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Quantum Mechanics and logic
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Quantum Mechanics and logic

DJPavel
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Posted 08/09/08 - 08:41 PM: Subject: Quantum Mechanics and logic	#1
While discussing the knowledge argument in a different thread, we got on a topic of Quantum Mechanics (QM) and I thought it would be more appropriate to discuss it under “science”. Basically, I’m interested in exploring the philosophical implications of breaking Bell’s inequalities by QM. (I’m a layman when it comes to QM, so please don’t bombard me with vector spaces and eigenvalues). If my understanding is correct, Bell’s inequality theorem is proven with reliance on classic logic and, to be more specific, set theory. However, QM violates the theorem, but only if you presume Local Realism. That is, QM can be implemented with classic logic, but you would have to allow non-locality. So, it seems we have two choices: EITHER say classic logic is not the only logic out there OR local realism is false. I’m not really sure which I like more. Being a pragmatist (and an embodied cognitivist (my new self-assigned label)), I don’t hold on dearly to the laws of logic as something fundamental in reality. To me, like anything else, logic is a tool to deal with a specific set of problems in a specific setting. QM is a different setting and therefore a different tool is more appropriate for the situation. Maybe that’s why there’s such a thing a Quantum Logic. But if I we stick with Classic logic, we have to conclude non-locality. But doesn’t that contradict Relativity, which experimentally has been extensively verified? If my understanding of the problem is accurate, which is the lesser of the two evils? I vote for Quantum Logic. A side question: since determinism in physics implies predictable evolution forward AND backward, how does entropy come into play? Classical mechanics is deterministic (with a few exceptions) and entropy means that in the next ‘tick of time’ there are more available states for the system to be in than a tick before. BUT, if it’s unique history, then what sense does it make to talk about “more states”. Such evolution is like a film roll: it’s one-to-one linear transition. How much sense does it make to talk about possibility of other slides being next? It sounds like all these available states are logical possibilities in some abstract space. What’s the story here? The rest is my reply to DeathMonkey on comments from a different thread: DeathMonkey wrote: I'm not sure what you mean by classic logic being "broken" at the quantum level. If you are referring to the odd things like a particle being able to be in a superposition state (which cannot be accurately described as being in both states, nor as being in neither state), then this is not an example of classical logic being "broken", but rather of applying classic logic to an incorrect set of assumptions. Namely, the assumption that the particle has a specific "state" at all. I was talking about Bell’s theorem ( (A and –B) + (B and –C) > (A and –C)), proven with classic logic, not working in QM. Or we can consider breaking the distributive law of classical logic. (p and (q or r) = (p and q) or (p and r)). The logic is not “broken” if you do not assume local realism. Is local realism that incorrect assumption you’re talking about? DeathMonkey wrote: It's not just logically possible. It is apperantly exactly the case. Local realism is inconsistent with our observations. Therefore it is wrong, just as flat-earth theory is wrong. Wait, I thought observations do tell us that local realism is true. Isn’t that Einstein’s accomplishment? In fact, it could probably be argued that it’s true a priori (analytically entailed from other facts about the world, like its spacial dimensions), although I won’t try to defend that claim. DeathMonkey wrote: You still have the statistical averaging effect. If every possible outcome is represented by a Universe (or more specifically, a branch of the wave function), then why does our universe appear to be one in which these classical deterministic rules seem to work out "on average"? The answer, of course, is that the set of universes where things average out to our normal deterministic laws is far larger than the set where strange things happen, like every carbon 14 atom in your body decaying in the space of 1 second. It's really the same statistical principle as in the other random interpretations. In the random interpretations, macroscopics appear to conform to deterministic laws because when you average over many many random events, the variation tends to grow very small relative to the total. This is just another way of saying that the overall outcome of those many events is far more likely to be one where the deterministic laws are approximately valid, than one where they are not. In the MW interpretation, the random events are effectively eliminated, but still, any single branch of the wave-function (universe) is far more likely to be one where the deterministic laws are approximately valid, than one where they are not. In effect, the "random" component in the MW interpretation is the selection of the specific branch of the wave function that we happen to be a part of. From the point of view of any observer in any one of those universes, the world will appear to be exactly the same as would be expected from a random interpretation. The same statistics apply, even though they are being used to represent different systems. I’m sorry, but I still don’t follow you. I don’t understand what you mean by “This is just another way of saying that the overall outcome of those many events is far more likely to be one where the deterministic laws are approximately valid, than one where they are not” with respect to the MWI. My understanding is there is no statistics or averaging in the MW interpretation. It’s purely deterministic. There are no possible states with one being collapsed to. The Universe simply diverges in its evolution into multitude of Universes and we’re in one of these paths. I don’t see how you can transcend these paths and talk about some averaging out. Can you please illustrate what you’re talking about with some dummed down thought experiment? Have Alice toss balls to Bob and have them count them or something DJP

swstephe
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Posted 08/09/08 - 11:44 PM:	#2
I think there is a "hidden variable" in the logic -- that Bell's theorem only says that physical models don't work intuitively. That doesn't mean the logic is broken, only that we have nothing in our general experience that behaves like quantum mechanics. You can't say "QM" is like this other well-known example which changes its physical characteristics based on how you measure it. If QM were not logical or truly indeterminate, then there wouldn't be any QM. Look at hypercubes as a simple example, (a cube extruded into a 4th cartesian direction). We are not able to deal with them, intuitively, because we don't have anything in our 3D world like hypercubes and are blind to any 4th dimension. We don't throw geometry out the window or try to come up with new forms of geography, in fact we can deduce so much using geometry and logic that hypercubes can be pretty intuitive and even "projected" so we can look at them. Overturning logic is possible, I suppose, if you can find everyday phenomena which deny concepts like identity and contradiction, but that would be more an exercise in redefinition. For example, suppose that we decided that two entangled particles were actually identically the same. Then you end up with a logic which is unintuitive, and then it fails as a model for rationalizing conclusions.
Ethics is the measuring of morality. Morality is the measuring of good. Good is the measuring of benefit. Benefit is the measure of values.

unenlightened
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Posted 08/10/08 - 12:14 AM:	#3
swstephe wrote: You can't say "QM" is like this other well-known example which changes its physical characteristics based on how you measure it. This probably isn't relevant, but the the other well known example is the mind.
The observer is the observed. J Krishnamurti "Philosophy, to the Philistine, is an evolutionary process, watched over by some sort of brisk dynamic Providence, and culminating in the supreme insight of modern thought." John Cowper Powys

Death Monkey
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Posted 08/10/08 - 02:44 AM:	#4
DJPavel, If my understanding is correct, Bell’s inequality theorem is proven with reliance on classic logic and, to be more specific, set theory. However, QM violates the theorem, but only if you presume Local Realism. That is, QM can be implemented with classic logic, but you would have to allow non-locality. So, it seems we have two choices: EITHER say classic logic is not the only logic out there OR local realism is false. I'm not even really sure what it would mean to try to say the above. After all, it is phrased in terms of "classical logic". Also, since the concept of local realism itself is formulated in terms of classical logic, I don't see how the above could be considered an "either/or" option anyway. Seems to me that if you toss classical logic, you lose local realism with it anyway. At best, you could replace it with some new form of local realism based on some new form of logic. But I don't see how that would be any improvement. Seems to me like it would just be describing the same thing in terms of a different formal language. But if I we stick with Classic logic, we have to conclude non-locality. But doesn’t that contradict Relativity, which experimentally has been extensively verified? Not at all. On the contrary, Relativity is why we have to conclude non-locality. Special Relativity is built-in to Quantum Mechanics. It is precisely the fact that movement faster than the speed of light is equivelent to travel backwards in time (a result of special relativity) that leads us to conclude that travel faster than C violates causality, which in turn leads to the loss of local realism. If my understanding of the problem is accurate, which is the lesser of the two evils? I vote for Quantum Logic. I don't understand this kind of reasoning? What is "evil" about