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Paradox of motion

  1. Sep 26, 2005 #1
    If an object goes from A to B it passes infinitely many points. How can the object pass an infinite number of points in a finite amount of time? If the object was at every moment (infinitley small time span) at a point (infinitely small distance), then how is motion possible?
    So that's the paradox of motion that mathematical thinking deals with.

    But then you got physics where when you go down in size you hit the quantum world.

    So what implication has physics for the paradox of motion? Is there no infinite small in physics, but small means quantum world here?

    ???
     
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  3. Sep 26, 2005 #2

    selfAdjoint

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    Classical motion is not paradoxical, it's your understanding of Measure that's inadequate. Vague talk about infinitely small distances only serves to confuse.


    Quantum physics has observation built right into it. So one time you observe the particle HERE, and some other time you observe it THERE. What happens in between is described as the evolution of the wave function, which doesn't happen in spacetime, but does behave classically (i.e. in accordance with measure theory).
     
  4. Sep 26, 2005 #3

    Hurkyl

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    Why couldn't it?

    In general, these questions are pseudoparadoxes -- they're statements that go against (some people's) intuition, but they are lacking in any actual reason why there's a problem.

    Since it seems that you

    (1) find it self-evident that objects do appear to pass an infinite number of points in a finite amount of time,

    and since you

    (2) have not given a reason why objects should not appear to pass an infinite number of points in a finite amount of time,

    your only reasonable conclusion is

    (3) objects can pass an infinite number of points in a finite amount of time.
     
  5. Sep 27, 2005 #4
    Vague talk? That's 'the arrow' version of Zeno's paradox. For Bertrand Russell "a plain statement of an elementary fact". For me, too.

    Yes, but wave functions ain't no classical trajectories of particles. What is motion, dynamical variables in the quantum world?

    In the real world, of course, things do move. But taking this self-evident fact together with Zeno's considerations makes it paradoxical and doesn't refutes it.

    Do you mind taking a look at this?
    http://www.mathpages.com/rr/s3-07/3-07.htm
     
  6. Sep 27, 2005 #5

    Hurkyl

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    Yes, I'll say vague -- I've never been impressed by any presentation of Zeno's arguments.

    Compare with, say, the Liar's paradox, in which I can formally derive a contradiction:

    If P is the statement "P is false", then we have:

    P = T or P = F
    P = T --> P = F
    P = T --> P = T and P = F
    P = F --> P = T
    P = F --> P = T and P = F
    P = T and P = F

    This is a real paradox. It, and other similar paradoxes, are the reason why the usual formal logic is designed in such a way that statements cannot refer to themselves (even indirectly!)


    I've never seen anyone do anything remotely similar with Zeno's paradoxes. I'm assuming, I suppose, that you're not taking as an postulate the statement:

    "objects cannot pass an infinite number of points in a finite amount of time"

    since it seems to me a rather silly argument to say:

    "Well mathematically, objects pass an infinite number of points in a finite amount of time... but I'm assuming objects cannot pass an infinite number of points in a finite amount of time... therefore there's a paradox!"

    but, I guess this would be an actual paradox if you are actually making this assumption.


    The problem is that people rarely give a reason for that assumption, and seem much more willing to discount hundreds of years of successful physics and mathematics instead of some postulate for which they cannot even begin to defend.

    I have seen it defended in terms of "tasks", but that approach begged a different question -- it assumed that in any "doable" collection of tasks, there must be a first and a last task. (Which is equivalent to assuming that only finite collections of tasks are "doable")
     
  7. Sep 27, 2005 #6
    Alright, I see your point here. Very interesting. I have to take this in.

    Let's forget about motion, paradoxes or pseudoparardoxes.
    What is about getting smaller and smaller? Conceptually and mathematically I can go down in scale as deep as I like. If there is a distance, area, volume or a period, I can keep on enlarging or slicing it up as much as I want. Never any dramatic happens, I can go on forever.
    But different story in the physical world. When going down here, dramatic things do happen.

    So what does that mean for math and physics and their relationship?
     
  8. Sep 27, 2005 #7

    selfAdjoint

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    Well, I assume you are talking about quantum mechanics? Or quantized spacetime? Neither one postulates discrete space or time (I assume you are still with Zeno). "Quantized" is not the same as "discrete". In quantized systems what we observe, through the reduction of the wave function is a possibly discrete set of OUTCOMES, for example in LQG they have such "eigenvalues" of area and volume. Quantum theories do not contain an account of what the quantized object is or does between observations. And they are all dependent on the continuous (indeed analytic) behavior of the wave function.

