter term has about it no less an air of tautology than does, for example, "absolutely perfect".
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Well, define it as all observe all :)
Then use 'c' to define the 'speed' by which we see action and reaction, in between 'observers' normally.
And leave quantum logic to the scale where it belongs. You might use decoherence for defining where it 'disappear' possibly? Doing so you get 'two' universes as I think, or two descriptions of one theoretical, co-existing. And what differ them is the scale you use. It's not too hard describing two planets macroscopically, or the earth and the moon orbiting. But try to do the same quantum mechanically, taking into account all possible interactions.
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Or we are walking on the edge of infinity, scale-wise :)
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I'm starting to look at it as a projection from infinity, and there we have scales, pointing us home. And that is where the men in white coats will smile..
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Maybe, as I wrote, it's not exactly a matter of "complexity" but of irreversibility / loss of coherence (which is related to complexity but not the same thing).
Well yes, but that's just restating wavefunction collapse.
In my opinion, not, because it would provide a (generic) model for the collapse.
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...what I actually have in mind is that things can maybe maintain a certain amount of superpositions until it reaches a point where it's too hard to maintain them all, at which point some kind of simplification must occur.
OK; that sounds like an Objective Collapse model, which is fine as far as it goes, but for me, it needs some meat on its bones to reduce its arbitrariness.
What we'd need to test this idea though is a way to measure the total amount of complexity involved in order to see if there is some consistent level where a collapse of the wavefunction becomes more likely than not.
The problem here is that we only become aware of wavefunction collapse when we measure/observe the system, and increasing complexity means more interactions, which makes the complexity collapse model increasingly indistinguishable from the interaction collapse model.
I envisage real material as being outside the universe and merely contacting with it at a multiplicity of points, a bit like a spider with many legs hanging onto a web. Outside of the universe where the spiders reside there is no speed limit of c, but the movement of all the points of contact with the web are limited by c. Each leg continually multiplies into many new legs, following the waves in the web and maintaining an external, instant communication system between all these points. When the wavefunction has to collapse due to complexity, the spider simply lets go of the web with many of its legs and absorbs them back into itself.
Yes, I've thought about entangled particles being locally connected in a higher spatial dimension, which is fundamentally not so different from your spider, but I don't know whether this would be covered by locality in a dimensionally extended version of relativity, and I don't have the maths to find out. It's the kind of thing a physicist would think of, and I've not heard it proposed as a solution, so I'm guessing that either it is covered by locality, in which case Bell theorem invalidates it, or it just doesn't work.
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Well yes, but that's just restating wavefunction collapse.
In my opinion, not, because it would provide a (generic) model for the collapse.
But since wavefunction collapse is decoherence observed, saying that wavefunction collapse may be a matter of decoherence is not a particularly useful generic model.
Perhaps I've missed something - can you clarify?
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What we really need to do is find experiments that allow us to push QM to the limits of how far it can sustain superpositions. I've found one that might fit the bill: http://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser (http://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser). The path to D0 can be kept very short while the other path to the D1, D2, D3, D4 cluster can be lengthened without limit, and the key point here is that it's what happens at this far end that dictates what "happened" a moment earlier at the D0 end. If the longer path can be stretched out to a year, you can appear to have backwards causation in time going back for a year. You can also bend the long path back on itself such that both ends are physically located right next to each other in the same lab.
Clearly it would be hard to stretch the path out to a year as the light would need to travel a whole lightyear to cover that distance, but a minute would probably be more than adequate, and a second might suffice. Even so, that's still going to be a long path. It may also be possible to slow down the light - I've read of materials in which it can be slowed down by 90% plus and even halted, so this could maybe enable very long time delays in a small space, hopefully without destroying the entanglements.
So, we have a setup in which there is a long delay between a future cause and its past effect, but I don't think there's really any backwards-in-time causation: what will actually happen is that the data received at D0 will be maintained in an state of superposition after it's been measured, and when the measurements are made later on at the far end, those states of the data can then simplify to remove the superpositions. However, if we take the data immediately after it's been taken generated at D0 and use it in complex calculations, we could maybe do something sufficiently complex with it to force it to lose its superpositions early, with the result that no interference patterns would be observed at the far end.
Addition to this post:-
To clarify a key feature of the experiment which I linked to, if you remove the beam splitter BSc, the interference patterns disappear at both ends of the experiment, so you can sit in a lab with the long path doubled back on itself (let's say a hour long) such that you can remove BSc and put it back in again and thereby dictate what happened at D0 an hour earlier.
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In my opinion, not, because it would provide a (generic) model for the collapse.
But since wavefunction collapse is decoherence observed, saying that wavefunction collapse may be a matter of decoherence is not a particularly useful generic model.
Perhaps I've missed something - can you clarify?
I added the concept of irreversibility, which is certainly far from being clarified in qm, but which is not simply decoherence.
But I admit I was quite criptic about it.
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I added the concept of irreversibility, which is certainly far from being clarified in qm, but which is not simply decoherence.
Ah, OK... so in what sense might the collapse of the wavefunction be a matter of irreversibility? irreversibility of what?
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What I've never understood is why the particle or wave question is linked to the actual act of choosing to observe or not observe and isn't a result of the system of measurement used to observe.
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What I've never understood is why the particle or wave question is linked to the actual act of choosing to observe or not observe and isn't a result of the system of measurement used to observe.
I don't quite follow you; whether you get particle or wave behaviour does depend on your measurement setup. The object itself has the properties of both a particle and a wave in some strange way (sometimes called a 'wavicle').
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I added the concept of irreversibility, which is certainly far from being clarified in qm, but which is not simply decoherence.
Ah, OK... so in what sense might the collapse of the wavefunction be a matter of irreversibility? irreversibility of what?
Making a measure means making a macroscopic registration of an event. Imagine a single photon hitting a fotomultiplier: something happens inside the macroscopic bulk of photo-sensitive metal, which then releases an electron, which then hits another electrode which releases 2 electrons and so on until a macroscopic current can be detected. I don't know what happens exactly, but certainly all the process is irreversible.
If, instead, a single photon hits a single atom and excites it, this proces is reversible. Maybe from the microscopic --> macroscopic some process becomes irreversible.
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... Imagine a single photon hitting a fotomultiplier: something happens inside the macroscopic bulk of photo-sensitive metal, which then releases an electron, which then hits another electrode which releases 2 electrons and so on until a macroscopic current can be detected. I don't know what happens exactly, but certainly all the process is irreversible.
If, instead, a single photon hits a single atom and excites it, this proces is reversible. Maybe from the microscopic --> macroscopic some process becomes irreversible.
OK. That sounds to me like statistical thermodynamics; all the underlying interactions are reversible, but entropy increases because disordered states are more likely than ordered states, hence the arrow of time, and macro irreversibility...
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You're perfectly correct Cheryl, although dlorde is too :) (a born diplomat, that's me)
As long as you add yourself as a part of the experiment, your observation of the setup defining the outcome, I hope we can agree on it. Remember that question if a tree falls in the wood, did it? If no one is there to see it? We are the ones observing outcomes, and our observations is what define them. You can use a lot of intricate logic and that way question a lot of outcomes. But in the end I expect it to come down to if a universe can be expected to 'work' without us, or not? I think it can, and in that way you should be right in that circumstances define outcomes.
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either my spelling, or my fingers, sux :)
Corrected though.
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OK, go for it :)
Some people just don't know when to run for the hills. :)
Actually, this thread has given me the impetus to pull together some of the scattered notes I have made over a few years, so I guess it's done me some good.
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The Infinite Cosmos Part 5
The apparently intrinsic indivisibility of infinity leads one to wonder if any “part” of infinity can be distinct from any other “part”. Is it in any way meaningful to talk of parts of infinity? If it is not, and if our Universe is “part” of this infinite cosmos, then we seem to have a problem. However, the problem may not be as difficult to solve as it at first appears. Consider the following possibility. The cosmos is infinite; therefore every part of the cosmos is the cosmos. Everything, including our apparently finite Universe, is infinite. The birth of the Universe and perhaps its ultimate death exist together in infinity, along with all the things that “happen” between those two points. It is all there, in eternity, in an all-embracing now. We perceive spatial differences, and the passage of time, because our minds need to make sense of the partial image to which we are restricted. This sounds like a recipe for predestination, but I am not suggesting that we should abdicate all responsibility for our actions; far from it. In eternity, things are as they are, permanently. However, we cannot entirely rule out the possibility that they are as they are, to some extent, because of the choices we seem to be making now. Without time "to keep everything from happening all at once", our familiar concept of causality needs some re-thinking. We cannot hope to stretch our finite understanding around infinity and eternity. Only if and when we realise the full potential of our oneness with everything in the cosmos could we hope to do that. In the meantime, we can reason that the finiteness and change that we perceive occur only in our finite frames of reference, beyond which there is no change, and therefore no passage of, or through, time because everything just “is”.