either option? The only thing "wrong" with either possibility is that they are both counter-intuitive. Anyway, as I mentioned above, I don't see how the two options are really different in anything more than a semantic sense. Either way, you are just using different terminology to describe the same thing. What difference does it make whether you deny "local realism", or redefine "realism" (by virtue of using a different formal logic), so that the same physical reality now qualifies as having "local realism"? Either way, what we intuitively think of as "local realism" is incorrect. A side question: since determinism in physics implies predictable evolution forward AND backward, how does entropy come into play? Classical mechanics is deterministic (with a few exceptions) and entropy means that in the next ‘tick of time’ there are more available states for the system to be in than a tick before. BUT, if it’s unique history, then what sense does it make to talk about “more states”. Entropy is a statistical mechanics concept. Even in a deterministic setting, it makes no sense to talk about it in terms of evolution of specific microstates. It only makes sense when talking about the evolution of macrostates, in which case you are talking about statistical properties of those macrostates. I'm not sure what you mean by classic logic being "broken" at the quantum level. If you are referring to the odd things like a particle being able to be in a superposition state (which cannot be accurately described as being in both states, nor as being in neither state), then this is not an example of classical logic being "broken", but rather of applying classic logic to an incorrect set of assumptions. Namely, the assumption that the particle has a specific "state" at all. I was talking about Bell’s theorem ( (A and ?B) + (B and ?C) > (A and ?C)), proven with classic logic, not working in QM. Or we can consider breaking the distributive law of classical logic. (p and (q or r) = (p and q) or (p and r)). The logic is not “broken” if you do not assume local realism. Is local realism that incorrect assumption you’re talking about? Yes. What local realism actually claims is that the various quantities we observe (momentum, energy, position, and so on), actually have exact, determined values at all times. It's not just logically possible. It is apperantly exactly the case. Local realism is inconsistent with our observations. Therefore it is wrong, just as flat-earth theory is wrong. Wait, I thought observations do tell us that local realism is true. Isn’t that Einstein’s accomplishment? No. Our observations only tell us that local realism tends to appear to be approximately true under so-called "classical" conditions. Namely, those conditions where quantum effects average out, and classical mechanics becomes a good approximation. In fact, it could probably be argued that it’s true a priori (analytically entailed from other facts about the world, like its spacial dimensions), although I won’t try to defend that claim. I don't think such a claim possibly could be defended. Local Realism is an a posteriori claim. After all, we only even thought to imagine that these particular quantities were somehow "real", because of our observations. I’m sorry, but I still don’t follow you. I don’t understand what you mean by “This is just another way of saying that the overall outcome of those many events is far more likely to be one where the deterministic laws are approximately valid, than one where they are not” with respect to the MWI. My understanding is there is no statistics or averaging in the MW interpretation. It’s purely deterministic. There are no possible states with one being collapsed to. The Universe simply diverges in its evolution into multitude of Universes and we’re in one of these paths. I don’t see how you can transcend these paths and talk about some averaging out. Can you please illustrate what you’re talking about with some dummed down thought experiment? Have Alice toss balls to Bob and have them count them or something The key is the fact that we are on one of these paths. So which one is it? Is that somehow determined? What would that even mean? You can consider it this way. At any given point in time, there are all these different "universes". The distribution of the states of those universes is the same as the random distribution of possible states would be for a random interpretation. The only difference is that instead of randomly selecting one of those states and "collapsing" the wave-function to that state, you have one universe for every single one of those possibilities. But for each of those universes, the effect is the same. DM
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ManiacJack
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Posted 08/10/08 - 12:05 PM: Subject: and in the back of my head is ...	#5