    Besides which your imagination is fine as far as it goes, but if you study topology a little more you'll be able to imagine a much richer class of spaces.
     
  9. Sep 27, 2005 #8

    Hurkyl

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    I suppose you mean the behavior of matter -- but that is a question about matter, not space-time itself. To date, there has not been one piece of evidence for space-time itself to look or behave any differently on small scales than on large scales.

    On the small end, as we continue to improve the resolution of our scanning devices to peer into smaller and smaller realms, it still looks like a continuum.

    On the large end, we don't see the blurring of incredibly distant objects that is predicted by many discrete theories.


    The only evidence we have is very indirect -- some of the work in attempting to develop a theory of quantum gravity results in mathematics, for which it might be reasonable to interpret as a discretized space-time. Of course, some other work suggests that what's going on is incredibly complicated, rather than something as nice as simple as discreteness.


    As for the relationship beween physics and mathematics, from some viewpoint, Physics is just the science of trying to select, among all possible mathematical models, which best represents the universe. If a continuum turns out to be a bad choice, then you just adopt some other mathematical model.
     
  10. Oct 2, 2005 #9

    It seems to me that the statement 'This statement is false' is not the same as 'The statement is false'. 'THIS statement is false' is a paradox while 'THE statement is false' refers to a statement that may or may not be false. I was wondering if you agree or disagree?
     
  11. Oct 3, 2005 #10

    Hurkyl

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    You mean P(Q) is the statement "Q is false"? It still leands to paradoxes in naive logic...

    Consider the statement G = P(G)

    If P(G) is true, that means G is false, which means P(G) is false.
    If P(G) is false, that means G is true, which means P(G) is true.


    In (the usual) modern approach, we have a strict hierarchy of formulae. A first-order formula is never allowed to take a formula as input. A second-order formula is allowed to take first-order formulae as input. A third-order formula is allowed to take first-order or second-order formulae as input, et cetera.

    This way, it is impossible to construct a literal self-reference like the liar's paradox, or the above variant.
     
  12. Oct 3, 2005 #11
    I've never read a particularly satisfactory solution to Zeno's paradox, but then I've never read a particularly satisfactory expression of it. Most solutions take the vantage that Zeno was unaware that an infinite series can sum to a finite number, and maybe that's where Zeno's problem lay, but this is basically the same as saying the particle DOES move through an infinite number of co-ordinates in a finite time. Zeno's point was to break the process down into more and more fundemental steps - an infinite sum just ignores the problem entirely by integrating it back to total distances and time intervals. I think Zeno's question was more profound than this.

    Zeno's question presupposes that Achilles and the tortoise have definite positions at any given time (cross-refer to his arrow paradox) and definite momenta, but does not consider the dimensions of the runners, suggesting he was thinking of them as particles. QM now teaches us that particles do not propagate in this way, but do so as waves. However, Zeno's posing of the question leads to incompatibilities with QM, such as the fact that it does not consider the possibility of NOT finding a particle at a given position and time, and does not foresee observation as having an effect on those particles, as well as HUP problems.

    On a classical, macroscopic level, would the following solution (that does not avoid the problem of increasingly fundemental intervals) be appropriate? Model the progress of Achilles and tortoise as six particles, two representing the legs of Achilles and four representing the legs of the tortoise. An observation of position at a given time is an observation of each of these six particles as being in contact with the ground, or not. This does two things: it removes the problem of an infinite number of points while honouring Zeno's approach to increasingly fundemental steps, but it also introduces the possibility of NOT finding a particle at a given time and, in the case of Achilles, not finding any of the particles at a given time. These non-observations simply have to be not counted when marking the progress of the runners as a whole and at least one further observation taken. There will be a minimum of one interval, possibly two or three, at the beginning of which Achilles is behind the tortoise and at the end of which Achilles is in front of it.

    The reason there may be three intervals is that you might have measurement 1 as both Achilles particles behind the closest tortoise particle, measurement 2 as either a) one Achilles particle level with/in front of any given tortoise particle and the other behind the closest, or b) one or no Achilles particle observed, and measurement 3 as one Achilles particle level with any given tortoise particle and the other in front of the foremost. Problem solved.