Julian Barbour writes of a realm – “Platonia” – in which movement and the apparent passage of time are illusions resulting from the way in which our minds interpret what he describes as a series of “snapshots” of a static, timeless cosmos. It is a short step, if indeed it is a step at all, from Platonia to an infinite cosmos. However, it would have to be acknowledged that in a truly infinite cosmos, the process of interpreting the snapshots, and the sequence of those snapshots, would also be illusions. In a truly infinite realm every snapshot is every other snapshot; they exist together with no semblance of order or chronology. We cannot move from one to another because there is no sequence and no passage of time in which to move. If we live in an infinite cosmos, then every change we perceive, every movement we detect and every second that “ticks past” on our clocks must be an illusion. Illusion is perhaps not the best word to use because if something exists in our reality, it is real for us. In no way am I denying the reality of our Universe. I am simply saying that our reality may not be "absolute" reality.
What is, perhaps, even harder to comprehend is that, because no “part” of infinity can be distinct from any other “part”, then, any distinction which we, as individuals, perceive between ourselves and other individuals – past or present, living or dead – must also be an illusion, as must be the apparent distinction between ourselves and other creatures and objects. Our individuality, our personality, that which we recognise as “I”, can be no more than an illusion created so that we may make sense of the limited perception which we have of “reality”.
Perhaps relativity is a more all pervading concept than we might imagine. It has to be possible that, not only are time and space relative within our perception, but also that time and space exist only in our current frame of reference. They are “real” within that frame of reference, but may be completely different, even non-existent, in another. In fact, this is the way in which we have to look at the things I have just referred to as illusions. It cannot be denied that they are real in our frames of reference, but reality is relative.
In an infinite cosmos existence is infinite; everything that exists shares that same existence. Each thing is everything. There are no divisions or distinctions, only an all-pervading oneness. Michael Talbot says this of the work of physicist David Bohm: “As he looked more carefully into the meaning of the quantum potential he discovered it had a number of features that implied an even more radical departure from orthodox thinking. One was the importance of wholeness. Classical science had always viewed the state of a system as a whole as merely the result of the interaction of its parts. However, quantum potential stood this view on its ear and indicated that the behavior of the parts was actually organized by the whole. This not only took Bohr’s assertion that subatomic particles are not independent ‘things’, but are part of an indivisible system one step further, but even suggested that wholeness was in some ways the more primary reality.” This fits well with the idea that infinity is the primary reality, and that our seemingly finite existence is a mere shadow of that reality
What happens if we apply this reasoning to life, as, of course, we must? If life is infinite, then it must be possessed by every “aspect” of the cosmos, whether or not we perceive it as being alive. Talbot (1996) again refers to Bohm: “…he believes that dividing the universe up into living and nonliving things also has no meaning. Animate and inanimate matter are inseparably interwoven, and life, too, is enfolded throughout the totality of the universe. Even a rock is in some sense alive, says Bohm, for life and intelligence are present not only in all of matter, but in ‘energy,’ ‘space,’ ‘time,’ ‘the fabric of the entire universe,’ and everything else we abstract out of the holomovement and mistakenly view as separate things”.
If all this is right, it might be tempting to say: “We are the cosmos”. However, the term “we” implies distinction, therefore it would be more correct to say: “I am the cosmos”, recognising, of course, that every “I” in the cosmos can rightly make the same claim. Talbot brings together two quotes from Whitman’s “Mystical Life” when he talks of “…feeling that ‘everything is everything’ and ‘I am that’.”
Can we talk about dimensions in infinity? Apparently some scientists feel that we can. Some of the current cosmological theories relating to dimensions suggest that our Universe has more than four dimensions, but that we are able to detect only three of space and one of time. Among the explanations offered for the fact that we are not aware of these other spatial dimensions is that they might be rolled up so tightly that our instruments cannot detect them. It is even claimed that these dimensions might be rolled so tightly as to be infinitely small. I have yet to find a definition of infinitely small that is able to distinguish it from non-existent. However, that is another matter. An alternative concept – the one that lets in the idea of dimensions of infinity – is that ours is a four dimensional Universe embedded in a higher dimensional cosmos, and that this cosmos might have infinite (sometimes stated as an infinite number of) dimensions.
Any attempt to calculate the dimensions of infinity must be a matter of conjecture. The most logical assumptions would seem to be that it might have infinite dimensions; or, perhaps, one infinite dimension. The latter possibility seems the more likely; because, if infinity had more than one dimension, each of the dimensions would have to be all of the others, and one has to wonder how this would differ from having just one dimension. The possibility must also be considered that dimensions are features of finite, temporal realms, and cannot be applied to infinity. Thus, infinity would be timeless and dimensionless. Those who have read Edwin Abbott’s “flatland” will undoubtedly notice a similarity between this interpretation of infinity and “Pointland”; although, it would have to be said that the occupant of “Pointland” was heard to be talking, which would indicate that “Pointland” must not have been timeless.
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I'm afraid I found that post confused, confusing, and somewhat incoherent. Confusion between a subjective and an objective view, unsubstantiated assertions, and leaps to unjustified conclusions...
For example:
The apparently intrinsic indivisibility of infinity leads one to wonder if any “part” of infinity can be distinct from any other “part”. Is it in any way meaningful to talk of parts of infinity?
Why does it seem apparently intrinsically indivisible to you? It seems to me that infinity is divisible into any number of parts, including an infinite number of parts. In any division, there will be at least two infinite parts. Consider a road that stretches away from you to infinity in either direction. You can paint a line across it and divide it into two infinite lengths, then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction.
It is all there, in eternity, in an all-embracing now.
It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.
This sounds like a recipe for predestination, but I am not suggesting that we should abdicate all responsibility for our actions; far from it. In eternity, things are as they are, permanently. However, we cannot entirely rule out the possibility that they are as they are, to some extent, because of the choices we seem to be making now.
These are two sides of the same coin in a 4D Parminidean block universe. Part of the future is dependent on the 'choices' we make now, but those choices are deterministic events like all others, from the 4D viewpoint. That we see them subjectively as free choices is a reflection of our ignorance of all the deterministic influences involved, including the processes in our own brains (and, of course, we can't see the future). Even in a deterministic universe we will act as if we have free will - we have no choice ;D
In a truly infinite realm every snapshot is every other snapshot; they exist together with no semblance of order or chronology.
Chronology is temporal order. But I don't see how it follows that infinity must be disordered. Consider the integers - a truly infinite extent of numbers in the plus and minus directions, and definitively ordered. All the evidence suggests that there is an ordering to time, at least at a macro scale; causality, statistical thermodynamics. There may be uncertainties at the quantum scale (see what I did there?), but they generally don't affect our subjective experience of chronology. The logical implication of your suggestion is that if we have chronology in this universe (which we appear to), it can't be infinite... I'm not averse to it being finite, but it ought to be for some coherent reason.
If we live in an infinite cosmos, then every change we perceive, every movement we detect and every second that “ticks past” on our clocks must be an illusion. Illusion is perhaps not the best word to use because if something exists in our reality, it is real for us.
So what are you saying? It isn't an illusion? And surely saying 'if something exists in our reality, it is real for us' is tautologous - what does it mean? what is 'our reality' but what is real for us?
no “part” of infinity can be distinct from any other “part”
Why? what makes you think so?
In no way am I denying the reality of our Universe. I am simply saying that our reality may not be "absolute" reality.
If reality is what is real to us, what is 'absolute reality'? what do you mean by it?
Perhaps relativity is a more all pervading concept than we might imagine. It has to be possible that, not only are time and space relative within our perception, but also that time and space exist only in our current frame of reference. They are “real” within that frame of reference, but may be completely different, even non-existent, in another.
Relativity is totally pervasive. Time and space are different for every observer. You could even say that from the 'point of view' of a particle travelling at light speed, there is no time or space (though how useful that would be isn't clear to me ;) ).
What happens if we apply this reasoning to life, as, of course, we must?
Why?
If life is infinite, then it must be possessed by every “aspect” of the cosmos, whether or not we perceive it as being alive.
What does 'if life is infinite' mean? and why does it imply that it must be a property of every aspect of the cosmos? Two different things can be infinite without one necessarily being an attribute of the other.
if infinity had more than one dimension, each of the dimensions would have to be all of the others...
Why?
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Dlorde, thanks for your response. One of the great things about being able to share ideas is that others usually think of things in a different way and add new perspectives.
Sorry you found the last post "confused and confusing". I've just had a look back through it, and I agree on both counts, although much less so if it is taken in conjunction with preceding posts.
As I mentioned, these posts are pulled together from scattered notes. It would undoubtedly have been better if I had spent more time organising them, but time is a bit short. There should be one more "Part" to come, which may touch on some of your questions, but I will address each of them separately, anyway.