Ternary Logic does describe the same thing in a different way; But it is also more accurate in it's logic [whatever that means]. From wiki: A ternary, three-valued or trivalent logic (sometimes abbreviated 3VL) is a term to describe any of several multi-valued logic systems in which there are three truth values indicating true, false and some third value. This is contrasted with the more commonly known bivalent logics (such as boolean logic) which provide only for true and false. As with bivalent logic, truth values in ternary logic may be represented numerically using various representations of the ternary numeral system. A few of the more common examples are: * 1 for true, 2 for false, and 0 for unknown, irrelevant, or both.[1] * 0 for false, 1 for true, with the third value being non-integer symbol such as # or ½.[2] * Balanced ternary uses −1 for false, +1 for true and 0 for the third value; these values may also be simplified to −, +, and 0, respectively.[3] This article mainly illustrates a system of ternary propositional logic using the truth values {false, unknown, and true}, and extends conventional boolean connectives to a trivalent context. Ternary predicate logics exist as well[citation needed]; these may have readings of the quantifier different from classical (binary) predicate logic, and may include alternative quantifiers as well. [picture] In this truth table, the UNKNOWN state can be metaphorically thought of as a sealed box containing either an unambiguously TRUE or unambiguously FALSE value. The knowledge of whether any particular UNKNOWN state secretly represents TRUE or FALSE at any moment in time is not available. However, certain logical operations can yield an unambiguous result, even if they involve at least one UNKNOWN operand. For example, since TRUE OR TRUE equals TRUE, and TRUE OR FALSE also equals TRUE, one can infer that TRUE OR UNKNOWN equals TRUE, as well. In this example, since either bivalent state could be underlying the UNKNOWN state, but either state also yields the same result, a definitive TRUE results in all three cases. I'm pretty sure Relativity operates on the Ternary.
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swstephe
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Posted 08/10/08 - 09:45 PM:	#6
unenlightened wrote: This probably isn't relevant, but the the other well known example is the mind. I was covering my ... self ... with "physical characteristics". I don't think minds have physical characteristics. I was thinking someone might throw up examples of rainbows, too. But QM is talking about apparently contradictory properties of the same physical thing at the same time. ManiacJack wrote: Ternary Logic describes the same thing in a different way; Bit it is also more accurate in it's logic [whatever that means]. The human brain seems to be biased toward thinking in terms of duality. The universe doesn't really work that way, but it generally works because the binary logic system can be used to describe general relationships. Ternary would be a single step in the right direction, if the right direction is toward "fuzzy logic". The brain subjectively experiences "fuzzy logic" just fine, and it is an essential part of our grammar, but we tend to avoid it in making logical assertions ... I guess because fuzzy premises lead to fuzzy conclusions. We seem to get more emotional satisfaction out of binary dualistic (absolute) answers. I imagine there is some connection to the way neurons fire. They receive some fuzzy input, and "activate" depending on a threshold. A single neuron uses binary logic, (either it is active or it isn't), even though the input is fuzzy and its activation in combination with other neurons is also fuzzy, (125 our 261 neurons are active with this stimulus). Fuzzy logic works by simply applying relative qualifiers, usually in natural distribution patterns. Something might be "mostly true", "somewhat true", "slightly true", etc. I guess it would be possible to build a system similar to classical logic based on fuzzy. I think I attempted it at one time in my college days, (fuzzy logic was a hot topic -- again -- at that time).
Ethics is the measuring of morality. Morality is the measuring of good. Good is the measuring of benefit. Benefit is the measure of values.

Simple Occam
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Posted 08/13/08 - 01:44 PM:	#7
Maybe are ARE non-local efficient causes at work in nature that are undiscovered because science does not know how to discover them.

starlarvae
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Posted 08/15/08 - 11:03 AM:	#8
DJPavel wrote: . . . Classical mechanics is deterministic (with a few exceptions) DJP Can you say more about the exceptions? Examples? DJPavel wrote: . . . and entropy means that in the next ‘tick of time’ there are more available states for the system to be in than a tick before. BUT, if it’s unique history, then what sense does it make to talk about “more states”. Such evolution is like a film roll: it’s one-to-one linear transition. How much sense does it make to talk about possibility of other slides being next? It sounds like all these available states are logical possibilities in some abstract space. What’s the story here? DJP Yes, the available states are on tap. The interesting question is that of causality: from among the available potential states, by what means is it decided which one will, as Whitehead said, "undergo the formality of actually occurring."?