    Treating these particles as quantum particles, would it be fair to say that Zeno's paradox, and in my opinion the actual crux of the problem (how do particles propagate through space and interact with observers) is similar, insofar as we would ignore anything that is not measured and only count observations that yield results? And treating Achilles and the tortoise as complex quantum systems, this question is still pending?

    Of course, this entire post takes the point of view that the infinite sum solution missed the point, about which I could be wrong.
     
  13. Oct 3, 2005 #12

    Hurkyl

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    The big thing missing from statements of Zeno's paradox, as I've said, is any reason to think that a problem has arisen.

    Some people just don't seem to like the "no problem has been demonstrated" response, even if it is accurate. *sigh* So, to get through to those people, a responder has to make their best guess at the missing reason why people think there is a problem.

    The typical guess is that someone is thinking "how can you accomplish infinitely many tasks in finitely much time?" and then one demonstrates how adding up the time it takes to do each individual task yields a finite result.

    The next most popular thought, I think, is to look at the "last task". Of course, the problem with that thought is that there isn't any particular reason to think that, in some collection of tasks, there must be a final task. (Except for hastily generalizing one's intuition about finite sequences)

    Of course, if someone is thinking there's a problem for yet another reason, yet a different response would be needed.
     
  14. Oct 13, 2005 #13
    I would like to come back to getting smaller and smaller in the physical world vs. in math.

    So what is with highly curved and dynamical at the planck scale? And again:

     
  15. Oct 15, 2005 #14
    That's your solution - the rest of your post is (with respect) redundant

    MF
     
  16. Oct 15, 2005 #15
    I don't understand where you are having a problem. Are you trying to compare a continuous function (the mathematical concept of the real number line) with a possibly discontinuous function (the quantum physical world)? If so, why on earth would you expect the same story, why would you expect a perfect relationship?

    MF

    ps - Zeno's paradox implicitly assumed a continuous function
     
  17. Oct 19, 2005 #16
    pardon my wandering thoughts but...

    ...my guess would be swirling bubbles of light surrounding a specific type of vacuum peculiar to our universe or section thereof ???

    this brings to mind exactly what is light made of does it have mass and therefore gravity and how does it travel in spacetime or what if it is spacetime itself ???

    I was also thinking, time must warp at the planck scale which suggests a blipping in and out either to another place in the megaverse or an alternately negatively charged negaverse as part of a multiverse...

    In that scenario of a negatively charged symmetric partner to our whole 3+1d universe does that imply background dependence and what of the ramifications for no locality???

    ...nothing moves only the background changes shape but beacause we are locked in the system and the universe reconstitutes itself at superluminal speed we can't tell the difference nor measure it

    but anyway back on topic...

    Why are there an infinite amount of points within a finite distance between A and B or is B an infinite length away from A in which case the time taken to travel between the 2 points would be infinite ???

    ...does non locality and blipping in and out of our universe mean that time is meaningless at superluminal speed and also at distances less than planck scale distances and greater than the radius of which the universe is inflating ???
     
  18. Oct 19, 2005 #17

    Hurkyl

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    Assume that there exists a line segment with only finitely many points on it.

    Then we can call that line segment L.

    Since L has only finitely many points on it, and we can order the points on L, we can find two points with no other points between them.

    Then we can call such a pair of points P and Q.

    We know from high school geometry that we can construct the midpoint between any two points.

    Then, there is a point that is half-way between P and Q.

    Let's call that point R.

    R cannot exist, because there are no points between P and Q. But we've proven R exists. This is a contradiction, so our initial assumption is incorrect.

    Therefore, all line segments have infinitely many points lying on them.
     
  19. Oct 19, 2005 #18
    ^^^surely the shortest distance between 2 points is zero if the 2 points are touching, making it impossible to add another point between finite points on a line...

    ...so you can't get an infinite number of points on a line !!!

    If 2 points are touching they become 1 as there is no distance between P & Q to have a halfway point between???

    does that make sense ???
     
  20. Oct 19, 2005 #19

    Hurkyl

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    Before I respond, do you understand the concept of a proof by contradiction, also known as reducto ad absurdum? (If I spelled it right)
     
  21. Oct 19, 2005 #20
    ^^^if by that, you mean that by taking arguments to an extreme where it becomes absurd and beyond the confines of logic just to prove a point then yeah i think so...

    ...kinda like if you can't dazzle em with brilliance baffle them with bull**** ???
     
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