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I was wondering if anyone here has read The Philosophy of Quantum Mechanics: The Interpretations of Quantum Mechanics in Historical Perspective by Max Jammer (Dec 3, 1974)? I was told my a physics historian friend of mine that it's an excellant book on the philosophy of quantum mechanics. If you're really interested in this subject then this is a must read on your list of books to consider buying and reading.
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Looks interesting; the question is how to find a copy...
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There's one on Amazon UK at a little over £790! I think I might try the local Library.
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The Infinite Cosmos Part 6 (Final Part)
What happens in infinity? Popular science books often point out that, in eternity, everything that can happen, must happen. Few, if any, take this to its logical conclusion. Because there can be no succession of events in eternity, everything that can happen, and must happen, must be now. Even the assertion that it must be happening now is misleading, because, to say that something is happening implies that it is undergoing a process of occurring, which requires a passage of time.
Consider events A, B and C. In linear time these might occur, one after the other, in that order. In eternity, though, they would all be present together. There could not have been a point in eternity when, for example, A had happened, but not B or C. The whole of eternity must contain A, B and C, in their entirety, for all eternity.
If the past - the period up to now - is infinite, then A, for example, has already happened, an infinite number of times, as have B and C. What sense does it make to claim that, because they occur in the sequence: A- B- C, there must have been a point at which A had happened more times than B or C?
If that were the case, it must be that if we had selected a point in the past as our "now", there would have been a chance that we opted for a point at which A had happened more times than B or C. This would mean that they had not happened an infinite number of times. Manifestly, this is not possible if the past is infinite, and if in infinity everything that can happen must happen an infinite number of times.
It is not possible to envisage a “part” of eternity that does not contain everything that is contained in eternity. I have been using the term “eternity” rather than “infinity” in order to stress the “everlasting” aspect, but we must not lose sight of the fact that it is only our time- and space-bound perception that persuades us that we should distinguish between space-like and time-like infinities. What applies to eternity applies equally to infinity.
We started by trying to apply rational thought to the origin of the cosmos and the position that our Universe might occupy within that cosmos. We are still left with one fundamental question: Is there any way in which we can work out whether our Universe was created, or whether it is simply part of an eternal cosmos? First, we have to ask if the cosmos was created, or if it is eternal.
If the cosmos was created, it must form part of the infinite realm of the creator. As discussed above, this implies that it has always been part of that realm; it, too, is eternal.
If the cosmos is eternal, and the Universe forms part of the cosmos, it follows that the Universe has always been part of the cosmos; the Universe is also eternal.
We seem to have reached a juncture at which we are saying that arguing about whether or not the Universe was created is totally pointless. If there can never have been a time when there was nothing, something must be eternal, and therefore infinite. Our Universe must be part of that infinite something, and therefore, according to the above reasoning, must be infinite.
Now I seem to be arguing that the Universe is finite, and infinite, at the same time. The logical way round this must, surely, be to assume that the Universe, and the whole cosmos, are infinite, and that our perception of differentiation of space, the “passage” of time and of any change is simply an illusion resulting from our very restricted viewpoint.
We should look briefly at the idea of the holographic universe. What is the holographic universe? Does it imply that the Universe is a hologram? I shall assume sufficient knowledge of holograms to make a description unnecessary.
When scientists, such as David Bohm, and authors, such as Michael Taylor describe the Universe as a hologram I very much doubt that they are suggesting that some other-worldly being is projecting laser images with an incredibly gigantic projector to produce what we experience as the Universe. I suspect that it would be more appropriate to say that the three-dimensional images we can produce with laser technology are as near as we can come to producing an effect that, to some extent, mimics the way in which our Universe works.
The kind of holographic image with which most people are familiar is that which is viewed by reflected light and produces a very limited three-dimensional image when observed directly. However, the kind of holographic image that is of interest in terms of the holographic universe model is that which is not directly observable simply by looking at the plate on which it is captured. Viewed by reflected light the plate seems to contain only vague, swirling the marks of interference. The true image can be seen only when the plate is illuminated by transmitted light. The image then stands out from the plate, forming a three-dimensional image that can be viewed from any angle. The object may appear real, but any attempt to touch it will reveal that it is not there, it is simply a product of the ability of our brains to interpret electromagnetic frequencies. One remarkable thing about these holographic plates is that if you cut one in half, each half will produce the same, complete, image. In fact, how ever many times you divide the plate, each fragment will produce the whole image. Only the quality of the reproduction will deteriorate as the fragments get smaller. The best explanation for this must be that the entire image is contained in every part of the plate. It is this quality of the holographic image that makes it particularly significant in the context of a possibly infinite universe. If our Universe is actually infinite, then the entire Universe is contained in every atom of what we perceive as a collection of divisible entities. William Blake’s “….World in a Grain of Sand” is no longer just poetic imagery; it is a small step towards seeing things as they really are.
When we talk of the holographic Universe we are suggesting two things. The first is that, as with the holographic images produced by lasers, it is our brains that interpret the “frequencies” of the Universe to produce the images we see and the reality we perceive around us. The second point is that every part of the Universe is, in a very real sense, the whole Universe, “….and I am that”.
Sighs of relief all round - it's over! Later I'll try to tackle some of the questions.
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There's one on Amazon UK at a little over £790! I think I might try the local Library.
If they don't have it ask them about an inter-library loan.
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Popular science books often point out that, in eternity, everything that can happen, must happen.
It's not quite as simple as that; it depends precisely what you mean by 'can happen'. There may well be possible states for a system (e.g. the universe) to be in that cannot be reached by any stepwise change or progression because there is a dependence on priors (see Does Everything Possible Have To Happen? (http://www.askamathematician.com/2012/12/q-in-an-infinite-universe-does-everything-thats-possible-have-to-happen-somewhere/)).
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Why does it seem apparently intrinsically indivisible to you? It seems to me that infinity is divisible into any number of parts, including an infinite number of parts. In any division, there will be at least two infinite parts. Consider a road that stretches away from you to infinity in either direction. You can paint a line across it and divide it into two infinite lengths, then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction.
It seems that, in Part 4, I didn't cover the reasoning behind the opening statement of Part 5. My bad!
" Consider a road that stretches away from you to infinity in either direction."
Mathematically, this may be an acceptable thing to ask, but in reality, you are asking the impossible.
How could you possibly know that the road went to infinity? There is certainly no way to prove that it does.
OK, you could argue that this is only a thought experiment, but it pertains to something that, almost certainly, cannot exist. However, let's stick with it for the time.
" You can paint a line across it and divide it into two infinite lengths,"
Literally, "infinite" means "without end". When you paint your line you mark an end to the first part of your quasi-infinite road. Beginning and end are dependent on subjective viewpoint, so all you need to do is turn round and your line marks an end to the other half of your road. It was not my intention to get into etymological discussion, but you rather invite it in your comment about time. :)
"..... then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction."
No, you can't. As you rightly point out; one length is finite, so however many times you repeat the action, you will never reach infinity, in fact, you will always be infinitely far from it. How could something finite become infinite?
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It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.
One of the difficulties involved in talking about infinity is that our terminology is rooted in linear time. Suggest a better term for a timeless state and that will be a big step in the right direction.
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(see Does Everything Possible Have To Happen?).
That's an interesting link, but most of what it says amounts to "everything that can happen, will happen, but not is it can't happen for some reason".
Also, it seems to assume that the same laws (e.g. gravity) that apply in our seemingly finite Universe would automatically in infinity. Can that be justified?
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" Consider a road that stretches away from you to infinity in either direction."
Mathematically, this may be an acceptable thing to ask, but in reality, you are asking the impossible.
How could you possibly know that the road went to infinity? There is certainly no way to prove that it does.
OK, you could argue that this is only a thought experiment, but it pertains to something that, almost certainly, cannot exist.
Well of course. All discussion about infinity is either mathematical or thought experiment.
Literally, "infinite" means "without end". When you paint your line you mark an end to the first part of your quasi-infinite road. Beginning and end are dependent on subjective viewpoint, so all you need to do is turn round and your line marks an end to the other half of your road.
That's a semantic straw man. An infinite extent can start wherever you like. Consider the integers, or the real numbers; consider Hilbert's Hotel.
It was not my intention to get into etymological discussion, but you rather invite it in your comment about time. :)
I don't follow you - which comment and how is it relevant?
"..... then paint another line across, making two infinite lengths and one finite length. You can do this an infinite number of times in either direction."
No, you can't. As you rightly point out; one length is finite, so however many times you repeat the action, you will never reach infinity, in fact, you will always be infinitely far from it. How could something finite become infinite?