www.starlarvae.org "Don't imagine that you can escape faith. Every science, every system of logic, has its axioms. Reason cannot move forward without some assumption upon which to base itself." -- The Rebbe, Menachem Schneerson "Seek simplicity, and distrust it." --A. N. Whitehead

DJPavel
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Posted 08/18/08 - 08:21 PM:	#9
DeathMonkey wrote: DJPavel wrote: So, it seems we have two choices: EITHER say classic logic is not the only logic out there OR local realism is false. I'm not even really sure what it would mean to try to say the above. After all, it is phrased in terms of "classical logic". Also, since the concept of local realism itself is formulated in terms of classical logic, I don't see how the above could be considered an "either/or" option anyway. Seems to me that if you toss classical logic, you lose local realism with it anyway. I know what you’re saying; I can’t deny something by employing in my denial the very thing I’m trying to deny. Let me address this philosophical part further down, but for now, let’s straighten out what’s going on. Please do me a favor. Check out this Standford lecture on Quantum Entanglement by Larry Susskind. Scroll up to 1h12m and listen for a few minutes when he talks about the implications of Bell’s theorem and answers questions (especially at 1h18m30s). http://www.youtube.com/watch?v=XlLsTaJn9AQ To me, the message is pretty clear: “Quantum Mechanics is not Classic logic”. He explains that any attempt to explain QM with classic logic comes at the sacrifice of the speed of light limit. This is why I formulated the either/or choice: you either drop some laws of classic logic (like the distributive law) or you drop locality (i.e. allow over the speed of light information transmission). The former sounds counterintuitive and contradictory, the latter – against the empirical evidence. That’s why I called them both “evil”. Now, I realize that it’s quite possible that I misinterpret what Susskind says and it seems like I just don’t follow what you say. Perhaps you give me too much credit for understanding the concepts you refer to and maybe it’ll be helpful if you spell them out in a “dumbed down” way, which I would appreciate. As an example, I finally got what you meant by statistical averaging, but by accidentally reading an article in the July issue of Scientific American on Quantum Gravity. This is what it said: SciAm wrote: The Euclidian Quantum Gravity is based on a fundamental principle from quantum mechanics: superposition. Any object, whether a classical or quantum one, is in a certain state—characterizing its position and velocity, say. But whereas the state of a classical object can be described by a unique set of numbers, the state of a quantum object is far richer. It is the sum, or superposition, of all possible classical states. For instance, a classical billiard ball moves along a single trajectory with a precise position and velocity at all times. That would not be a good description for how the much smaller electron moves. Its motion is described by quantum laws, which imply that it can exist simultaneously in a wide range of positions and velocities. When an electron travels from point A to point B in the absence of any external forces, it does not just take the straight line between A and B but all available routes simultaneously. This qualitative picture of all possible electron paths conspiring together translates into the precise mathematical prescription of a quantum superposition, formulated by Nobel laureate Richard Feynman, which is a weighted average of all these distinct possibilities. Interestingly, the article says that when the ‘fundamental principle of QM” is applied to the evolution of the Universe (as the principle applies to ALL objects), the computer simulation shows that such Universe would collapse in a infinite amount of dimensions. The reason, they speculate, is that Euclidian quantum gravity treats space and time equally – there’s no causality built in the structure; causality is emergent at macro level. So, they propose a model, Causal Dynamical Triangulation, which puts temporal sequence of events (causality) into the structure, which prevents the collapse and produces a nice 4D space-time. Anyway, this is a digression, but as soon as we’re done with this, I’ve been wanting to discuss the philosophy of emergence with you, since you’re in the field most appropriate for this kind of conversation. In the same article, there’s something else that caught my attention: SciAm wrote: The tiny building blocks (comprising spacetime) have no direct physical meaning. If one could examine real spacetime with an ultrapowerful microscope, one would not see small triangles. They are merely approximations. The only physically relevant information comes from the collective behavior of the building blocks imagining that each one is shrunk down to zero size. In this limit, nothing depends on whether the blocks were triangular, cubic, pentagonal or any mixture thereof to start with. The insensitivity to a variety of small-scale details also goes under the name of "universality." It is a well-known phenomenon in statistical mechanics, the study of molecular motion in gases and fluids; these substances behave much the same whatever their detailed composition is. Universality is associated with properties of systems of many interacting parts and shows up on a scale much larger than that of the individual constituents. The analogous statement for a flock of starlings is that the color, size, wingspan and age of individual birds are completely irrelevant in determining the flying behavior of the flock as a whole. Only a few microscopic details filter through to macroscopic scales. This puzzles me. I don’t believe (intuitively) there is no information “hidden” on the micro level to give novelty ‘out of the blue’ on the macro level. Or is this the same phenomenon of “no hidden variables” that we observe in entaglement? That also seem to break determinism to me. If the same stuff can give rise to different macro properties, there’s no unique evolution, hence no determinism. I was also wondering if there are systems (at least in principle) that are identical on macro level but which are constructed by micro levels with different properties; a kind of ‘inverse supervinience’ if you will . That would kill reductionism I guess. Anyway, this is a digression again; I’m just giving heads-up of things that I’d like to ask about in the next topic. Unfortunately, there are not enough hours in a day to discuss everything I want. Ok, now to the Classic logic(CL). I’m not sure how to express myself well without too much hand waving, but I no longer believe in the validity of holding Classic logic as an absolute reference bar “out there”, despite the fact that such denial produces seeming paradoxes. It reminds me of some (ontological) arguments that denying the existence of God is self-contradictory. Sure, but we still manage to get things done and pass on our genes to get more things done. First, holding CL as absolute reference bar leads to all kinds of Russelian paradoxes. Seems like any time you postulate absolute truths and rules, you either can’t prove (some) of them or you end up with contradictions. It’s much less problematic to postulate that CL is only a formal language, like Peano’s arithmetic rules, that is there to resolve only some classes of problems. If the problem is of a different class, a different tool might be more appropriate. So, in a sense, the CL is more of a heuristic. It’s a very powerful one; it allows to launch man in space, but it’s nevertheless only a means of communication and a reflection of how we, humans, process information. Humans, after all, are not logical computers that would stop breathing on encountering a logical error. We’re full of logical inconsistencies and we communicate with each other by means of prototypes, frames, and metaphors, not classical logic. Sure, the metaphor construction has to obey some structural laws to be meaningful, but such laws do not have to obey the classical logic rules in every context. Yes, in a sense I’m picking and choosing when I want to apply specific rules of logic. It all depends on what I’m trying to accomplish and picking the right tool to do the job. It seems unreasonable to me to postulate something absolute by jumping outside the context of human cognition and make statements from such a vantage and privileged point, it’s paradoxical by your own rules. The best we can do is to agree on how we’re going to approach and solve the problems we have. The classic logic is not an exception. If we can’t understand QM with the CL, heck with it. Let’s devise a system that will allow us to understand it in a meaningful and intuitive way. I actually tend to believe that because of the fact that you can’t reconcile Relativity with QM, the problem is with how we conceptualize these theories. I bet there soon will be some genius, the next Einstein, who will say “you know, we’ve been looking at it the wrong way. Forget about the superpositions and tensors, here’s a new picture of reality for you that makes it all consistent”. The history is filled with such examples and I don’t see any reason to believe that we, right now, live in some privileged epoch where we finally see the light and all we have is to work out those pesky details. I'm betting on much more sophisticated and different ways of looking at reality to come in the future. With this in mind, part of the confusion in the discussions of QM here, is that we're equating solving physical problems with metaphysical ones. When we look at entagled particles and Bell's theorem, are we making the statement about how reality is or are we making a statement about how we can meaninfully explain a set of observations? This is a very important distinction, although most physicists don't like it because they consider their theories to be the descriptions of how reality really IS. I think if the physicists stop making the metaphysical claims and learn a lesson from Ptolemy's epicycles, the QM will be more accessible and less controversial. DJP

DJPavel
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Posted 08/18/08 - 08:55 PM:	#10