I didn't say you will reach infinity, simply that you can repeat the operation an infinite number of times in either direction. Consider the integers as an analogy for the road. You can move in the positive and negative directions, 'marking' every 5th integer to infinity in either direction. You'll get an infinite number of finite sequences of 5 integers. You might also consider that between every pair of integers there is an infinite number of real numbers. The integers are countably infinite, the reals are uncountable; the positive integers make the smallest ordinal infinity, the uncountably infinite real numbers are a bigger ordinal infinity. It's fascinating stuff.
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One of the difficulties involved in talking about infinity is that our terminology is rooted in linear time. Suggest a better term for a timeless state and that will be a big step in the right direction.
What's wrong with 'timeless state', or '4D block'? I sometimes use 'Parminidean block universe', because it's a reminder that these ideas are ancient, but it's a bit clumsy and can sound pompous. When you stand outside time in this way, you need to look at time as just another dimensional axis; 'now' and 'then' and 'future' and 'past' are points and directions relative to observers on that axis.
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(see Does Everything Possible Have To Happen?).
That's an interesting link, but most of what it says amounts to "everything that can happen, will happen, but not is it can't happen for some reason".
Not really. It's main point is that not all possible states of a system will necessarily occur even given an infinite time. That's why I linked it.
Also, it seems to assume that the same laws (e.g. gravity) that apply in our seemingly finite Universe would automatically in infinity. Can that be justified?
It just took a particular example that used gravity to illustrate the point; in that hypothetical universe familiar laws applied (surely it would only confuse matters to try and illustrate a point with totally unfamiliar physical laws?).
Is there any reason to suppose that the physical laws familiar to us would not operate in a universe of infinite extent? As far as I know, we still have no definitive evidence that our own universe isn't infinite in extent beyond the observable horizon.
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Well of course. All discussion about infinity is either mathematical or thought experiment.
True, but this misses the salient point that infinite roads, infinite divisions and all other forms of the infinite series exist only in the (presumably finite) minds of those who think about these things.
That's a semantic straw man.
No. something that may be considered to have no end in one direction, but be clearly limited in the other may be said to be unbounded in one direction, but not infinite. It is possible to argue reasonably and logically that something is unbounded, but to describe any physical thing, in our 4D reality, as infinite, without stipulating that you are talking about a mathematical, or pseudo, infinity is presumptuous and usually inaccurate.
I recall that a few years ago I wrote some notes about the Hilbert hotel. Unfortunately I can't find them at the moment. However, my recollection is that it is a clever mathematical illusion.
Time has caught up with me again, but I'll try to pick up the thread tomorrow. Sorry that responses are rather bitty, but that's how things are at the moment.
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Well of course. All discussion about infinity is either mathematical or thought experiment.
True, but this misses the salient point that infinite roads, infinite divisions and all other forms of the infinite series exist only in the (presumably finite) minds of those who think about these things.
By pointing out that it's either mathematical or thought experiment, I was emphasising the point that they are abstractions and not (necessarily) real-world considerations. But, whatever.
No. something that may be considered to have no end in one direction, but be clearly limited in the other may be said to be unbounded in one direction, but not infinite. It is possible to argue reasonably and logically that something is unbounded, but to describe any physical thing, in our 4D reality, as infinite, without stipulating that you are talking about a mathematical, or pseudo, infinity is presumptuous and usually inaccurate.
There is a difference between unboundedness and infinity; infinity has the property of unboundedness in some respect (e.g. along some particular vector), but not all unboundedness is infinite, e.g. the surface of a sphere is unbounded but not infinite. Coincidentally, I'm currently on a short course at the University of Cambridge, on 'Philosophical Paradoxes', and we've just had a session on paradoxes of infinity. In particular, talking about Kant's paradox that there are compelling arguments both that the universe must be infinite in time of existence (the requirement for 'sufficient reason' for starting at some point), and that it cannot be infinite in time (this would make the present the end of an infinite series of events, and an infinite series cannot be completed). Much of the discussion of the second argument involved the choice of starting point for infinite sequences, in time (events), spatial extent, and in numbers (e.g. the positive integers start at 1 (or 0) and extend to infinity in unit increments; there is a lower bound, but no upper bound). We had no problem with starting points for an infinite series, sequences, or extents. Nobody suggested that they necessarily corresponded to any real-world contexts, these were all metaphysical abstractions, thought experiments.
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sounds fun dlorde. And on a totally unrelated question made by Pete. Try to search on < 'jammer.pdf' Max Jammer > for a taste. Not that I would advice anything more than a search naturally, but I did find something from that book in the middle of the search.
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There is a difference between unboundedness and infinity; infinity has the property of unboundedness in some respect (e.g. along some particular vector), but not all unboundedness is infinite, e.g. the surface of a sphere is unbounded but not infinite
I couldn't have put it better myself. :)
Unfortunately, it is quite common to see the surface of a sphere referred to as infinite.
this would make the present the end of an infinite series of events, and an infinite series cannot be completed
Since directionality is subjective, what sense does it make to talk of being able to start an infinite series?
Nobody suggested that they necessarily corresponded to any real-world contexts, these were all metaphysical abstractions, thought experiments.
In maths and philosophy reality can be ignored.
"Gradually mathematicians lighted upon a new concept of existence. Mathematical ‘existence’ meant only logical self-consistency and this neither required nor needed physical existence to complete it. If a mathematician could write down a set of non-contradictory axioms and rules for deducing true statements from them, then those statements would be said to ‘exist’." John D Barrow.
Enjoy your course.
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Since directionality is subjective, what sense does it make to talk of being able to start an infinite series?
I don't see what direction has to do with it - e.g. the positive integers are an infinite series starting at 0 (or 1); likewise the negative integers are an infinite series starting at 0 (or -1). Why should starting an infinite series be a problem?
Enjoy your course.
Thanks, it's been good so far...
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I don't see what direction has to do with it - e.g. the positive integers are an infinite series starting at 0 (or 1)
Consider your positive integers; move from 0 to 100. Now turn round and go back the other way. When you reach 0, you have come to the end of an infinite series. It should take infinite time to reach the end of an infinite series.
There are two problems here:
1. There is no such thing as an infinite series.
2. There is no such thing as infinite time.
It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.
Does that not support statement 2?
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Consider your positive integers; move from 0 to 100. Now turn round and go back the other way. When you reach 0, you have come to the end of an infinite series. It should take infinite time to reach the end of an infinite series.
As I said, it's the start of an infinite series. Just as 3.1415... is the start of the infinite series of digits that is the decimal representation of pi.
There are two problems here:
1. There is no such thing as an infinite series.
2. There is no such thing as infinite time.
I wouldn't mention 1. to a mathematician, they use infinite sequences and series (http://en.wikipedia.org/wiki/Infinite_series) all the time. Archimedes had a method for summing a decreasing infinite series before 212 BC; that task is now done with calculus. Perhaps you're taking the Aristotlean/Intuitionist view that infinities are potential rather than actual?
Statement 2. is debatable - that's what Kant's paradox is about - he argued that logically, time both must be and could not be infinite. But he used separate arguments to do so.
Do you have any argument to support assertions 1. or 2. ?
It's not an 'all embracing now', because you're outside of time in that perspective; 'now' is a subjective experience of observers traversing the time dimension.
Does that not support statement 2?
I don't see it; how?
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I wouldn't mention 1. to a mathematician
Too late! as far back as the late 1960s, when I worked in a residential school, I had several discussions with the maths teacher about infinite series. Eventually he conceded that although the infinite series was a valid mathematical concept, with which I have no problem, the concept was not valid in the real world.
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One of the problems with long posts, especially multiples of long posts, is that people are inclined to skim over them, or even lose patience and not read them. I am ashamed to confess, I sometimes do that myself when time is very short.
We may have to be more selective in our points, and work towards added clarity, if we are ever to get anywhere with this.
Perhaps we could start with an opinion from any interested poster on the following:
Do the positive integers 1,2,3...., the negative integers -1,-2,-3..... and the real numbers (eg) between 0 and 1 constitute three infinite series?
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Do the positive integers 1,2,3...., the negative integers -1,-2,-3..... and the real numbers (eg) between 0 and 1 constitute three infinite series?
As I understand it, yes.
However I would much prefer that you outline your arguments to support statements 1 and 2 than go through some Socratic dialogue; perhaps the dialogue could follow the arguments so we know what we're debating.
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Do you have any argument to support assertions 1. or 2. ?
Let's start with an extract from "Part 5". If you can't find what you want we can progress from there:
"
Consider Cantor's infinities: the whole numbers constitute an infinite series, so do the even numbers and the odd numbers. Thus, Cantor demonstrated that, not only were there numerous infinities, but they were not all the same size. It is evident that the infinity containing the even, or odd, numbers must be half the size of the infinity containing the whole numbers. Could it be that question is answered, that we can divide infinity and that any parts into which we divide it will be infinite? There seem to be at least two reasons why this cannot be the case. The first is that even Cantor does not seem to have performed mathematical calculations with the infinite set of all infinities; this appears to be the only one of his infinities that is not actually a mathematical infinity. The other is that, practically there is the complication that anything that is truly infinite must contain everything; there cannot be two infinities, because each would have to contain the other.