Hi Heresiarch, welcome to the forum. I was wondering if you were going to join. I think you’ll find plenty of interesting discussions here. starlarvae wrote: DJPavel wrote: . . . Classical mechanics is deterministic (with a few exceptions) DJP Can you say more about the exceptions? Examples? It probably would be best if I find a reference. Here’s one: http://arxiv.org/ftp/physics/papers/0506/0506108.pdf Robert Bishop wrote: … Still, there are some surprises and controversy regarding the judgment that classical mechanics is a deterministic theory. For example, as Keith Hutchinson (1993, p. 320) notes, if the force function varies as the square root of the velocity, then a specification of the initial position and velocity of a particle does not fix a unique evolution of the particle in state space (indeed, the particle can sit stationary for an arbitrary length of time and then spontaneously begin to move). Hence, such a force law is not deterministic. There are a number of such force functions consistent with Newton’s laws, but that fail to satisfy unique evolution. Therefore, the judgment that classical mechanics is a deterministic theory is false. Newtonian Gravity. One might think that the set of force functions leading to violations of unique evolution represents an unrealistic set so that all force laws of classical mechanics really are deterministic. However, worries for determinism await one even in the case of point particles interacting under Isaac Newton’s force of gravity, the paradigm case of determinism that Laplace had in mind. In 1897 the French mathematician Paul Painlevé conjectured that a system of pointparticles interacting only under Newton’s force of gravity could all accelerate to spatial infinity within a finite time interval. (The source of the energy needed for this acceleration is the infinite potential well associated with the inverse-square law of gravitation.) If particles could disappear to ‘spatial infinity,’ then unique evolution would break down because solutions to the equations of motion no longer would be guaranteed to exist. Painlevé’s conjecture was proven by Zhihong Xia (1992) for a system of five point-masses. Though provocative, these results are not without controversy. For example, there are two interesting possibilities for interpreting the status of these particles that have flown off to spatial infinity. On the one hand, one could say the particles have left the universe and now has some indefinite properties. On the other hand, one could say that the particles no longer exists. Newton’s mechanics is silent on this interpretive question. Furthermore, are events such as leaving the universe to be taken as predictions of Newton’s gravitational theory of point particles, or as indications that the theory is breaking down because particle position becomes undefined? Perhaps such behavior is an artifact of a spatially infinite universe. If the universe is finite, particle positions are always bounded and such violations of unique evolution are not possible. Diagnosis. Other failures of unique evolution in classical mechanics can be found in John Earman’s (1986) survey. What is one to say, then, about the uniqueness and existence theorems for the equations of motion, the theorems that appear so suggestive of unique evolution? The root problem of these failures to satisfy unique evolution can be traced back to the fact that one’s mathematical theorems only guarantee existence and uniqueness locally in time. This means that the equations of motion only have unique solutions for some interval of time. This interval might be short and, as time goes on, the interval of time for which such solutions exist might get shorter or even shrink to zero in such a way that after some period solutions cease to exist. So determinism might hold locally, but this does not guarantee determinism must hold globally. starlarvae wrote: Yes, the available states are on tap. The interesting question is that of causality: from among the available potential states, by what means is it decided which one will, as Whitehead said, "undergo the formality of actually occurring."? But that still doesn’t answer the question of what it means to be “available states” in the Universe where there’s only one path. Now, the question only makes sense under the assumption of classical physics. QM changes that picture, but raises another question, as you point out, by what means is it decided what the next state will be? Well, after my conversation with Deathmonkey and reading SciAm, I would answer that there is no decision to be made. The existing state is a weighted average of all the possible states of the system. That’s what the superposition means. The next state is determined by the evolution of the wave function for that system. The wave function is itself deterministic, but it doesn’t determine a specific macro state of the system, it determines the probabilities of all possible states, and thus the overall classical macro state as, yet again, weighted average of all possibilities. DJP Edited by DJPavel on 08/18/08 - 09:10 PM

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