Applying the Reflection Principle to the infinite set of all infinities would lead to the following contradiction: The reflection principle holds that within a universal set, containing all sets, it must be possible to find a set that contains any property found in the universal set. The obvious contradiction is that the universal set contains all other sets (that is one of its properties), but this property cannot be found in any of the other sets.
Wikipedia says: " In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers. In number systems incorporating infinitesimals, the reciprocal of an infinitesimal is an infinite number, i.e., a number greater than any real number. Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[2] For example, the set of integers is countably infinite, while the set of real numbers is uncountably infinite."
Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers, whereas an uncountable infinity cannot. Useful as these concepts may be to the mathematician, none is an "absolute" infinity, and cannot therefore be considered as more than "unbounded"."
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... It is evident that the infinity containing the even, or odd, numbers must be half the size of the infinity containing the whole numbers.
You may think it's evident, but what is intuitive isn't necessarily correct. The definition of an infinite set is that any proper subset has the same size as the whole set. The elements of the subset can be mapped one-to-one with the members of the whole set.
Could it be that question is answered, that we can divide infinity and that any parts into which we divide it will be infinite?
I already covered this.
... even Cantor does not seem to have performed mathematical calculations with the infinite set of all infinities; this appears to be the only one of his infinities that is not actually a mathematical infinity.
I don't know whether Cantor used the set of all infinite sets in his calculations (do you have a source for this?), but there are an infinite number of infinite sets, so it must be a mathematical infinity. What is your argument that it is not?
...anything that is truly infinite must contain everything; there cannot be two infinities, because each would have to contain the other.
What precisely does the truly in 'truly infinite' mean? It is generally accepted that there are multiple infinite sets; e.g. the real numbers are infinite, the whole numbers are infinite, neither set contains the other. If you introduce your own concept of 'truly infinite' that way, you're not talking about the same thing; and I don't see how it has any coherent meaning - can you explain?
Applying the Reflection Principle to the infinite set of all infinities would lead to the following contradiction: The reflection principle holds that within a universal set, containing all sets, it must be possible to find a set that contains any property found in the universal set. The obvious contradiction is that the universal set contains all other sets (that is one of its properties), but this property cannot be found in any of the other sets.
That simplistic version of the Reflection principle is clearly self-contradictory for all classes of universal sets, and so is useless in that form. A description of a non-contradictory formulation is given here: Motivation for reflection principles (http://en.wikipedia.org/wiki/Reflection_principle#Motivation_for_reflection_principles).
Cantor defined a countable infinity to be one that can be put into one-to-one correspondence with the list of natural numbers, whereas an uncountable infinity cannot. Useful as these concepts may be to the mathematician, none is an "absolute" infinity, and cannot therefore be considered as more than "unbounded".
Another custom infinity... so what does the qualifier 'absolute' mean in this context?
Infinity is a particular kind of unboundedness, it doesn't require 'more' than that.
Cantor himself defined an Absolute Infinite as that which transcended the transfinites (all other infinities). He said:
"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type".
For him it was a kind of mathematical deity, possessing a reflection principle that every property of the Absolute Infinite is also held by some smaller object. Personally, I think this is a step beyond the coherent, but I'm no set theorist.
Was that what you had in mind? if not, what? and is your 'absolute' infinity different from what you call 'truly' infinite? if not, why use two names for it?
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It should take infinite time to reach the end of an infinite series.
I can do it in a finite time, and I can even prove it.
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It should take infinite time to reach the end of an infinite series.
I can do it in a finite time, and I can even prove it.
Cool, let's see the proof.
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It should take infinite time to reach the end of an infinite series.
I can do it in a finite time, and I can even prove it.
Cool, let's see the proof.
Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.
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Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...Summing all the times, I reach the end of the series in 2 seconds.
No; because when you reach the n-th term, or any other, you are still infinitely far from the (non-existent) end.
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Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...Summing all the times, I reach the end of the series in 2 seconds.
No; because when you reach the n-th term, or any other, you are still infinitely far from the (non-existent) end.
While you worry about it, I have already reached the end in 2 seconds, as I wrote. If you don't believe it, explain how you could still be in some term of the series after 10 seconds...
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While you worry about it, I have already reached the end in 2 seconds, as I wrote. If you don't believe it, explain how you could still be in some term of the series after 10 seconds...
Presumably, you can't; especially if your name happens to be Zeno, but then, you have not explained how you actually reach 2 seconds.
I am in the fortunate position of not having to explain anything, because I don't accept the infinite series outside mathematics, nor do I believe you can reach infinity. If you can arrive at the end of it, it is not infinite.
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The definition of an infinite set is that any proper subset has the same size as the whole set. The elements of the subset can be mapped one-to-one with the members of the whole set.
Surely this is the definition of a countably infinite set, so, at best it is part of the definition of a mathematical infinity.
I don't know whether Cantor used the set of all infinite sets in his calculations (do you have a source for this?),
Incautious wording on my part. It would have been better to have said something like: "In my limited knowledge of 2the work of Cantor, I am not aware that he actually used the set of all infinite sets in his calculations. If anyone can provide an example of his so doing I would give it my attention". That would make more sense than my trying to give examples of where he didn't. :)
What precisely does the truly in 'truly infinite' mean? It is generally accepted that there are multiple infinite sets; e.g. the real numbers are infinite, the whole numbers are infinite, neither set contains the other. If you introduce your own concept of 'truly infinite' that way, you're not talking about the same thing; and I don't see how it has any coherent meaning - can you explain?
I used the term "truly" infinite so as not to confuse what I was talking about with "absolutely" infinite. I did this because it is easy to argue, as you did, that absolute infinity is a mathematical infinity. Indeed, I was not " talking about the same thing".
There is also something of a question mark over the concept of absolute infinity. "... when it is realized in the most complete form". What is the most complete form of infinity? Does it contain all other infinities? If not, in what sense is it absolute? If it does, how can Barrow say:
"Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist. There is no biggest of all that can contain them all. There is no Universe of universes that we can write down and capture."
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Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.
Ah, Zeno would be proud ;)
Problem is, you're trying to do an infinite number of actions in a finite time, and each action takes a finite time (also a finite amount of energy).
From a physics viewpoint, the quantisation of energy (and probably time) prevents you going beyond a certain point.
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Surely this is the definition of a countably infinite set, so, at best it is part of the definition of a mathematical infinity.
The strict definition is 'A set is infinite if and only if for every natural number the set has a subset whose cardinality is that natural number'. So it covers all infinite sets.
I used the term "truly" infinite so as not to confuse what I was talking about with "absolutely" infinite. I did this because it is easy to argue, as you did, that absolute infinity is a mathematical infinity. Indeed, I was not " talking about the same thing".
You have yet to define, describe, or distinguish between them. Were you talking about Cantor's Absolute infinity?
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Ok. I have set to take 1 second to reach the first term of the series, 0.5 seconds to reach the second term, ... (1/2)n-1 seconds to reach the n-th term, ...
Summing all the times, I reach the end of the series in 2 seconds.
Ah, Zeno would be proud ;)
Problem is, you're trying to do an infinite number of actions in a finite time, and each action takes a finite time
Infact it was exactly showing that the total time were finite, that Zeno solved the paradox [:)]
Every action takes a finite time, indeed. Infact (1/2)(n-1) is *always* a finite time. But their sums is finite as well...
You have to go over calculus?
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I admit I haven't been following this thread that closely, so if I'm repeating someone else's thoughts, I apologize.
Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite. The reason its counterintuitive is that our brains aren't built to naturally handle the concept of infinity, so it takes training and practice to get comfortable with these ideas. (If we could deal easily with infinities, calculus wouldn't be such a painful subject!)
The other question being discussed here seems to be whether infinity is physically real or not: can we find examples of infinity in nature or is the mathematics of infinity just a useful approximation to very big or very small things? We don't know the answer to that. What we do know is that current models do assume infinitely small things exist since space is continuous. If I give you a length, you can always chop it in half in our current models and that is the definition of something getting infinitesimally small. From that, Lightarrow's example/Zeno's paradox follows automatically. Since we can't currently measure things past some very small scale, we don't know if space and time are really continuous or if there are some smallest building blocks. In the latter case, infinitesimally small might not exist.
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Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite.
But, of course, you can't complete an infinite series...
What we do know is that current models do assume infinitely small things exist since space is continuous.
Not all current models assume that; for example, Loop Quantum Gravity is quite popular, and potentially resolves the problem of singularities (by removing them).
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You have to go over calculus?
??
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Lightarrow is correct that at any particular step you take a finite amount of time, but the total time taken if you complete the infinite number of steps is also finite.
But, of course, you can't complete an infinite series...
Of course you can! Lightarrow just did so mathematically. You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.
What we do know is that current models do assume infinitely small things exist since space is continuous.
Not all current models assume that; for example, Loop Quantum Gravity is quite popular, and potentially resolves the problem of singularities (by removing them).
True, but those are speculative hypotheses currently. The primary models that have been well tested and accepted (the standard model, general relativity) do assume continuous space and time. That doesn't mean anything more than that if they are granular, the granularity is small enough that we haven't seen it yet on the scales of those models. But on the other hand there's no evidence for granularity.
Taking a firm stance the existence or nonexistence of physical infinities is stating an opinion without evidence to back it up. Mathematical infinities, on the other hand, clearly exist.
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Of course you can! Lightarrow just did so mathematically. You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.
Summing it, or writing it symbolically isn't what I had in mind. What exactly do you mean by 'complete'? It seems to me that if you could complete it, you could give me the value of the final term - but there isn't one.
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You can take a stretch of line, 13 cm, then split it in 13 even chunks, then split those into 26, then those into 52. Where should it end? Doesn't this depend on what mathematics you use. Probably also on where those mathematics stop producing meaningful answers. But I think you can define a infinity to a finite stretch.
=
It also goes back to if there is a 'granularity' to SpaceTime I think. If it is a 'flow' then you won't find a stop, and it should be possible to zoom in for ever. Or you can define it from what is meaningful, in which case you get two answers. One where it stops being meaningful at some scale, another that ignores what we find meaningful.
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There is actually a third too, as I think, and that would be if it was a fractal describing a SpaceTime. On the other tentacle that should be real close to a definition of a flow.
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Of course you can! Lightarrow just did so mathematically. You can't write down all the terms explicitly, but you can complete it or write the series symbolically as he did.
Summing it, or writing it symbolically isn't what I had in mind. What exactly do you mean by 'complete'? It seems to me that if you could complete it, you could give me the value of the final term - but there isn't one.
He completed it by writing down an expression for all the terms and then summing them. The set 1/2n, when n is a whole number is the expression for all the terms. Like I said, explicitly writing out all the terms is impossible physically, since we can't write arbitrarily fast. But since both of these methods are abstract mathematical notation for the series, why should we afford one more importance than the other in a mathematical sense? In a physical sense, the existence of infinities is a matter of opinion whether continuous things exist or not or whether the universe is infinitely large or not--at least until we get some evidence to back up theories one way or the other.
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He completed it by writing down an expression for all the terms and then summing them. The set 1/2n, when n is a whole number is the expression for all the terms. Like I said, explicitly writing out all the terms is impossible physically, since we can't write arbitrarily fast. But since both of these methods are abstract mathematical notation for the series, why should we afford one more importance than the other in a mathematical sense? In a physical sense, the existence of infinities is a matter of opinion whether continuous things exist or not or whether the universe is infinitely large or not--at least until we get some evidence to back up theories one way or the other.
OK, fair enough. Defeated by the semantics of 'complete', I withdraw my statement.
However, I still can't see how an infinite number of actions can be completed in a finite time, since an action requires energy and there is a minimum energy of action (i.e. the quantum of energy transfer). Though my understanding of 'action' in this context (a force doing work on an object) may be astray too.
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Certainly there's no problem with the mathematics, since mathematics exists in a world of thought and doesn't necessarily reflect anything physical.
Physically speaking, if space and time are continuous, then lightarrow's example is physical--if you move 2 meters, you move 1 meter + 1/2 meter +1/4 meter + 1/8 meter, etc. Moving 1/2n meters takes 1/2n time. Since the time and distance get arbitrarily short, there's no problem doing this. Is space and time have some granularity and a smallest unit, then lightarrow's example will run into problems since there exists some 1/2n length that is a minimum and the smaller terms in the series don't represent anything physical.
Whether arbitrarily short time and distance can physically exist in nature is another question, and I agree with you that it may not be physical. Personally, I tend to think it is not physical and that time and distance are granular on some scale, but I have to admit that there isn't any good evidence to support my opinion, although some hypotheses in physics propose it. Our best well-supported models right now assume continuous space and time, but we haven't checked that they're valid on arbitrarily short length and time scales.
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Just letting everyone know I have not abandoned this thread. I've been off line for a few days, and may continue to be for a couple more. However, I'm looking forward to catching up, and joining in again asap.
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Were you talking about Cantor's Absolute infinity?
What is Cantor's Absolute infinity? Is it the ultimate infinity, beyond which it is not possible to go? If not, as I already asked, in what sense is it absolute? If it is, then it does not exist, because Cantor established that there was no overarching infinity.
For him it was a kind of mathematical deity, possessing a reflection principle that every property of the Absolute Infinite is also held by some smaller object.
If " every property of the Absolute Infinite is also held by some smaller object", then there must be a "smaller object" that is absolutely infinite, which, at best, causes problems; and, at worst, makes no sense.
Personally, I think this is a step beyond the coherent, but I'm no set theorist.
I am no set theorist, either, but, apart possibly from the creation of a mathematical deity, I see no problem with the concept of an infinity that is transcendent. It must contain all other infinities, because it must contain everything. It cannot be manipulated by mathematics, because it must contain mathematics.
I feel sure you will object to that last assertion, but think about it; it may be perceived by us as capable of being manipulated mathematically, but we are finite, or so it appears to us, so how can we make infinite judgements about something which according to Cantor's insights into mathematical infinities, cannot exist?
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If " every property of the Absolute Infinite is also held by some smaller object", then there must be a "smaller object" that is absolutely infinite, which, at best, causes problems; and, at worst, makes no sense..
Yes. That's why I said I thought it was a step beyond the coherent. It is possible he was poorly paraphrased and meant each property of the Absolute Infinite is also held by some smaller object, but even this seems incoherent unless you explicitly exclude the property of 'absoluteness' (and probably some others).
... I see no problem with the concept of an infinity that is transcendent. It must contain all other infinities, because it must contain everything. It cannot be manipulated by mathematics, because it must contain mathematics... I feel sure you will object to that last assertion
It's not my cup of tea, but - fine as long as you acknowledge that necessarily makes it metaphysics, not mathematics. Cantor thought Absolute Infinity was mathematical.
... we are finite, ... so how can we make infinite judgements about something which according to Cantor's insights into mathematical infinities, cannot exist?
I don't know what you mean by 'infinite judgements'.
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fine as long as you acknowledge that necessarily makes it metaphysics, not mathematics.
Interesting that you say " metaphysics, not mathematics", rather than "metaphysics, not physics". Have we reached a point where physics is so ruled my mathematics that a mathematical "reality" automatically becomes a physical reality?
mathematics exists in a world of thought and doesn't necessarily reflect anything physical.
It seems very easy to lose sight of that fact.
Mathematics is, undoubtedly, the language of nature, but I suspect that is because mathematics is the best language we have found to describe nature, rather than because it actually governs nature.
Cantor thought Absolute Infinity was mathematical.
He also established that it could not exist.
This is why I have been trying to stress the difference between mathematical infinities and physical infinity.
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Interesting that you say " metaphysics, not mathematics", rather than "metaphysics, not physics".
Why interesting? It seems to me that infinity is a mathematical concept that can be used in physics. There may or may not be infinities in the real world - I don't see how we could know.
Have we reached a point where physics is so ruled my mathematics that a mathematical "reality" automatically becomes a physical reality?
Depends what you mean by a 'mathematical "reality"'; but assuming your question isn't tautologous (if something is 'real', that usually means physically real), I'd say no, not at all - that's why I said 'mathematics' rather than 'physics'.
mathematics exists in a world of thought and doesn't necessarily reflect anything physical.
It seems very easy to lose sight of that fact.
If you say so.
Mathematics is, undoubtedly, the language of nature, but I suspect that is because mathematics is the best language we have found to describe nature, rather than because it actually governs nature.
I wouldn't argue with that, although it seems slightly loaded - i.e. if the universe does fundamentally operate in a mathematical way (or if maths does describe it's behaviour precisely), is it governed by maths or does it define maths? It seems a question of semantics. After all, where did the fundamentals of mathematics originate, if not observation of, and interaction with, the universe.
Eugene Wigner had quite strong views on this: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html).
Cantor thought Absolute Infinity was mathematical.
He also established that it could not exist.
Do you have a source for that? my understanding as that he initially called the set-theoretic universe (including transfinites) 'the Absolutely Infinite', and thought that it could be acknowledged but not known, or even approximated. Later, he revised this, calling the set-theoretic universe and other proper classes 'inconsistent multiplicities' or 'absolutely infinite': A multiplicity can be of such nature, that the assumption of the togetherness/combining of its elements leads to a contradiction, so that it is impossible to conceive the multiplicity as a unity, as a finished/completed thing. I call such multiplicities absolutely infinite or inconsistent multiplicities.
[Letter to Dedekind]
I haven't seen anything to suggest he abandoned it altogether. Not that it really matters...
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I haven't seen anything to suggest he abandoned it altogether
Look closely at the wording of the quote, it would be possible to say almost anything without abandoning it altogether, :)
Have you read: " Barrow. John D. The Infinite Book. Vintage, Random House, London 2005"?
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Have you read: " Barrow. John D. The Infinite Book. Vintage, Random House, London 2005"?
No.
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It is not unusual to find references to "infinite speed". How would one define infinite speed? Can it exist?
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It is not unusual to find references to "infinite speed". How would one define infinite speed? Can it exist?
Speed is distance over time, and increases as time decreases. So speed tends to infinity as time tends to zero; although mathematically dividing by zero is undefined, not infinity, one could define infinite speed as traversing a distance in no time, i.e. instantaneously.
If by 'can it exist?', you're asking if something in the real world can traverse a distance instantaneously, there are relativistic considerations. No physical object can accelerate to or past the speed of light, but if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous. The only other apparently instantaneous effect over distance I can think of is the decoherence of quantum entanglement.
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Average speed is distance traveled over time taken to travel that distance. Instantaneous speed is distance over time in the limit as time goes to zero. As always, dealing with limits requires care. As time decreases to zero, distance traveled also decreases to zero. Their ratio as they get tiny determines instantaneous speed. An "infinite" speed would correspond to an object whose distance traveled did not decrease to zero as the time interval over which it was traveling did decrease to zero. As an example, if an object moved at least 1 meter no matter how tiny a time interval you measured, that object would have infinite speed. This would violate relativity and doesn't seem particularly physical, so I doubt it exists in reality.
I'm not sure where "infinite speed" gets referenced, but I can't recall seeing any references to it.
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I'm not sure where "infinite speed" gets referenced
Here's one: “So if a tachyon were created in some violent event in space, it would radiate energy away furiously…..and go faster and faster, until it had zero energy ……and was travelling at infinite speed”.
Gribbin John. Companion to the Cosmos. Phoenix (Orion Books Ltd.), London. 1996.
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One of the characteristics of infinite speed must be that it would be immeasurable. Consider what this implies: Prior to becoming infinite the tachyon’s speed would be measurable. It seems hardly credible that it would suddenly reach a point where its speed would no longer be measurable. What could happen to bring about this change?
We might argue that we already have the answer to that: " An "infinite" speed would correspond to an object whose distance traveled did not decrease to zero as the time interval over which it was traveling did decrease to zero." [JP]. Would that not imply that it must be everywhere at once? Could we not argue that if tachyons exist, either there is only one, that is everywhere; or, every tachyon in existence is "here" all the time?
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What if we consider infinite acceleration? In that scenario the tachyon would just continue to accelerate for ever. Certainly this would make more sense, but then, it would never actually reach infinite speed, it would simply continue striving for it, and a more accurate term might be unlimited/ unbounded acceleration, rather than infinite acceleration, because, although we could imaging the acceleration going on for ever, it could never reach a point where we could say: "now it is infinite", unless we accept JP's definition (above), in which case, there are some scientists who would equate this with the speed of light. That would lead to the absurd situation in which a tachyon would accelerate from the speed of light to the speed of light.
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One of the characteristics of infinite speed must be that it would be immeasurable. Consider what this implies: Prior to becoming infinite the tachyon’s speed would be measurable.
A tachyon, in this context, is an hypothetical faster-than-light particle with imaginary mass. How would its speed be measurable at all?
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... That would lead to the absurd situation in which a tachyon would accelerate from the speed of light to the speed of light.
As I understand it, a tachyon would never be able to reach the speed of light; its energy-velocity relation would be a mirror of normal particles, its velocity increasing as its energy decreases. It would require infinite energy to decelerate to c (just as a normal particle would require infinite energy to accelerate to c), so it could only exist by moving FTL.
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A tachyon, in this context, is an hypothetical faster-than-light particle with imaginary mass. How would its speed be measurable at all?
I'll rethink!
Tachyons are hypothetical particles that, hypothetically, travel faster than light. Any measurement of their hypothetical speed would, hypothetically, have to be made by some hypothetical measuring device that, hypothetically, inhabited the hypothetical realm of which tachyons are hypothetical denizens.
Does that put us on the same page? :D
Back to the serious stuff a little later, I hope.
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Our classical concept of speed probably doesn't work that well for tachyons, even if they somehow did exist. All Bradyons (particles with real, non-zero mass) travel slower than light. One explanation for this is that they can achieve zero momentum but still have energy (from E=mc2). An object with energy moves through time and an object with momentum moves through space. So a Bradyon always moves through time but can stop moving through space, obtaining a zero velocity.
A tachyon is in some ways the opposite. It can obtain zero energy, but it always has non-zero momentum. At zero energy, it satisfies (mc=p where p is momentum). This means it can somehow stop moving through time and move only through space. I'm not sure what this means intuitively, but that's what the equations say.
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As I understand it, a tachyon would never be able to reach the speed of light; its energy-velocity relation would be a mirror of normal particles, its velocity increasing as its energy decreases. It would require infinite energy to decelerate to c (just as a normal particle would require infinite energy to accelerate to c), so it could only exist by moving FTL.
we're definitely on the same page here. However, you seem to have missed the point, perhaps because I was not clear enough.
The point I was aiming for was that if we accept JP's definition of infinite speed, which seems quite reasonable to me, a tachyon (at full speed?!) would not experience time. Some people, including some scientists, believe that this is the case with the photon, which travels at c. Since we cannot prove, either that the photon does not experience time, or that the tachyon, if it exists, has mass, it must be acceptable to theorise that the tachyon accelerates away from c, where the photon, and possibly the tachyon, experience no time, and arrives at a point where its experience of time is identical to that at its starting point. Does that make sense?
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I'm not sure what this means intuitively, but that's what the equations say.
Intuitively, it means very little to me, but, sadly, the equations would probably mean even less.
Congratulations on yet another explanation that does mean something!
I've been having some thoughts about tachyons and their relationship to the Universe. If they still make sense to me when I get them together, and if I can work out how to include diagrams in posts, :) I might have a go at a new theory.
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if it exists, has mass...
Imaginary mass (http://en.wikipedia.org/wiki/Tachyon#Mass).
... the tachyon accelerates away from c, where the photon, and possibly the tachyon, experience no time, and arrives at a point where its experience of time is identical to that at its starting point. Does that make sense?
It's a bit opaque (see bolding - surely either it 'experiences' time or it doesn't). If you're saying the tachyon arrives at some point having accelerated from c, but no time has elapsed in its frame of reference, I would have to query your definition of acceleration. Acceleration apart, it looks like a way of saying for a zero-energy tachyon what I proposed earlier for the photon: "... if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous".
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.....and possibly the tachyon, experience no time, and arrives at a point where its experience of time is identical.....
What is identical to experiencing no time? It must be experiencing no time.
That's why I asked: "Does that make sense?"
If you're saying the tachyon arrives at some point having accelerated from c, but no time has elapsed in its frame of reference, I would have to query your definition of acceleration.
Rightly so. I was simply pointing out the two definitions in question.
"... if you consider a photon to have its own valid frame of reference, its 'journey' in that frame would appear to be instantaneous".
Convenient as it would be, at times, to consider a photon to have its own valid frame of reference, such would seem not to be the majority view in scientific circles.
"Therefore, just as bradyons are forbidden to break the light-speed barrier, so too are tachyons forbidden from slowing down to below c, because infinite energy is required to reach the barrier from either above or below." (From your link)
Presumably, they would be able to travel at c if they were massless; but then, would they be able to travel at any other speed?
Of course, we must not forget that this is all hypothetical, but I think I'm beginning to like the possibility of a link between c and infinite speed. That could be good for some crackpottery! :)
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I've mentioned this before, but you have to be very careful about applying the idea of reference frame to a photon (and presumably tachyons). The reason we can discuss what something experiences is that we can compare it's reference frame (a frame in which it is at rest) to another reference frame (in which it's in motion). Length and times change as measured from the ref. frame of the moving observer. We cannot do this for a photon since there exists no reference frame in which its at rest--we can't find such a frame to compare to other observers. It doesn't exist in special relativity. Perhaps a post-SR theory will describe it.
The usual claim that photons are timeless (which I've occasionally seen in pop-sci books) comes from mis-applying the equations of special relativity which assume that the photon is at rest in some reference frame.
I'm not sure how to think about a tachyon's experience, since it can't ever exist in the reference frame of a bradyon (which we are). However, there should be frames in which it's at rest, so you might be able to compare it to other tachyons and figure out what it experiences. I'll have to think about it more.
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I've mentioned this before, but you have to be very careful about applying the idea of reference frame to a photon (and presumably tachyons).
I agree - perhaps I should have emphasised the 'if' in ".. if you consider a photon to have its own valid frame of reference...". I suppose it's an intuitive attempt to understand photons in familiar terms.
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I've mentioned this before, but you have to be very careful about applying the idea of reference frame to a photon (and presumably tachyons).
I agree - perhaps I should have emphasised the 'if' in ".. if you consider a photon to have its own valid frame of reference...". I suppose it's an intuitive attempt to understand photons in familiar terms.
What I'm trying to point out is that there is a scientific problem with doing so. If we're going to talk about the properties of this hypothetical frame in scientific terms, we need to have a scientific theory that covers it, i.e. the theory needs to describe the properties of that frame and be testable somehow. Special relativity explicitly does not cover that reference frame and there is no way we know of to test the properties of such a frame (if it were to exist). The best we can say scientifically is that we don't know if such a frame exists and don't know its properties if it were to exist.
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What did cantor do?
Cantor revolutionised the mathematical concept of infinity. He lifted infinity from the realms of philosophy and theology and handed it to mathematicians and scientists. He identified infinities that were "countable" and "uncountable"; that is, infinities that could be put in one-to-one correspondence with the list of natural numbers 1,2,3,4,5,6… , and infinities that could not. So, for example, the even numbers are countably infinite, so are all the odd numbers.
Cantor defined all countable infinities as being the same size. This seems to introduce something of a paradox, because the list of natural numbers is infinite, and the list of odd numbers is infinite, intuition would suggest that one of these must be half the size of the other. We know that intuition is not always our best guide, so we should put aside intuition and look at how we can claim that these two infinities are the same size. the obvious answer is that we can go on putting one into one-to-one correspondence with the other for ever and we will not run out of either; but is that what infinity is really about? We can do this in theory, but never in practice. These infinities, undoubtedly, have their uses in the more esoteric realms of set theory and other branches of mathematics, but are of very dubious value when applied to the real world. One of the major problems with trying to integrate general relativity and quantum theory is that the equations of one, when applied to the other tend to lead to infinities, so the equations become nonsense.
Initially, the mathematics community was not over enthusiastic about Cantor's work, but after a time, mathematicians said "cool", or "hoc frigidulum est" or whatever the expression of the time might have been. Since then, Cantor has been widely quoted as having established that infinity was mathematically manipulatable.
What did Cantor really do to infinities? He discovered that there were ways of making infinities manageable by mathematicians. However, even he accepts that this is only a partial victory.
Barrow says: “Cantor’s most dramatic discovery was that infinities are not only uncountable, they are insuperable. He discovered that a never-ending ascending hierarchy of infinities must exist. There is no biggest of all that can contain them all. There is no Universe of universes that we can write down and capture."
Cantor called this "Absolute infinity", he likened it to "God", but at the same time established that this absolute infinity did not exist. Here is another paradox. If we argue that "There is no Universe of universes that we can write down and capture." There is no greatest infinity. Then, surely, we must argue that our so-called infinite series are not infinite, because they lead to no infinity "that we can write down and capture". They are unbounded, because we can neither see nor imagine an end, but we cannot say that they are infinite in any sense other than mathematical.
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I suppose picking a "largest infinity"is a bit like picking a "largest number." No matter what number or infinity you tell me, I can give you a bigger one. You can get around this by defining a process rather than a number, such as "my number is X, which is defined as X=1/Y as Y tends to zero." I can no longer pick a single number that is bigger than your X. I wonder if you could find a "biggest infinity" in a loose sense through a limiting process like this. For example, start with the set of all real numbers, and replace each number with the set of real numbers--then repeat without end. I don't know if there are bigger infinite sets--my set theory is a bit limited and rusty. :)
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my set theory is a bit limited and rusty. :)
You're lucky, I don't have enough set theory to go rusty. :(
I suspect you are right about applying the same technique to infinities as to numbers. Of course, you would not identify a "biggest infinity", any more than you would identify a biggest number.
If cantor is right, there is no "biggest infinity", there are mathematical infinities, which are actually unbounded mathematical concepts. Then there is "absolute infinity" which cannot be reached by any finite process, so, as far as mathematics is concerned, does not exist.
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A frame of reference practically, is you measuring locally (clock and ruler) relative some other. If you you use 'c' for the one measuring, then nothing makes sense anymore. The arrow of time must 'disappear', as the Lorentz contraction should be infinite, etc. But we 'see' photons, and they follow causality, macroscopically defined. It's all about coordinate systems, and their limits. 'c' is one.
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Possibly the discussion of infinity is wearing a bit thin. No anticipated broadside from Pete as yet. :)
I think this (in part) is where I am at present.
1. Infinity is not just a very big number.
2. Eternity is not just a very long time.
3. Something that is finite can never become infinite.
4. Mathematical infinities are theoretical concepts that are unbounded, but not infinite.
5. Cantor’s “absolute infinity” may be infinite, but this cannot be proved nor disproved.
6. Unbounded entities may be subjected to mathematical processes, but attempting to do this to infinity leads to nonsensical answers.
7. There cannot be more than one true (absolute?) infinity, because it must contain everything.
8. Multiplying or dividing infinity makes no practical sense because the result would have to be infinite, and there cannot be more than one "everything".
9. Practically, nothing can be added to infinity, because it is already everything.
10. Nothing can be taken away from infinity, because the remaining quantity would still be infinite, therefore it makes no sense to talk of something being taken away.
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Possibly the discussion of infinity is wearing a bit thin. No anticipated broadside from Pete as yet. :)
Huh? I don't get it. Fill me in on the joke. :)
I have strong feelings about a lof of this stuff. Some of it's based in the article
Quantum Theory Needs No 'Interpretation, Christopher A. Fuchs and Asher Peres, Physics Today, March 2000
For anyone who'd like to read it PM your e-mail address to me and I'll e-mail it to you.
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Possibly the discussion of infinity is wearing a bit thin. No anticipated broadside from Pete as yet. :)
I think this (in part) is where I am at present.
I can give you a broadside on this one. :)
The problem with many of your points is that you're not being precise. There are different ways to get at the concept of infinity. Most common in physics is to mean something very big, which is represented by allowing numbers to increase without bound. Very small is another option, in which things decrease without bound. It's unknown if these concepts match nature: is the universe infinite in size, or can space be broken down into infinitesimally small pieces? Regardless, the theories based upon them are accurate enough that we can get away with these uses of infinity.
1. Infinity is not just a very big number.
2. Eternity is not just a very long time.
Yes, I agree. They are more along the lines of concepts of things or times increasing without bound.
3. Something that is finite can never become infinite.
This is a blanket claim and I'm not sure it's provable. I suspect that in physics, this is probably the case--at least for most things, but making a blanket statement like this is metaphysics, not physics.
4. Mathematical infinities are theoretical concepts that are unbounded, but not infinite.
I'm not sure what you mean by "mathematical" infinities. There are certainly infinities that are in a sense bounded but infinite: for example the set of real numbers between 0 and 1 is bounded below by 0 and above by 1, but has an infinite number of elements.
5. Cantor’s “absolute infinity” may be infinite, but this cannot be proved nor disproved.
I don't know enough math to derive a proof either way, but if absolute infinity contains all other infinite sets, then it has to at least be as big as any single one of them. Since all those sets are infinite, absolute infinity must be infinite, if it exists.
6. Unbounded entities may be subjected to mathematical processes, but attempting to do this to infinity leads to nonsensical answers.
As stated, this claim is false. It is a mathematical process to add elements to a set. I can add all real numbers to the set of all rational numbers and end up with a valid set, for example. What you can't do is to pretend that "infinity" can stand in as a real number, since it's a concept, not a number. 1+infinity doesn't make sense since the "+" operation isn't defined for the concept infinity.
7. There cannot be more than one true (absolute?) infinity, because it must contain everything.
Don't know on this one.
8. Multiplying or dividing infinity makes no practical sense because the result would have to be infinite, and there cannot be more than one "everything".
Kind of. It's more general to go back to the idea that infinity is a concept, not a number, so you can't expect to apply operations which are defined for numbers to it.
9. Practically, nothing can be added to infinity, because it is already everything.
This is false. You can add elements to an infinite set. This is precisely what Cantor realized when he established different types of infinity.
10. Nothing can be taken away from infinity, because the remaining quantity would still be infinite, therefore it makes no sense to talk of something being taken away.
Again, false. You can take an element out of an infinite set and are left with an infinite set without that element.
Edit: Realized I had quoted your list twice. :p
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Thanks JP.
Bit pushed tonight, but will have some comments asap.
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Fill me in on the joke.
No joke. Just thought you would have damning things to say. :)